var textForPages =["SCHREIER SPLIT EPIMORPHISMS IN MONOIDS AND IN SEMIRINGS
Dominique Bourn, Nelson Martins-Ferreira, Andrea Montoli and Manuela Sobral
","Comissa˜o Editorial/Editorial Board
Ana Paula Santana
Ju´lio Severino Neves
Maria Paula Martins Serra de Oliveira
Financial support from Centre for Mathematics of the University of Coimbra (CMUC) and from FCT for the edition of this volume of Textos de Matem´atica is gratefully acknowledged.
","Dominique Bourn, Nelson Martins-Ferreira, Andrea Montoli and Manuela Sobral
SCHREIER SPLIT EPIMORPHISMS IN MONOIDS AND IN SEMIRINGS
Textos de Matem´atica/Mathematical Texts
Volume 45
Departamento de Matem´atica da Universidade de Coimbra Portugal 2013
","DL: 369076/14
ISBN-13: 978-972-8564-49-0
","Financial support by the Department of Mathematics of the University of Coimbra, the Centro de Matema´tica da Universidade de Coimbra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT - Funda¸ca˜o para a Ciˆencia e a Tecnologia under the project PEst-C/MAT/UI0324/2013 and grants number PTDC/MAT/120222/2010 and SFRH/BPD/69661/2010, and also by ESTG and CDRSP from the Polytechnical Institute of Leiria is gratefully acknowledged.
Dominique Bourn
Laboratoire de Math´ematiques pures et appliqu´ees, Universit´e du Littoral Coˆte d’Opale, Calais, France
bourn@lmpa.univ-littoral.fr
Nelson Martins-Ferreira
Escola Superior de Tecnologia e Gesta˜o,
Centro para o Desenvolvimento Ra´pido e Sustentado do Produto, Instituto Polit´ecnico de Leiria, Leiria
Portugal
martins.ferreira@ipleiria.pt
Andrea Montoli
CMUC,
Universidade de Coimbra, 3001-501 Coimbra, Portugal
montoli@mat.uc.pt
Manuela Sobral
CMUC and Departamento de Matema´tica, Universidade de Coimbra, 3001-501 Coimbra, Portugal
sobral@mat.uc.pt
","","Contents
Introduction 1
1 Unital categories and intrinsic commutation 5
1.1 Notation............................... 5
1.2 Unitalcategories .......................... 5
1.3 C′-unitalcategories ........................ 8
1.4 Thefibrationofpoints.................
1.5 Mal’tsevcategories...................
2 Schreier and homogeneous split epimorphisms in monoids
...... 8 ...... 9
11
2.1 Definitionsandfirstproperties .................. 11
2.2 Examples of Schreier and homogeneous split epimorphisms . . 14
2.3 First properties of Schreier and homogeneous split epimorphisms 17
2.3.1 Stabilityproperties..................... 17
2.3.2 TheSchreiersplitshortfivelemma . . . . . . . . . . . . 22 2.4 The fibrations of Schreier and homogeneous points . . . . . . . 23
3 Schreier internal relations, categories and groupoids 29
3.1 Schreierreflexiverelationsandgraphs . . . . . . . . . . . . . . 30 3.2 Schreierinternalcategories .................... 35 3.3 Schreierinternalgroupoids .................... 38
4 Mal’tsev aspects of Mon related to Schreier split epimorphisms 43 4.1 Mal’tsevcategories......................... 43 4.2 Firstobservations.......................... 44 4.3 CentralityforSchreierrelations.................. 44 4.4 Doublecentralizingrelation .................... 47
5 Schreier and homogeneous representability 49
5.1 TheSchreiersplitextensionclassifier . . . . . . . . . . . . . . . 49 5.2 Semidirectproducts ........................ 52
v
","vi
5.3 Thehomogeneoussplitextensionclassifier . . . . . . . . . . . . 52 5.4 Centralizers of Schreier reflexive relations . . . . . . . . . . . . 54 5.5 Thecaseofcommutativemonoids ................ 55
6 Semirings
6.1 Properties of Schreier split epimorphisms of semirings
6.2 ThefibrationofSchreierpointsinsemirings . . . . . .
6.3 Schreierinternalstructuresinsemirings . . . . . . . .
6.4 Mal’tsev aspects of SRng related to Schreier split epimorphisms 69
6.5 Schreieraccessibility ........................ 70
6.6 Centralizers of Schreier reflexive relations . . . . . . . . . . . . 76
6.7 SemidirectproductsinSRng ................... 77
7 Special Schreier and special homogeneous surjections 79
7.1 SpecialSchreiersurjectionsinMon. . . . . . . . . . . . . . . . 79
7.2 SpecialSchreiershortfivelemma ................. 81
7.3 Special Schreier extensions with abelian kernel . . . . . . . . . 83
7.4 ThedirectionfunctorinMon................... 86
7.5 BaersumsinMon ......................... 89
7.6 SpecialSchreiersurjectionsinSRng . . . . . . . . . . . . . . . 90
7.7 Special Schreier extensions with trivial kernel . . . . . . . . . . 91
7.8 BaersumsinSRng......................... 95
8 Conclusion 97
8.1 Partialprotomodularity ...................... 98 8.2 BacktoC′-unitalcategories.................... 99 8.3 Leftexactconservativeforgetfulfunctors. . . . . . . . . . . . . 100
9 Appendix 101
9.1 Pullbacks .............................. 101 9.1.1 Leftexactconservativefunctors. . . . . . . . . . . . . . 102
9.2 Regularepimorphisms ....................... 102
9.3 Fibrations.............................. 103
9.4 Simplicialobjects.......................... 104
9.5 Internal preorders, categories and monoids . . . . . . . . . . . . 106
9.6 Internal groupoids, equivalence relations, groups . . . . . . . . 108
9.7 Theshiftfunctor .......................... 110
Bibliography 111 Index 115
..... 59 . . . . . 61 . . . . . 64
57
","Introduction
Many intrinsic properties of the category Gp of groups have already been pointed out (for instance, see [4]), emphasizing and explaining the similari- ties with other algebraic structures, like rings and Lie algebras. The category Mon of monoids, however, did not seem to have been already investigated for itself, namely from the point of view of its internal structural properties. One of the aims of this work is to fill this gap.
It is true that, besides the property of being unital, a property which, among other things, controls the algebraic notion of commutative pair of sub- objects and, more generally, of commutative pair of morphisms [8], this cate- gory was not yet showed to satisfy strong categorical schemes. However, recent works about some aspects of the semidirect product in this setting [26] brought us a new pertinent tool with the notion of Schreier split epimorphism, which will make explicit a strong and meaningful parallelism with the category Gp of groups.
On one hand, from this point of view, in the same way as the group AutG of automorphisms of a group G classifies the class of split exact sequences with kernel G (see [5], [6] and also [3]), the monoid End(M) of endomorphisms of a monoid M will appear to classify a certain class of so-called Schreier split exact sequences with kernel M, while the group Aut(M) will appear to classify the subclass of so-called homogeneous split exact sequences with kernel M. In other words, the notions of Schreier split epimorphism and homogeneous split epimorphism allow us to read, in the category Mon, the tracks of the property of action representability satisfied by the category Gp.
On the other hand, the category Gp of groups is a Mal’tsev category in the sense of [16, 17], namely a category in which any reflexive relation is an equivalence relation; this, among other things, forbids the existence of internal preorders. On the contrary, there are preorders in the category Mon which reveal that its weaker categorical structuration gives it an interesting flexibility in comparison with the rigidity of the paradigmatic category Gp.
1
","2
However the flexibility in question will be showed to maintain some im- portant aspects of the Mal’tsev structural organization. For instance we shall show that in Mon there are pairs of equivalence relations which centralize each other, and that a certain class of equivalence relations admits centralizers. Ex- actly in the same way of what happens in Gp we shall show, as well, that certains classes of exact sequences with abelian kernel in Mon are endowed with a natural structure of abelian group.
Beyond this, recall that the category Rng of non-commutative rings also satisfies a property dealing with classification of split exact sequences, namely action accessibility [15], of which action representability appeared to be a par- ticular case. A semiring is only a commutative monoid with an associative and distibutive multiplication. We shall show, in the same way, that the category SRng of semirings inherits a notion of Schreier split epimorphism which cap- tures, as well, the tracks of the property of action accessibility of the category Rng. Again, from this, we shall get results about centrality and centralizers of equivalence relations and again an abelian group structure on certain classes of extensions with trivial kernel. So that our approach gives a strong structural meaning to the intuitive proportion:
Mon = SRng Gp Rng
Up to now the structural attempt to characterize a group (resp. a ring), as an object in Mon (resp. in SRng), by an intrincsic property inside the category Mon (resp. in the category SRng) has failed. The notion of Schreier split epimorphism allows us to characterize groups (resp. rings) inside Mon (resp. SRng). Unfortunately the notion of Schreier split epimorphism itself is not yet showed to be quite intrinsic to Mon (resp. to SRng), it seems to need the use of the forgetful functor Mon → Set (resp. SRng → Set); among other things, we do hope that this work will constitute a step towards this expected intrinsic characterization.
This work is organized along the following lines: in Chapter 1 we briefly re- call the notion of unital category and its main properties related to the commu- tation of subobjects. In Chapter 2 we introduce Schreier (resp. homogeneous) split epimorphisms, and in Chapter 3 we introduce Schreier (resp. homoge- neous) equivalence relations, internal categories and groupoids, and in both chapters we study the first stability properties of these new tools. In Chapter 4 we investigate their remaining associated Mal’tsev aspects in terms of cen- tralization of equivalence relations. Chapter 5 is devoted to the Schreier and homogeneous representabilities and their consequences dealing with the exis- tence of centralizers. Chapter 6 develops the same kind of structural analysis for the semirings and investigates Schreier accessibility. Chapter 7 is devoted to
","3
the abelian group structure which is given to some specific class of extensions (called special Schreier extensions) with abelian kernel in Mon and to the par- allel specific class in SRng. Chapter 8 gives a synthetic structural presentation of what is involved under the Schreier split epimorphic process. There is an Appendix which collects some basic categorical notations and facts.
The authors are grateful to Alfredo Costa for the careful reading of the preliminary version of this book.
","","Chapter 1
Unital categories and intrinsic commutation
1.1 Notation
Let E be a finitely complete category. Given any map f : X → Y , we denote the kernel equivalence relation of this map by R[f]. Given the following right hand side commutative square, we denote by R(x) the map induced by the map x between the respective kernel equivalences:
p1 R(x)
p0
// f R[f]oo s0 // X // Y
′ oo p′0 R[f]s′0 //X′
p′ 1
See the Appendix for further details.
1.2 Unital categories
//Y′.
//
xy
f′
In this section, C will be a pointed category, i.e. a category with a zero object 0 (in the category Mon of monoids, it is given by the trivial monoid with only one element). Let us recall from [8]:
5
","6 Chapter 1. Unital categories and intrinsic commutation
Definition 1.2.1. Let C be a pointed category with finite products. Given two objects A and B in C, consider the following diagram
πB //
// A×B oo B. (1.2.1)
⟨0,1B ⟩
oo πA ⟨1A ,0⟩
A
The category C is said to be unital if, for every pair of objects A, B ∈ C, the
morphisms ⟨1A,0⟩ and ⟨0,1B⟩ are jointly strongly epimorphic.
When C is finitely complete, this is equivalent to the fact that the object A×B is the supremum of the two subobjects ⟨1A,0⟩ and ⟨0,1B⟩; namely, any monomorphism j : J A × B containing A and B, as in the following diagram
<< J cc
j
ooπA πB//
A // A × B oo B, ⟨1A ,0⟩ ⟨0,1B ⟩
is an isomorphism.
Example 1.2.2. The category Mon of monoids is unital.
Proof. It suffices to prove that, for any pair of monoids A and B, any submonoid
M → A×B containing all the elements (a,1) and (1,b) is equal to A×B. Unital categories are a setting where it is possible to express a categorical
notion of commutativity.
Definition 1.2.3 ([10]). Let C be a unital category. Two morphisms with the same codomain f: X → Z and g: Y → Z are said to cooperate (or to com- mute) if there exists a morphism φ : X × Y → Z such that both triangles in the following diagram commute:
X ⟨1X ,0⟩// X × Y oo⟨0,1Y ⟩ Y
φ
## {{ Z.
The morphism φ is necessarily unique, because ⟨1X , 0⟩ and ⟨0, 1Y ⟩ are jointly (strongly) epimorphic, and it is called the cooperator of f and g.
The uniqueness of the cooperator makes commutativity a property and not an additional structure in the category C.
f
g
","1.2. Unital categories 7 Definition 1.2.4. An object A of a unital category C is said to be commutative
if the identity 1A cooperates with itself.
The cooperator m: A × A → A of a commutative object A endows A with a canonical structure of internal commutative monoid in C, i.e. the binary operation m satisfies the axioms of an internal commutative monoid (see the Appendix).
Proposition 1.2.5. In Mon, two morphisms f and g, as in Definition 1.2.3, cooperate if and only if
f(x)g(y) = g(y)f(x) for all x ∈ X, y ∈ Y. (1.2.2) Proof. If Condition 1.2.2 is true, we can define the cooperator as
φ(x, y) = f (x)g(y).
It is easy to show that φ is a morphism and φ⟨1X,0⟩ = f, φ⟨0,1Y ⟩ = g.
Conversely, if a cooperator φ exists, then we have
φ(x, y) = φ((x, 1)(1, y)) = φ(x, 1)φ(1, y) = f (x)g(y),
and
foranyx∈X andy∈Y,andhencewegetCondition1.2.2.
Corollary 1.2.6. A monoid A is commutative in the categorical sense if and only if it is commutative in the classical sense, i.e. xy = yx for any x, y ∈ X.
Any functor U : C → D between unital categories which preserves products preserves the cooperating pairs as well.
Definition 1.2.7. A split epimorphism (A, B, f, s) in C is said to be a strongly split epimorphism if the pair (k, s) in the associated split sequence
f
Lemma 1.2.8. ([26], Proposition 2.6 and [12], Proposition 1.9) If (A, B, f, s) is a strongly split epimorphism, then f is the cokernel of k. In other words the split sequence
//k//oos //
0 K[f] A ////B 0
f
φ(x, y) = φ((1, y)(x, 1)) = φ(1, y)φ(x, 1) = g(y)f (x)
k // oo s K[f] A
// // B,
with fs = 1B and k = ker f, is jointly strongly epimorphic.
is exact.
","8 Chapter 1. Unital categories and intrinsic commutation Proof. Given a morphism g : A → D such that gk = 0, we have that gs makes
the triangle below commutative:
K[f] k //Aoo f //B
s
gs
g
D.
Indeed:
and since k and s are jointly (strongly) epimorphic, we have that gsf = g.
gsfs=gs and gsfk=0=gk, Moreover, given any h: B → D such that hf = g, we have that
h = hfs = gs.
Definition 1.3.1. The category C is said to be C′-unital when, for any object A ∈ C′ and any object B ∈ C, the morphisms ⟨1A,0⟩ and ⟨0,1B⟩ in the following diagram are jointly strongly epimorphic:
1.3 C′ -unital categories
Let C′ be a full subcategory of a pointed category C with finite products.
A
oo πA ⟨1A ,0⟩
πB //
// A×B oo B. (1.3.3)
⟨0,1B ⟩
In a C′-unital category we can still speak of cooperating pairs (f,g) of morphisms, provided that the domain X of f belongs to C′. More generally, X × Y being isomorphic to Y × X, we can speak of cooperating pair as soon as the domain of one of the two maps is in C′. Accordingly, we can still speak of commutative objects in C′.
Proposition 1.3.2. Suppose that C is C′-unital and that C′ is closed under finite products (and thus it contains the zero object 0). Then C′ is unital.
Proof. Straightforward. 1.4 The fibration of points
Let E be any category. We shall denote by PtE the category whose objects are the split epimorphisms in E and whose arrows are the commuting squares
","1.5. Mal’tsev categories 9
between such split epimorphisms. We denote by ¶E : P tE → E the functor associating with any split epimorphism its codomain. As soon as the category E has pullbacks, this functor is a fibration, which is called the fibration of points (see the Appendix). Given any object X in E, the fiber PtXE is the category of split epimorphisms with codomain X, while, given any map h: X → Y , the change-of-base functor h∗ : P tY E → P tX E is given by the pullback along the map h.
Proposition 1.4.1. If E is a unital category, then so is PtE.
Proof. Given two split epimorphisms (A,B,f,s) and (A′,B′,f′,s′) in E, their product in PtE is the split epimorphism (A×A′,B×B′,f×f′,s×s′). Consider then the following diagram:
Aoo πA //A×A′ oo πA′ //A′
OO ⟨1A,0⟩ OO ⟨0,1A′⟩ OO
f s f×f′ s×s′ f′ s′ oo πB ′ πB′ // ′
B
//B×B oo B ⟨1B ,0⟩ ⟨0,1B′ ⟩
Since the two pairs (⟨1A,0⟩,⟨0,1A′⟩) and (⟨1B,0⟩,⟨0,1B′⟩) are jointly strongly epimorphic in E, the pair of morphisms in PtE they define is jointly strongly epimorphic, too.
1.5 Mal’tsev categories
We briefly recall here some concepts that will be developed later in more details. A category C is said to be a Mal’tsev category [16, 17] when any internal reflexive relation R on an object X in C is an equivalence relation. The category
Gp of groups is a Mal’tsev one. The preorder ON on the natural numbers:
where
p0 // ON oo s0 // N
p1
ON ={(x,y)∈N×N|x≤y}
shows that the category Mon is no longer a Mal’tsev one.
From [8], it appears that being a Mal’tsev category is equivalent to the
property that any fiber PtXC of the fibration of points ¶C is unital. Conse- quently it is not true that any fiber PtX(Mon) is unital.
However we shall show here that these fibers are unital relatively to a certain full subcategory of specific split epimorphims and we shall make explicit what is remaining of the classical Mal’tsev results in this new structural context.
","","Chapter 2
Schreier and homogeneous split epimorphisms in monoids
2.1 Definitions and first properties
From now on, C will be the category Mon of monoids. Let us introduce the following definition:
Definition 2.1.1. A split epimorphism (A, B, f, s) of monoids is said to be right homogeneous when, for any element b ∈ B, the map μb : K[f] → f−1(b) de- fined by the multiplication on the right by s(b), as μb(k) = k · s(b), is bijective. Similarly, by duality, we can define a left homogeneous split epimorphism. (A,B,f,s) is said to be homogeneous when it is both right and left homoge- neous.
In [26] (Definition 2.6) was introduced the following
Definition 2.1.2. A split epimorphism (A,B,f,s) of monoids is said to be a Schreier split epimorphism when, for any a ∈ A, there exists a unique α in the kernel K[f] of f such that a = α · sf(a).
In other terms, a Schreier split epimorphism is a split epimorphism (A, B, f, s) equipped with a unique set-theoretical map q: A K[f] with the property that, for any a ∈ A, we have:
a = q(a) · sf(a).
Proposition 2.1.3. A split epimorphism (A, B, f, s) is right homogeneous if and
only if it is a Schreier split epimorphism.
11
","12 Chapter 2. Schreier and homogeneous split epimorphisms in monoids Proof. Given a right homogeneous split epimorphism (A, B, f, s), the unique α
that appears in Definition 2.1.2 is given by α = μ−1 (a). Conversely, given a f (a)
Schreier split epimorphism, for any b ∈ B the inverse of the map μb : K[f] → f−1(b) is defined in the following way: if a ∈ f−1(b), μ−1(a) is the unique
b
α ∈ K[f] such that a = α · sf(a).
Proposition 2.1.4. A split epimorphism (A, B, f, s) is a Schreier split epimor-
phism if and only if there exists a set-theoretical map q: A K[f] such that:
q(a) · sf(a) = a q(α · s(b)) = α
for every a ∈ A, α ∈ K[f] and b ∈ B. Dually, a split epimorphism is left homogeneous if and only if there exists a set-theoretical map q¯: A K[f] such that:
sf(a)·q¯(a) = a q¯(s(b) · α) = α
for every a∈A, α∈K[f] and b∈B.
Proof. Suppose that for every a ∈ A, there exists a unique α ∈ K[f] such that a = α·sf(a). This property defines a map q: A → K[f], by q(a) = α such that a = q(a)·sf(a), for every a ∈ A. In order to prove that q(α·s(b)) = α for any α ∈ K[f], it suffices to observe that sf(α · s(b)) = s(b).
Conversely, given a set-theoretical map q : A → B satisfying the asserted identities, we can choose α = q(a) for every a ∈ A by the first identity; suppose now that a = α′ · sf(a), then we get:
q(a) = q(α′ · sf(a)) = α′
by the second identity. There is a similar proof for the left homogeneous split
epimorphisms.
We shall call the following diagram:
phism and q the associated Schreier retraction. In fact, we have: Proposition 2.1.5. Given a Schreier split epimorphism (A, B, f, s), we have:
// B,
the canonical Schreier split sequence associated with the Schreier split epimor-
oo q
K[f] // //A
oo s oo kf
(a) qk = 1K[f];
","2.1. Definitions and first properties 13
(b) qs = 0;
(c) q(1) = 1;
(d) if b∈B and α∈K[f], then q(s(b)·α)·s(b)=s(b)·α; (e) for every a,a′ ∈A q(a·a′)=q(a)·q(sf(a)·q(a′)).
Proof. (a) it is a straighforward consequence of the second identity in Propo- sition 2.1.4.
(b) forb∈Bwehave:
s(b) = 1 · sf(s(b))
and the uniqueness of q gives that qs(b) = 1 for every b ∈ B.
(c) obviously we have 1 = 1 · sf (1).
(d) foranyb∈Bandanyα∈K[f]wehave:
s(b) · α = q(s(b) · α) · sf(s(b) · α) =
= q(s(b) · α) · sfs(b) · sf(α) = q(s(b) · α) · s(b).
(e) q(a · a′) is the unique element of K[f] such that
a·a′ =q(a·a′)·sf(a·a′)=q(a·a′)·sf(a)·sf(a′),
so it suffices to prove that
q(a) · q(sf(a) · q(a′)) · sf(a) · sf(a′) = a · a′.
By point (d), we have that
q(sf(a) · q(a′)) · sf(a) = sf(a) · q(a′)
and hence
q(a) · q(sf(a) · q(a′)) · sf(a) · sf(a′) = q(a) · sf(a) · q(a′) · sf(a′) = a · a′.
We end this section with two important observations:
Lemma 2.1.6. A Schreier split epimorphism (and, a fortiori, a homogeneous split epimorphism) is a strongly split epimorphism.
Proof. Given a Schreier split epimorphism, the formula a = q(a) · sf (a) proves that A is the supremum of the subobjects k: K[f] A and s: B A.
","14 Chapter 2. Schreier and homogeneous split epimorphisms in monoids Lemma 2.1.7. Given a Schreier split epimorphism (A,B,f,s), the following
diagram
// B,
makes s: B A the kernel of q in the category of pointed sets.
Proof. We already know that qs = 0 (Proposition 2.1.5 (b)), hence B is con- tained in the kernel of q. Conversely, if a belongs to the kernel of q, then
a = q(a) · sf(a) = 1 · sf(a) = sf(a),
and hence a is in the image of s.
2.2 Examples of Schreier and homogeneous split epi- morphisms
Proposition 2.2.1. Given any direct product diagram
oo q oo s oo K[f] // //A
ooπX ⟨1X ,0⟩
// X × B oo
πB // ⟨0,1B ⟩
kf
B,
the canonical split epimorphism (X × B, B, πB , ⟨0, 1B ⟩) is homogeneous.
Proof. The equality (x, 1) · (1, b) = (x, b) shows that it is right homogeneous, while (1, b) · (x, 1) = (x, b) shows that it is left homogeneous. Here the Schreier retraction πX is a monoid homomorphism.
Corollary 2.2.2. The terminal split epimorphism X oo // 1
is homogeneous.
Corollary 2.2.3. The identity split epimorphism
Proposition 2.2.4. If B is a group, then every split epimorphism (A, B, f, s) is homogeneous.
X
//
and more generally any isomorphism, is homogeneous.
X oo
X,
1X 1X
","2.2. Examples of Schreier and homogeneous split epimorphisms 15 Proof. Given a split sequence of the form
k // oo s K[f] A
f
let us define, for any a ∈ A, q(a) = a · sf(a)−1 ∈ K[f]. This map q clearly satisfies the conditions of Proposition 2.1.4, showing that it is a Schreier split epimorphism. The map q¯(a) = sf(a)−1 · a ∈ K[f], which satisfies the dual conditions, shows that it is left homogeneous.
We will prove later (Corollary 3.1.7) that the converse is also true: if any split epimorphism with codomain B is a Schreier one, then B is a group.
Corollary 2.2.5. If B is a group and (A,B,f,s) a split epimorphism in Mon, A is a group if and only if K[f] is a group.
Proof. Since B is a group, the split epimorphism (A, B, f, s) is a Schreier one. If K[f] is a group and a = q(a) · sf(a), then a−1 = sf(a)−1 · q(a)−1. The other implication is obvious, since (A, B, f, s) turns out to be a split epimorphism of groups.
Example 2.2.6. From Proposition 2.3.4 below, given any split epimorphism be- tween groups and any submonoid M of its codomain B, the following pullback produces a homogeneous split epimorphism in Mon:
f−1(M) // //YOO OO j
f′s′ fs
// // MB i
Example 2.2.7. Consider the internal order in Mon given by the usual order between natural numbers:
p0 // ON oo s0 // N,
p1
// // B,
where
is a submonoid of the direct product N×N (with the usual sum), and the monoid
homomorphisms p0, p1 and s0 are given by:
p0(x,y) = x, p1(x,y) = y, s0(x) = (x,x).
Then the split epimorphism (ON, N, p0, s0) is homogeneous.
ON = {(x, y) ∈ N × N | x ≤ y}
","16 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
Proof. The equality (x, y) = (0, y−x)+(x, x), for any (x, y) ∈ R proves that the split epimorphism in question is right homogeneous, while the commutativity implies that it is left homogeneous.
Notice that, in the last example, the split epimorphism (p1,s0) is not a Schreier one. Indeed, there is no natural number n such that (0, 1) = (n, 0) + (1, 1).
Example 2.2.8. We denote by Z∗ the monoid of non-zero integers with the usual multiplication, and by N∗ its submonoid whose elements are the numbers greater than 0. Then the split epimorphism
Z∗oo i //N∗, abs
where i is the inclusion and abs associates with any integer its absolute value, is a homogeneous split epimorphism. In fact K[abs] = {±1}, and it is immediate to see that any non-zero integer z can be written in a unique way as z = ±1 · |z| = |z| · ±1.
Example 2.2.9. Let us fix a natural number n. We can define on the cartesian product N × N the monoid structure given by
(x1, b1) · (x2, b2) = (x1 + nb1 x2, b1 + b2).
(when both n and b1 are 0, we use the convention 00 = 1). We denote this monoid by N n N. The projection π1 : N n N → N defined by π1(x,b) = b is a monoid homomorphism, in the same way as the section σ1 : N → N n N defined by σ1(b) = (0, b). For any n ∈ N, the split epimorphism (N n N, N, π1, σ1) is a Schreier split epimorphism: the (unique) map q: N × N K[π1] is just the first projection π0 : N × N N. If n = 0, this split epimorphism is not homogeneous (because it is not left homogeneous): for example, for b = 1, the map λ : K[π ] π−1(1) defined by
λ1(x, 0) = (0, 1) · (x, 0) = (0, 1)
is clearly not bijective.
The last example is an instance of the semidirect product construction, whose relationship with the Schreier and homogeneous split epimorphisms will be explored in Chapter 5.
111
","2.3. First properties of Schreier and homogeneous split epimorphisms 17 2.3 FirstpropertiesofSchreierandhomogeneoussplit
epimorphisms
Proposition 2.3.1. Consider a (vertical) map (h, l) in P tC:
oo s
X
// // Y
f lh ′oo s′ ′
X
Suppose that the two rows are Schreier split epimorphisms, then the Schreier retractions are compatible, i.e. the following leftward left hand side diagram commutes (in the category Set of sets):
ooq oos K[f] // X // // Y
kf
K(l) l h
′oo s′ ′
f′
// // Y .
′ ooq′ K[f ]
// X
Proof. We have to show that q′l(x) = lq(x) for any x in the monoid X. It is
true since we have:
lq(x) · s′f′l(x) = lq(x) · lsf(x) = l(q(x) · sf(x)) = l(x) = q′l(x) · s′f′l(x)
2.3.1 Stability properties
We are going to investigate here what are the main stability properties of the Schreier split epimorphisms.
Proposition 2.3.2. Schreier split epimorphims are stable under composition. When the composite (gf,st) of two split epimorphisms (f,s) and (g,t) is a Schreier one, so is (g, t). The same is true for homogeneous split epimorphisms.
Proof. Let be given a pair of composable split epimorphisms: Xoo s //W
YY f
gf t
st g
Y
k′
f′
// // Y .
EE
","18 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
Suppose that qf and qg are the Schreier retractions of the Schreier split epi- morphisms (f,s) and (g,t), respectively. Let us set qgf(x) = qf(x)·sqg(f(x)) and show it satisfies the required identities. First we have:
qf (x) · sqg(f(x)) · stgf(x) = qf (x) · s(qg(f(x)) · tgf(x)) = qf (x) · sf(x) = x. Then, given any (α, y) ∈ K[gf] × Y , notice that f(α) is in K[g]. Hence:
qgf (α · st(y)) = qf (α · st(y)) · sqg(f(α · st(y))) =
= qf (α) · qf (sf(α) · qf (st(y))) · sqg(f(α) · t(y)) =
= qf(α)·qf(sf(α)·1)·sf(α) = qf(α)·qf(sf(α))·sf(α) = qf(α)·1·sf(α) = α.
Suppose now that the composite is a Schreier split epimorphism with Schreier retraction qgf . Let us set qg (w) = f qgf (s(w)) and show it satisfies the required identities. First we have:
qg(w) · tg(w) = fqgf (s(w)) · tg(w) = fqgf (s(w)) · fstg(w) =
= f (qgf (s(w)) · stg(w)) = f (qgf (s(w)) · stgf s(w)) = f (s(w)) = w.
Now, given any (α, y) ∈ K[g] × Y , we have s(α) ∈ K[gf] and get:
qg(α · t(y)) = fqgf (s(α · t(y))) = fqgf (s(α) · st(y)) = f(s(α)) = α.
The results concerning the homogeneous split epimorphisms are obtained du- ally.
Proposition 2.3.3. Schreier split epimorphisms are stable under products, i.e. the product of two Schreier split epimorphisms is a Schreier one. The same is true for homogeneous split epimorphisms.
Proof. Consider the two Schreier split exact sequences
K[f]
oo q oo s // A
kf
// // B
// // B′,
oo s×s′ f ×f ′
and
K[f′] Their term by term product
// A′
oo q′ k′
oo q×q′ K[f] × K[f′]
k×k′
oo s′ f′
// A × A′
// // B × B′,
clearly satisfies the conditions of Proposition 2.1.4. Dually we have the same result for left homogeneous split epimorphisms, and hence for homogeneous split epimorphisms.
","2.3. First properties of Schreier and homogeneous split epimorphisms 19 Proposition 2.3.4. Schreier (resp. homogeneous) split epimorphisms are stable
under pullbacks along any morphism.
Proof. Consider the following diagram, where the lower row is a Schreier split
sequence and the right-hand side square is a pullback:
oo q′
πY
≃ πA h
ooq oos K[f] // A // // B.
kf
The map q′ defined by q′(a, y) = (q(a), 1) satisfies the conditions of Proposition 2.1.4 since:
(a, y) = (q(a), 1) · (sf (a), y) = (q(a), 1) · (sh(y), y)
for any (a, y) ∈ A×B Y . Moreover, the elements of K[πY ] are of the form (α, 1),
with α ∈ K[f], and then:
q′((α, 1) · (sh(y), y)) = q′(α · sh(y), y) = (q(α · sh(y)), 1) = (α, 1)
Accordingly the upper row is a Schreier split epimorphism. So the right ho- mogeneous split epimorphisms are stable under pullbacks. Dually, the same holds for the left homogeneous split epimorphisms, and consequently for the homogeneous ones.
Proposition 2.3.5. Consider the lower right hand side vertical morphism of split epimorphisms:
R[K(g)] //oo qR // R[g] oo R(s′) // R[h] OO R(kf′) OO R(f′) OO
K[πY ]
// A ×B Y
⟨oosh,1Y ⟩
// // Y
⟨k,0⟩
p0 p1 oo
K[f′] //
p0 p1
qf′ oo s′
//A′
kf′ f′
p0 p1 //B′
g
K[f] A B
K(g)
h // oos //
//kf
UU f
qf
Complete it with the horizontal kernels and the vertical kernel equivalence rela- tions. Commutativity of limits makes the upper row a kernel diagram. Suppose that g is a regular epimorphism (= a surjective homomorphism). Suppose that moreover K(g) is a regular epimorphism. If the two upper split epimorphisms are Schreier (resp. homogeneous) ones, so is the lower one.
","20 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
Proof. First notice that when g is surjective, so is f g and hence h. Let us denote by qf′ and qR the Schreier retractions associated with the two upper Schreier split epimorphisms. According to Proposition 2.3.1, we have p0qR = p0qf′ and p1qR = p1qf′ . Since the surjective homomorphisms are the quotients in Set of their kernel equivalence relations, there is a factorization qf in Set making the lower left hand side leftward square commute. The fact that h, g and K(g) are surjective allows to transfer the required conditions from qf′ to qf. The same facts hold for left homogeneous split epimorphisms by duality; consequently they hold for homogeneous split epimorphisms.
Corollary 2.3.6. Suppose that the right hand side square is a pullback, and h is a regular epimorphism.
k′
K[f′] // f //A′
K(g) ≃ g
oo s′ f′
//B′
h // // oos//
K[f] A B kf f
If the upper split epimorphism is a Schreier (resp. homogeneous) one, so is the lower one.
Proof. When the right hand side square is a pullback, the factorization K(g) is an isomorphism and consequently a regular epimorphism. When h is a regular epimorphism, so is g (because regular epimorphisms, i.e. surjective morphisms, are stable under pullbacks, see the Appendix). Moreover, all the commutative diagrams determined by the kernel equivalence relations R[g] and R[h] are pullbacks, and consequently the split epimorphism (R[g], R[h], R(f′), R(s′)) is a Schreier (resp. homogeneous) one. According to the previous proposition, so is (A,B,f,s)
Theorem 2.3.7. Consider a commutative diagram of split epimorphisms:
k // oo s K[f] A
// // B
f
K(u) u v
′ ooq′ ′oo s′ ′
K[f ] // A k′
// // B f′
where the lower row is a Schreier split sequence and where the map K(u) is an isomorphism. The following conditions are equivalent:
(a) the pair (u,f) is jointly monomorphic;
","2.3. First properties of Schreier and homogeneous split epimorphisms 21 (b) the commutative square f′u = vf is a pullback square;
(c) the upper row, with the map q = K(u)−1q′u, is a Schreier split epimor- phism.
Proof. (a)⇒(b) Given l: Z →A′ and g: Z →B, such that f′l = vg, we can define the map h: Z → A by
h(z) = K(u)−1q′l(z) · sg(z), z ∈ Z. Then uh = l and fh = g, indeed
u(K(u)−1q′l(z) · sg(z)) = q′l(z) · s′vg(z) = q′l(z) · s′f′l(z) = l(z), and
f(K(u)−1q′l(z) · sg(z)) = g(z).
Moreover, h is a morphism: in order to show this fact, since u and f are jointly monomorphic, it suffices to show that uh(z·z′) = uh(z)·uh(z′) and fh(z·z′) = fh(z) · fh(z′). We have that
u(K(u)−1q′l(z · z′) · sg(z · z′)) = q′(l(z) · l(z′)) · s′vg(z) · usg(z′)
= q′l(z) · q′(s′f′l(z) · q′l(z′)) · s′vg(z) · usg(z′)
= q′l(z) · q′(s′f′l(z) · q′l(z′)) · s′f′l(z) · usg(z′) = q′l(z) · s′f′l(z) · q′l(z′) · usg(z′)
= u(K(u)−1q′l(z) · sg(z) · K(u)−1q′l(z′) · sg(z′)) = uh(z) · uh(z′)
fh(z · z′) = f(K(u)−1q′l(z · z′) · sg(z · z′)) = g(z) · g(z′) =
and
The uniqueness required for the universal property of the pullback follows from the fact that the pair (u,f) is jointly monomorphic.
(b)⇒(c) follows from Proposition 2.3.4.
(c)⇒(a) Consider l,g: Z →A with fl = fg and ul = ug. If (f,s) is a
Schreier split epimorphism with the map q = K(u)−1q′u then we have l(z) = ql(z) · sfl(z)
and
g(z) = qg(z) · sfg(z).
Since fl = fg, in order to prove that l = g it suffices to show that ql(z) = qg(z), but this follows directly from the fact that q = K(u)−1q′u and ul = ug.
= f(K(u)−1q′l(z) · sg(z) · K(u)−1q′l(z′) · sg(z′)) = fh(z) · fh(z′).
","22 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
2.3.2 The Schreier split short five lemma
We can now assert the Schreier split short five lemma and its variations.
Corollary 2.3.8. Consider a commutative diagram of Schreier split epimor- phisms:
K[f]oo q //Aoo f //B
s
′ f′ //
// A oo B
k′
the map u is an isomorphism if and only if K(u) is an isomorphism.
Proof. The fact that K(u) is an isomorphism as soon as so is u holds in any pointed category with kernels. The converse is an immediate consequence of the previous theorem since, the right hand square being a pullback, u appears to be the pullback of the isomorphism 1B.
Corollary 2.3.9. Consider a commutative diagram of split exact sequences of the form
Xoo q //Aoo s ////B kf
φ
Proof. If φ is an isomorphism, then the lower split epimorphism is obviously a Schreier one: the unique map q′ satisfying the Schreier condition is qφ−1. The converse is an immediate consequence of the previous corollary.
More generally we have:
Proposition 2.3.10. Consider the following commutative diagram, where the two rows are Schreier split sequences:
K[f]oo q //Aoo s ////B kf
K(u) u v
k ′ooq′
u
K(u) K[f ]
s′
oo q′ k′
′oo s′
// A // // B,
X
where the upper split sequence is a Schreier one. The morphism φ is an iso-
f′
morphism if and only if the lower split sequence is a Schreier one.
′ oo q′
K[f ]
k′
′ oo s′ ′
// A
f′
// // B .
","2.4. The fibrations of Schreier and homogeneous points 23 (i) u is a regular epimorphism (i.e. a surjective homomorphism) if and only
if so are v and K(u);
(ii) u is a monomorphism if and only if so are v and K(u).
Proof. (i) If u is a regular epimorphism, then so is vf = f′u, and this implies that v is a regular epimorphism. The map q′ is surjective in Set, then so is q′u = K(u)q, and this implies that K(u) is surjective. Conversely, suppose that v and K(u) are regular epimorphisms. Consider a′ ∈ A′. There are α ∈ K(f) and b ∈ B such that u(α) = q′(a′) and v(b) = f′(a′). Accordingly u(α·s(b)) = u(α)·us(b) = q′(a′)·s′v(b) = q′(a′)·s′f′(a′) = a′. Hence the homomorphism u is surjective.
(ii) If u is a monomorphism, then so is us = s′v, which implies that v is a monomorphism. Similarly uk = k′K(u) is a monomorphism which implies that K(u) is a monomorphism. Conversely, suppose that K(u) and v are monomorphisms. Suppose now that u(a1) = u(a2). Then we have f′u(a1) = f′u(a2) and then vf(a1) = vf(a2), whence f(a1) = f(a2). From u(a1) = u(a2), we can conclude uq(a1) = q′u(a1) = q′u(a2) = uq(a2). Since the restriction K(u) of u to K[f] is a monomorphism, we get q(a1) = q(a2). With f(a1) = f(a2), we get a1 = a2.
2.4 ThefibrationsofSchreierandhomogeneouspoints
Let us consider the category Pt(Mon) and its full subcategories SPt(Mon) and HPt(Mon) whose objects are respectively the Schreier split epimorphisms and the homogeneous ones. According to Proposition 2.3.4, the restrictions ¶S and ¶H of the fibration ¶: Pt(Mon) → Mon to the full subcategories SPt(Mon) and HPt(Mon) are still fibrations, which we shall call the fibrations of Schreier points and of homogeneous points.
Proposition 2.4.1. The change-of-base functors of the fibrations ¶S and ¶H are conservative (i.e. they reflect isomorphisms).
Proof. The Schreier split short five lemma is equivalent to the fact that the change of base functor along the initial map αB∗ : SPtB(Mon) → Mon is con- servative. Now, given any homomorphism h : A → B we have hαA = αB and thus αA∗ h∗ = αB∗ , and since αB∗ is conservative, so is h∗.
Theorem 2.4.2. Given any monoid B, the fiber PtB(Mon) is SPtB(Mon)- unital and consequently HPtB(Mon)-unital.
","24 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
Proof. Consider the following pullback of split epimorphisms (which gives the product in the category P tB (M on)), where the lower horizontal one is a Schreier split epimorphism:
(2.4.1)
oo eC
A ×B C // COO
OO πC πAeA gt
oos
A
// B.
The homomorphisms eA and eC are defined by eA(a) = (a,tf(a)) and eC(c) = (sg(c),c). We have to show that they are jointly strongly epimorphic in the fiber PtB(Mon), which is equivalent to be jointly strongly epimorphic in Mon; in other words we have to show that the only submonoid of A ×B C containing these two classes of elements is A ×B C itself. This is a consequence of the following calculation for any element (a¯, c¯) ∈ A ×B C;
(a¯, c¯) = (q(a¯), 1) · (sf (a¯), c¯) = (q(a¯), 1) · (sg(c¯), c¯) = (q(a¯), tf q(a¯)) · (sg(c¯), c¯) since we have f(a¯) = g(c¯).
Then, according to Section 1.3, we can define the commutativity of two morphisms (m,n) in PtB(Mon) provided that the domain of one of the two maps is a Schreier split epimorphism.
Proposition 2.4.3. Suppose the domain (A, B, f, s) of m is a Schreier split epi- morphism and denote by (C, B, g, t) the domain of n, as in the diagram below. The cooperator φ: A ×B C → X of (m,n) in PtB(Mon), when it exists, is necessarily (being unique) defined by φ(a, c) = mqf (a) · n(c). It is actually a cooperator if and only if we have mqf (sg(c)·α)·n(c) = n(c)·m(α) for α ∈ K[f] and c ∈ C.
A ×B C
φ
;;
cc
f
eA
eC
m // oo n AXC cc OO ;; fg
s
Proof. We must have φeA = m and φeC = n. Moreover, for any (a, c) ∈ A×B C, we have
(a, c) = (qf (a) · sf (a), c) = (qf (a) · sg(c), c) = (qf (a), 1) · (sg(c), c).
If φ is a monoid homomorphism, we get: φ(a, c) = mqf (a) · n(c). When it is a monoid homomorphism, the identity (sg(c), c) · (α, 1) = (sg(c) · α, c), for
## {{ B
t
","2.4. The fibrations of Schreier and homogeneous points 25 α ∈ K[f] and c ∈ C, forces the condition in question. Conversely it is easy to
check that this condition implies that φ is a monoid homomorphism.
Proposition 2.4.4. For any monoid B, the fibers SPtB(Mon) and HPtB(Mon) contain the terminal object and are closed under finite products. Hence, the fibers SPtB(Mon) and HPtB(Mon) are unital.
Proof. The fact that SPtB(Mon) and HPtB(Mon) contain the terminal ob- ject of PtB(Mon) is Corollary 2.2.3. Let us show the closedness under finite products. Starting with two Schreier split epimorphisms (f, s) and (f′, s′) above B, the product in the fiber, which is nothing but the pullback of f along f′, as seen in Theorem 2.4.2, is consequently given by the following pullback, where s0 is the diagonal (defined by s0 (b) = (b, b)):
A ×BOO A′ //
// // A × OOA′ (2.4.2) f ×f ′ s×s′
//
B s0 B×B
Now the Schreier (resp. homogeneous) split epimorphisms are stable under products (Proposition 2.3.3), so that the right hand side vertical split epimor- phism is a Schreier (resp. homogeneous) one. And they are also stable under pullbacks, so that the left hand side split epimorphism is a Schreier (resp. homogeneous) one. The fact that the fibers in question are unital is now a consequence of the previous proposition and Proposition 1.3.2.
Proposition 2.4.5. For any monoid B, the fibers SPtB(Mon) and HPtB(Mon) are closed under equalizers as well, and thus under finite limits.
Proof. Given any parallel pair (h,h′) of morphisms in PtB(Mon):
// ′
// K[f ]
// ′ //66A
s′
consider its equalizer j in Mon; it determines a split epimorphism (φ,σ) which is the equalizer in P tB (M on). Complete the diagram by the vertical kernels; by commutation of limits, this produces the upper horizontal equalizer diagram. Since the set-theoretical Schreier retractions qf and qf′ make the upward right
66 K[φ] //
K[f] OO
K(h ) h
OO kf′ qf′
// K(j) kK(j)
//
K(h) ′
q
kf qf //
I hh
j
AOO
f sf′
σ
(( vv B
φ
h′
","26 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
hand side squares commute, and, since equalizers in Mon are equalizers in Set, this produces a factorization q which satisfies the conditions of a Schreier retraction for the split epimorphism (φ,σ) and makes it a Schreier split epi- morphism.
We get now the main result of this section:
Theorem 2.4.6. The kernel functor reflects commutativity. Given any pair of maps in PtB(Mon) with their two domains in SPtB(Mon):
U
ff
// OOA oo
w v′
l
v
n 88 U′ && xx w′
sf
B
l and n cooperate in the fiber PtB(Mon) if and only if their images by the kernel functor (which is the change of base functor αB∗ : PtB(Mon) → Mon along the initial map αB : 1 → B) cooperate in Mon.
Proof. Bycommutativityoffinitelimits(see,forexample,[25]formoredetails), the kernel functor is left exact (i.e. it preserves finite limits). This implies that it preserves the cooperating pairs (since the cooperator is constructed using only finite limits). Conversely suppose that the following pair cooperates in M on:
K[w] K(l) // K[f] oo K(n) K[w′]
Let us denote by qw : U → K[w] (resp. qw′ : U′ → K[w′]) the Schreier retraction of the kernel kw (resp. kw′ ). We have to define a cooperator φ: U ×B U′ → A in the fibre PtB(Mon). Let us set φ(u,u′) = lqw(u)·n(u′). We check easily that
φ(u, v′w(u)) = lqw(u) · nv′w(u) = lqw(u) · sw(u) = lqw(u) · lvw(u) = = l(qw(u) · vw(u)) = l(u)
and that
φ(vw′(u′), u′) = lqwvw′(u′) · n(u′) = 1 · n(u′) = n(u′).
It remains to show that φ is a monoid homomorphism when the restrictions of
l and n to the kernels cooperate. We have
φ(u·u¯,u′ ·u¯′)=lqw(u·u¯)·n(u·u¯′)=
while
= l(qw(u) · qw(vw(u) · qw(u¯))) · n(u′) · n(u¯′), φ(u, u′) · φ(u¯, u¯′) = lqw(u) · n(u′) · lqw(u¯) · n(u¯′),
","2.4. The fibrations of Schreier and homogeneous points 27
so we have to show that:
lqw(vw(u) · qw(u¯)) · n(u′) = n(u′) · lqw(u¯)
First notice that w(u) = w′(u′) since (u,u′) in U ×B U′, so we have to show that:
lqw(vw′(u′) · qw(u¯)) · n(u′) = n(u′) · lqw(u¯)
We have:
lqw(vw′(u′) · qw(u¯)) · n(u′) = lqw(vw′(u′) · qw(u¯)) · nqw′ (u′) · nv′w′(u′) = (∗).
According to the assumption lqw(vw′(u′)·qw(u¯)) and nqw′ (u′) commute. Whence (∗) = nqw′ (u′) · lqw(vw′(u′) · qw(u¯)) · nv′w′(u′) =
= nqw′ (u′) · lqw(vw′(u′) · qw(u¯)) · lvw′(u′) =
= nqw′ (u′) · l(qw(vw′(u′) · qw(u¯)) · vw′(u′)) = nqw′ (u′) · l(vw′(u′) · qw(u¯)) = = nqw′ (u′) · nv′w′(u′) · lqw(u¯) = n(qw′ (u′) · v′w′(u′)) · lqw(u¯) = n(u′) · lqw(u¯).
We noticed that the fiber SPtB(Mon) is unital. Accordingly, there is, inside this fiber, a natural notion of internal commutative object.
Corollary 2.4.7. A Schreier split epimorphism Aoo s ////B,
f
seen as an object in the fiber P tB (M on), is commutative if and only if the kernel K[f] of f is a commutative monoid. Accordingly any Schreier split epimorphism has at most one (commutative) monoid structure in this fiber.
Proof. As we already observed in the proof of the previous theorem, the ker- nel functor preserves finite limits. Consequently, it preserves the commutative objects. So that, when the Schreier split epimorphism is commutative, so is its kernel. The converse comes from the reflection of commutativity for Schreier split epimorphisms. The last point comes from the fact that, in a unital cate- gory, an object is commutative if and only if it has an internal monoid structure (see for example [4], Theorem 1.4.5).
Proposition 2.4.8. Suppose that the Schreier split epimorphism (A,B,f,s) is commutative in the fiber SPtB(Mon). Then the unique possible multiplication is:
m(a,a′)=q(a)·a′, for (a,a′)∈R[f].
","28 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
Proof. We have that
(a, a′) = (q(a) · sf(a), a′) = (q(a) · sf(a′), a′) = (q(a), 1) · (sf(a′), a′).
Hence, if m gives a monoid structure in PtB(Mon), it satisfies the following equalities :
m(a, a′) = m(q(a), 1) · m(sf(a′), a′) = q(a) · a′,
having m(sf(a′),a′) = a′ since m, being a binary operation on the object
(A,B,f,s) in the fiber SPtB(Mon), has to preserve units. Corollary 2.4.9. A Schreier split epimorphism (A,B,f,s) is endowed with a
group structure in PtB(Mon) if and only if its kernel K[f] is an abelian group.
Proof. The kernel functor, being left exact, preserves the group structure. Con- versely suppose K[f] is an abelian group. It is a commutative monoid, and con- sequently the Schreier split epimorphism (A, B, f, s) is endowed with a monoid structure. Moreover the kernel functor SPtB(Mon) → Mon is left exact and conservative; accordingly it reflects pullbacks, and consequently its extension to internal monoids reflects groups. See the Appendix for more details.
","Chapter 3
Schreier internal relations, categories and groupoids
We recall that an internal reflexive graph in a category E is a diagram of the form
d0 // X1oos0 //X0
d1
such that d0 s0 = 1X0 = d1 s0 . A reflexive relation is a reflexive graph such that the pair (d0,d1) is jointly monomorphic. If the category E has binary products,thismeansthattheinducedmorphism⟨d0,d1⟩:X1 →X0×X0 isa monomorphism.
Definition 3.0.10. An internal reflexive graph in the category Mon of monoids d0 //
X1oos0 //X0 d1
is a Schreier reflexive graph if the split epimorphism (d0 , s0 ) is a Schreier one. It is homogeneous if (d0, s0) is homogeneous.
As a consequence of Proposition 2.3.3, we have that Schreier (resp. homo- geneous) reflexive graphs are closed under finite products inside the category of internal reflexive graphs. The same is true for Schreier (resp. homogeneous) reflexive relations and for Schreier (resp. homogeneous) equivalence relations.
29
","30 Chapter 3. Schreier internal relations, categories and groupoids 3.1 Schreier reflexive relations and graphs
Example 3.1.1. For every monoid X, the discrete internal equivalence relation ∆X:
1X // Xoo1X //X
1X
is a homogeneous internal equivalence relation. It is explicitly given by: x1∆Xx2
if and only if x1 = x2.
Example 3.1.2. Example 2.2.7 shows that the internal order in Mon given by
the usual order between natural numbers:
p0 // ON oo s0 // N,
p1
where
is a homogeneous order relation.
ON = {(x, y) ∈ N × N | x ≤ y},
Example 3.1.3. Since Z, with the usual sum, is a group, the internal order in
Mon given by the usual order between integers:
p0 // OZ oo s0 // Z,
p1
where
is a homogeneous order relation (thanks to Proposition 2.2.4).
OZ = {(x, y) ∈ Z × Z | x ≤ y},
The previous example can be obviously generalized to the case of the or-
der relation OG associated with any ordered group G.
Given any Schreier reflexive relation R on a monoid M and any pair xRy,
let us denote by 1Rq(xRy) the value of the Schreier retraction at xRy. Lemma 3.1.4. The characteristic equations of a Schreier retraction become
q(xRy)·x=y and q(xR(t·x))=t.
Proof. The first equality comes from the fact that s0(x) = xRx, hence
xRy = 1Rq(xRy) · xRx = xR(q(xRy) · x).
Concerning the second one, an element of K[d0] is of the form 1Rt, and so we have:
1Rt = 1Rq(1Rt · xRx) = 1Rq(xR(t · x)).
","3.1. Schreier reflexive relations and graphs 31
Proposition 3.1.5. Any Schreier reflexive relation R is transitive, and we have q(yRz) · q(xRy) = q(xRz). It is an equivalence relation if and only if K[d0] is a group.
Proof. Suppose that xRy and yRz. Since the reflexive relation is a Schreier one, we have 1Rq(xRy) and 1Rq(yRz), whence 1R(q(yRz) · q(xRy)). From that we get xR(q(yRz) · q(xRy) · x), with q(yRz) · q(xRy) · x = q(yRz) · y = z thanks to the previous lemma, whence xRz. The uniqueness of the decomposition implies that q(yRz) · q(xRy) = q(xRz).
Suppose moreover that R is symmetric. Then, for any y such that 1Ry, we have that yR1. Whence q(yR1) · y = 1 for any y ∈ K[d0], which implies that K[d0] is a group. Conversely, suppose that K[d0] is a group and that we have xRy. Then we have q(xRy) · x = y with q(xRy) ∈ K[d0]. Whence x = q(xRy)−1 ·y with 1Rq(xRy)−1; so that yR(q(xRy)−1 ·y), which is yRx.
Proposition 3.1.6. Given a monoid X, the indiscrete equivalence relation ∇X given by:
p0 // X×Xoo s0 //X
p1
is a Schreier equivalence relation if and only if X is a group. In this case ∇X is actually homogeneous. The equivalence relation ∇X is explicitly given by: x1∇Xx2 forallx1,x2∈X.
Proof. If X is a group, it is an immediate consequence of Proposition 2.2.4; ac- tually the equivalence relation ∇X is even homogeneous. Conversely, suppose the indiscrete equivalence relation is a Schreier equivalence relation. Then, ac- cording to the previous proposition, K[p0] = X is a group.
Corollary 3.1.7. Given any monoid B, the three following conditions are equiv- alent:
(a) the monoid B is a group
(b) any split epimorphism with codomain B is homogeneous
(c) any split epimorphism with codomain B is a Schreier split epimorphism.
Proof. (a) ⇒ (b) is given by Proposition 2.2.4.
(b) ⇒ (c) holds since any homogeneous split epimorphism is a Schreier one. (c) ⇒ (a) Condition (c) implies that the indiscrete relation ∇B is Schreier and that consequently B is a group.
Corollary 3.1.8. Given a group B, any equivalence relation R on B in Mon is homogeneous, and R is itself a group.
From that, it is easy to build some examples:
","32 Chapter 3. Schreier internal relations, categories and groupoids Example 3.1.9. Consider the homogeneous split epimorphism
Z∗oo i //N∗ abs
described in Example 2.2.8. The kernel equivalence relation R[abs], defined by:
xR[abs]y ⇔ |x| = |y|
is homogeneous, as well. This can be seen directly, but it also comes from Propo- sition 3.1.12 below, since the kernel of abs is K[abs] = {±1}, which is a group.
Example 3.1.10. Let G be a group and H be a normal subgroup; denote by
RH the equivalence relation on G associated with H, given by the following:
ifg,g ∈G,thengR g ifandonlyifgg−1 ∈H.GivenanymonoidM 12 1H2 12
such that H ⊆ M ⊆ G, then the restriction RM of RH on M is homogeneous. Indeed, the Schreier retraction q′ of the relation RM is nothing but the restric- tion of the Schreier retraction q of RH (since the kernels of the corresponding split epimorphisms are both isomorphic to H); this proves that RM is right homogeneous. The proof that it is also left homogeneous is analogous.
Example 3.1.11. Some particular cases of the general situation described in Example 3.1.10 are obtained when G = C∗ is the multiplicative group of non- zero complex numbers, M ={a+ib | a,b∈Z, (a,b)̸=(0,0)} and H is either {±1} or {±1, ±i}.
Proposition 3.1.12. Given a split epimorphism (X,Y,f,s), its kernel equiva- lence relation R[f] is a Schreier one if and only if the split epimorphism is itself a Schreier one and the kernel K[f] is a group. Its kernel equivalence rela- tion is homogeneous if and only if the split epimorphism is itself homogeneous and the kernel K[f] is a group.
Proof. Let (X,Y,f,s) be a split epimorphism. Suppose R[f] is a Schreier equiv- alence relation, then K[p0] ≃ K[f] is a group. On the other hand, consider the following diagram of split epimorphisms, which is always a pullback:
X s1 // R[f] (3.1.1)
f s p0 s0 //
Y s X,
where the morphism s1 is given by s1(x) = (sf(x),x) (see the Appendix). Then the left hand side split epimorphism is a Schreier one, since so is the right hand side one. In the same way, when R[f] is a homogeneous equivalence relation, the split epimorphism in question is homogeneous.
OO
OO
","3.1. Schreier reflexive relations and graphs 33 Conversely, suppose that (X, Y, f, s) is a Schreier split epimorphism and
K [f ] is a group. Define qp0 (x, x′ ) = (1, qf (x′ ) · qf (x)−1 ). We can check that qp0(x,x′)·s0p0(x,x′) = (1,qf(x′)·qf(x)−1)·(x,x) = (x,qf(x′)·qf(x)−1 ·x) =
= (x, qf (x′ ) · sf (x)) = (x, qf (x′ ) · sf (x′ )) = (x, x′ ),
qp0 ((1, k) · s0(x)) = qp0 ((1, k) · (x, x)) = qp0 (x, k · x) = (1, qf (k · x) · qf (x)−1) =
= (1, qf (k) · qf (sf(k) · qf (x)) · qf (x)−1) = (1, qf (k) · qf (x) · qf (x)−1) = (1, k). If moreover (X,Y,f,s) is left homogeneous, in the same way we can define
q¯ (x, x′ ) = (1, q¯ (x)−1 · q¯ (x′ )) which satisfies the desired conditions for a left p0 ff
homogeneous split epimorphism. Accordingly R[f] is a homogeneous equiva- lence relation.
We conclude this section with a result which will be useful later on.
and
Proposition 3.1.13. Let
d0 // X1oos0 //X0
d1
be a Schreier reflexive graph. The split epimorphism (d1,s0) is also a Schreier one if and only if, for any x ∈ K[d0], the element d1(x) is invertible in X0. In particular, when K[d0] is a group, the split epimorphism (d1,s0) is also a Schreier one.
Proof. Suppose that (d1, s0) is a Schreier split epimorphism. Let us denote by q0 and q1 the Schreier retractions associated with (d0, s0) and (d1, s0), respectively. Then, for any x ∈ K[d0], the inverse of d1(x) is d0q1(x). Indeed:
and hence
and moreover hence
x = q1(x) · s0d1(x)
1 = d0(x) = d0q1(x) · d0s0d1(x) = d0q1(x) · d1(x), q1(x) = q0q1(x) · s0d0q1(x),
1 = d1q1(x) = d1q0q1(x) · d1s0d0q1(x) = d1q0q1(x) · d0q1(x),
","34 Chapter 3. Schreier internal relations, categories and groupoids
but q0q1(x) = q0(x), because, thanks to Proposition 2.1.5, we have q0(x) = q0(q1(x) · s0d1(x)) =
= q0q1(x) · q0(s0d0q1(x) · q0s0d1(x)) = q0q1(x) · 1 = q0q1(x); since q0(x) = x (because x ∈ K[d0]), we get
1 = d1(x) · d0q1(x).
Conversely, suppose that d1(x) has an inverse for any x ∈ K[d0]. We have to define a map q1 : X1 K[d1] satisfying the properties of the Schreier retraction of (d1,s0). This map can be defined as:
q1(x) = q0(x) · s0(d1q0(x))−1,
where q0 is the Schreier retraction of (d0, s0). First of all, q1(x) ∈ K[d1] for any
x ∈ K[d0], since:
d1q1(x) = d1q0(x) · d1s0(d1q0(x))−1 = d1q0(x) · (d1q0(x))−1 = 1.
Moreover, x = q1(x) · s0d1(x) for any x ∈ X1, because:
q1(x) · s0d1(x) = q0(x) · s0(d1q0(x))−1 · s0d1(x) =
= q0(x) · s0(d1q0(x))−1 · s0d1(q0(x) · s0d0(x)) = = q0(x) · s0(d1q0(x))−1 · s0d1q0(x) · s0d0(x) =
= q0(x) · s0d0(x) = x.
Finally, we have that q1(x · s0(b)) = x for any x ∈ K[d1] and any b ∈ X0,
because:
where the second equality holds because
q0(x · s0(b)) = q0(x) · q0(s0d0(x) · q0s0(b)) = q0(x)
q1(x · s0(b)) = q0(x · s0(b)) · s0(d1q0(x · s0(b)))−1 =
= q0(x) · s0(d1q0(x))−1 = q0(x) · s0(d1q0(x))−1 · s0d1(x) = = q0(x) · s0(d1q0(x))−1 · s0d1q0(x) · s0d1s0d0(x) =
= q0(x) · s0d0(x) = x,
thanks to Proposition 2.1.5. This concludes the proof.
","3.2. Schreier internal categories 35 We conclude this section by observing that, in a reflexive graph
d0 // X1oos0 //X0,
d1
the fact that both (d0, s0) and (d1, s0) are Schreier split epimorphisms does not imply that K[d0] is a group. As an example we can take, for any monoid M, the Schreier reflexive graph given by:
M
⟨1,0⟩ p1 //
// M × M oo ⟨0,1⟩ // M,
p1
where both the domain and the codomain are given by the morphism p1 such that p1(x,y) = y, according to the simplicial notation (see the Appendix). Then the two split epimorphisms are both Schreier, but M, which is the kernel of p1, is not necessarily a group. Observe that the image by p1 of any element in K[p1] is invertible in M because it is the unit element. In general, given any Schreier split epimorphism (A, B, f, s), the following is a Schreier reflexive graph:
f //
Aoo s //B.
f
3.2 Schreier internal categories
Let us recall that an internal category X1 in any category E is a reflexive graph:
d0 // X1oos0 //X0
d1
such that the following pullback of split epimorphisms, which defines X2 as the
internal object of the composable pairs
oo s0
X2 OO
// X1 OO
(3.2.2)
d0
d2 s1 d1 s0
oos0
X1
// X0
d0
is endowed with a composition map d1 : X2 → X1 satisfying the remaining simplicial identities (see the Appendix):
","36 Chapter 3. Schreier internal relations, categories and groupoids
(1) d0d1 = d0d0, d1d1 = d1d2 (incidence axioms)
(2) d1s0 = 1X1 , d1s1 = 1X1 (composition with identities)
This composition must satisfy the associativity axiom; for that consider the following pullback of split epimorphisms:
X3 OO
oo s0
// X2 OO
(3.2.3)
d0
d3 s2 d2 s1
oos0
X2
// X1
d0
The composition map d1 induces a couple of maps (d1,d2): X3 ⇒ X2 such that d0d1 = d0d0, d2d1 = d1d3 and d0d2 = d1d0, d2d2 = d2d3. The associativity is given by the remaining simplicial axiom
(3) d1d1 = d1d2.
So an internal category in E is a 3-truncated simplicial object such that the two squares above are pullbacks. Moreover, given an internal category X1, consider the following diagram:
d0 // Dec1X1 : X2 oo s0 // X1
ε1X1 d2
(3.2.4)
d1 X1: X1oo s0 // X0
d1 d0
//
then the upper reflexive graph is still a category denoted by Dec1X1, while the vertical diagram determines an internal functor denoted by ε1X1 (see the Appendix). According to the pullback defining the object X2, when we work in Mon, the upper category is a Schreier one as soon as so is the lower one.
According to Patchkoria [28], we give the following:
Definition 3.2.1. An internal category in the category of monoids
d0 // X1oos0 //X0
d1
is a Schreier internal category if the split epimorphism (d0,s0) is a Schreier
one. It is a homogeneous internal category if (d0,s0) is homogeneous.
Again, as consequence of Proposition 2.3.3, we have that Schreier (resp. homogeneous) internal categories are closed under finite products inside the
d1
","3.2. Schreier internal categories 37 category of internal categories.
Actually the notion of Schreier internal category in Mon was already introduced, under the name of homogeneous category, in [22], where it was proved to be equivalent to the notion of crossed module in Mon.
Proposition 3.2.2. On a Schreier reflexive graph there is at most one structure of category. It is sufficient to have the composition map d1 : X2 → X1 with axiom (2), the axioms (1) and (3) come for free.
Proof. When the reflexive graph is a Schreier one, the pullback 3.2.2 is actually a product in the fiber P tX0 (M on). Since the lower horizontal split epimorphism is a Schreier one, the pair (s0,s1) is jointly epimorphic, because PtX0(Mon) is SP tX0 -unital. Accordingly there is a unique possible map satisfying the axioms (2). Actually when we have a map d1 satisfying axiom (2), we can check axiom (1) by composition with the same pair (s0,s1). The pullback 3.2.3 is such that the lower split epimorphism is a Schreier one, and hence the pair (s0,s2) is jointly epimorphic. We get axiom (3) by composition with this pair.
So, given a Schreier reflexive graph, being an internal category is a prop- erty and not an additional structure.
Proposition 3.2.3. Given a Schreier reflexive graph: d0 //
with q: X1 → K[d0] its associated Schreier retraction, it is an internal category if and only if, for any pair (σ, α) ∈ X1 × K[d0], we have:
q(s0d1(σ) · α) · σ = σ · α.
When it is the case, the composition of a composable pair (τ,σ) is given by:
d1(τ,σ)=q(τ)·σ, which implies that K[d0] and K[d1] commute in X1. Proof. Since, for any (τ,σ) ∈ X2, we have (τ,σ) = (q(τ),1)) · (s0d0(τ),σ) the
compositioninthecategoryX1 mustbed1(τ,σ)=q(τ)·σ.Indeed: d1(τ,σ) = d1(q(τ),1)·d1(s0d0(τ),σ) = q(τ)·d1(s0d1(σ),σ) = q(τ)·σ.
On the other hand, in X2, when α is in K[d0], we have that (s0d1(σ), σ) · (α, 1) = (s0d1(σ) · α, σ)
and hence
q(s0d1(σ) · α) · σ = d1(s0d1(σ) · α, σ) = d1(s0d1(σ), σ) · d1(α, 1) =
X1oos0 //X0 d1
","38 Chapter 3. Schreier internal relations, categories and groupoids = qs0d1(σ) · σ · q(α) = σ · α.
It is easy to check that, when this formula holds, the composition becomes a monoid homomorphism. Moreover, when σ is in K[d1], we get
σ · α = q(s0d1(σ) · α) · σ = q(α) · σ = α · σ.
3.3 Schreier internal groupoids
An internal category X1 in a category E is a groupoid when, moreover, the following square determined by the composition map is a pullback:
X d1 //X 21
d0 X1
d0 X0,
// d0
or, in other words, when the following diagram is the kernel equivalence relation of d0:
d0 // d0 X2oos0 //X1 //X0
d1
or equivalently the natural comparison functor θ1 : Dec1X1 → R[d0] is an
isomorphism.
Definition 3.3.1. An internal groupoid in the category of monoids d0 //
is a Schreier internal groupoid if the split epimorphism (d0,s0) is a Schreier one. It is a homogeneous internal groupoid if (d0,s0) is homogeneous.
Again, as consequence of Proposition 2.3.3, we have that Schreier (resp. homogeneous) internal groupoids are closed under finite products inside the category of internal groupoids.
Proposition 3.3.2. Given a Schreier internal category X1, the following condi- tions are equivalent:
(a) X1 is a Schreier groupoid;
(b) R[d0] is a Schreier equivalence relation;
(c) K[d0] is a group.
X1oos0 //X0 d1
","3.3. Schreier internal groupoids 39
Proof. (b) ⇔ (c) is given by Proposition 3.1.12. Suppose X1 is a Schreier groupoid, then Dec1X1 is a Schreier category and we have Dec1X1 ≃ R[d0] since X1 is a groupoid. Accordingly R[d0] is a Schreier equivalence relation. Conversely suppose that X1 is a Schreier category and R[d0] is a Schreier equivalence relation; then consider the following diagram
X s1 //X OO1 OO2
s1
DD d0s0
d0
θ1 // ##
X1 s0
R[d0 ] BB
d0 s0 d0
//
X0 s0 1
X
s0
where any commutative square of split epimorphisms is a pullback. This says that the image by the change-of-base functor s∗0 of θ1 is an isomorphism. Since the two right hand side split epimorphisms are Schreier ones and the functor s∗0:SPtX1(Mon)→SPtX0(Mon)isconservative(seeProposition2.4.1),θ1 is itself an isomorphism; we have then Dec1X1 ≃ R[d0] and X1 is a groupoid.
Example 3.3.3. Let L and M be monoids, and d: L → M be a morphism between them. Suppose that d is central, i.e. d(L) is contained in the center of M (This means that d(l)·m = m·d(l) for any l ∈ L and any m ∈ M). Then we get the following Schreier internal reflexive graph:
πM //
M × L oo ⟨1,0⟩ // M,
⟨1,d⟩
where the morphism ⟨1, d⟩ is defined by ⟨1, d⟩(m, l) = m · d(l). It is a Schreier internal category if and only if L is a commutative monoid. Indeed, thanks to Proposition 3.2.3, we have that the unique possible composition of arrows is givenbythemorphism1M ×(·L):M×L×L→M×L,associatingwithany triple (m, l1, l2) the pair (m, l1 · l2). It is immediate to see that it is actually a morphism if and only if L is commutative. Moreover, according to the previous proposition, it is an internal groupoid if and only if L (which is the kernel of πM ) is a group.
We can also observe that if the Schreier reflexive graph
πM //
M × L oo ⟨1,0⟩ // M.
⟨1,d⟩
is a Schreier reflexive relation, then d is a monomorphism. This implies that L is commutative, and hence the relation is actually a preorder. According to Proposition 3.1.5, it is an equivalence relation if and only if L is a group.
","40 Chapter 3. Schreier internal relations, categories and groupoids
Example 3.3.4. An interesting particular case of the Schreier reflexive graph described in the previous example is given when L = M is a commutative monoid and d = 1M ; in this case, the codomain morphism ⟨1, d⟩ is nothing but the multiplication in M:
π1 //
M × M oo ⟨1,0⟩ // M.
·M
The following proposition is related to a known result about internal cat-
egories and groupoids in the category Gp of groups (see [23]). d0 //
Proposition 3.3.5. Let X1 oo s0 // X0 be a Schreier reflexive graph such that d1
K[d0] is a group. The following conditions are equivalent: (a) the kernels of d0 and d1 cooperate in Mon;
(b) the reflexive graph is an internal category;
(c) the reflexive graph is an internal groupoid.
Proof. We already proved (see the end of the proof of Proposition 3.2.3) that, if a Schreier reflexive graph is an internal category, then the kernels of the do- main and of the codomain cooperate. Hence (b) implies (a).
Conversely, suppose that we have a Schreier reflexive graph
oo q k
d0 //
// X1 oo s0 // X0,
K[d0]
with the Schreier split epimorphism associated with the codomain d1 given by
oo t oo s0 K[d1] // X1
// X0,
such that k and l cooperate (we recall that (d1,s0) is a Schreier split epimor-
phism thanks to Proposition 3.1.13). This means that x·y=y·x forallx∈K[d0], y∈K[d1].
We first show that the map defined by
m(a′, a) = q(a′) · a,
d1
l d1
","3.3. Schreier internal groupoids 41
which is the unique possible multiplication, is a morphism. We have that m((a′1, a1) · (a′2, a2)) = m(a′1 · a′2, a1 · a2) =
=q(a′1 ·a′2)·a1 ·a2 =q(a′1)·q(s0d0(a′1)·q(a′2))·a1 ·a2 = = q(a′1) · q(s0d1(a1) · q(a′2)) · t(a1) · s0d1(a1) · a2 =
= q(a′1) · t(a1) · q(s0d1(a1) · q(a′2)) · s0d1(a1) · a2 =
= q(a′1) · t(a1) · s0d1(a1) · q(a′2) · a2 =
= q(a′1) · a1 · q(a′2) · a2 = m(a′1, a1) · m(a′2, a2). Moreover, m respects domain and codomain, indeed:
and
d0m(a′, a) = d0q(a′) · d0(a) = 1 · d0(a) = d0(a),
d1m(a′, a) = d1q(a′) · d1(a) = d1q(a′) · d0(a′) =
= d1q(a′) · d1s0d0(a′) = d1(q(a′) · s0d0(a′)) = d1(a′).
It is associative, because
m(m(a′′, a′), a) = m(q(a′′) · a′, a) = q(q(a′′) · a′) · a =
= qq(a′′) · q(s0d0q(a′′) · q(a′)) · a = q(a′′) · q(1 · q(a′)) · a = = q(a′′) · q(a′) · a = m(a′′, m(a′, a)).
Finally, m preserves identities, since
m(s0d1(a), a) = qs0d1(a) · a = 1 · a = a,
and
m(a, s0d0(a)) = q(a) · s0d0(a) = a. This proves that (a) implies (b).
The equivalence between (b) and (c) comes from Proposition 3.3.2.
Gran proved in [20] that, in a Mal’tsev category, internal groupoids are closed under quotients inside the category of internal reflexive graphs. Here we have a similar result:
","42 Chapter 3. Schreier internal relations, categories and groupoids Proposition 3.3.6. Consider the following commutative diagram
d′0 //
′′ X1′oos0 X
// 0 d′1
uv d0 //
X, X1 oo s // 0
0
d1
where the two rows are Schreier reflexive graphs, u is a regular epimorphism, and the kernels of d0 and d′0 are groups. If the upper row is an internal groupoid, then so is the lower one.
Proof. Thanks to Proposition 3.3.5, we only have to show that k: K[d0] → X1 and l: K[d1] → X1 cooperate, where k is the kernel of d0 and l is the kernel of
k′ : K[d′0] → X1′ and l′ : K[d′1] → X1′ cooperate, where k′ is the kernel of d′0 and l′ is the kernel of d′1. This means that
x′·y′ =y′·x′ forall x′ ∈K[d′0], y′ ∈K[d′1].
Consider the following diagrams, where i and j are induced by u via the uni-
versal property of the kernels:
d′0 // ′′
d1. Since X1′ oo s0 // X0 is a Schreier internal groupoid, we already know that d′1
s′ ′ k′ //′oo0
′
s′
′ l′ //′oo0 ′
K[d0] X1
d′0
// X0
K[d1] X1
// X0
d′1 iuvjuv
// oo s0
K[d0] k X1
d0
// X0,
K[d1]
// oo s0
l
X1
// X0.
Since u is a regular epimorphism, i.e. it is surjective, also i is. Indeed, there exists a set-theoretical map r : X1 → X1′ such that ur = 1X1 . Then the map q′rk: K[d0] → K[d′0] (where q′ is the map given by the Schreier condition for the split epimorphism d′0) is such that
iq′rk = qurk = qk = 1K[d0],
where q is the map given by the Schreier condition for the split epimorphism d0, and this proves that i is surjective. Similarly we prove that j is surjective, too. Now, given x ∈ K[d0] and y ∈ K[d1], there exist x′ ∈ K[d′0] and y′ ∈ K[d′1] such that i(x′) = x and j(y′) = y. Then we have:
x·y=i(x′)·j(y′)=u(x′ ·y′)=u(y′ ·x′)=j(y′)·i(x′)=y·x,
and this proves that k and l cooperate.
d1
","Chapter 4
Mal’tsev aspects of Mon related to Schreier split epimorphisms
4.1 Mal’tsev categories
We recalled in Section 1.5 that a category C is a Mal’tsev category [16, 17] when any reflexive relation is an equivalence relation; this fact appears to be equivalent to the property that any fiber PtXC of the fibration ¶C is unital, see [8]. The category Gp of groups is a Mal’tsev one. The preorder ON on the natural numbers (Example 2.2.7) shows that the category Mon is not a Mal’tsev one. However, we pointed out a meaningful structural observation: the category Mon of monoids is equipped with a fully faithful subfibration ¶S of the fibration of points ¶:
SPtMon //
¶S && yy
Mon
such that any fiber PtBMon is SPtBMon-unital and, consequently, any fiber SPtBMon is unital. In this chapter, we shall be interested in what is remaining of the Mal’tsev results in this new structural context.
// PtMon
43
¶
","44 Chapter 4. Mal’tsev aspects of Mon 4.2 First observations
First, we observed that, in the category Mon, any Schreier reflexive relation (d0,d1): R ⇒ X is only necessarily transitive (Proposition 3.1.5). Example 2.2.7 gives a Schreier reflexive relation which is not an equivalence relation. A Schreier reflexive relation R is an equivalence relation if and only if K[d0] is a group.
In a Mal’tsev category, on a reflexive graph there is at most one structure of internal category, which is necessarily an internal groupoid. In the previous chapter we showed that, on a Schreier reflexive graph, there is again at most one structure of internal category, but there are Schreier internal categories which are not internal groupoids. A Schreier internal category is a Schreier internal groupoid if and only if K[d0] is a group.
4.3 Centrality for Schreier relations
More importantly, the Mal’tsev context guarantees some nice centrality prop- erties of the equivalence relations. The equivalence relations R on an object X, coinciding with the reflexive relations on X, are just subobjects of the object (p0 , s0 ) : X × X X in the fibre P tX C:
R //(d0,d1)//X×X
cc
p0
OO
s0
s0 d0
## X
It appears that two equivalence relations R and S on X centralize each other in a Mal’tsev category C when the subobjects (d1,d0): R X × X and (d0,d1): S X × X commute in the fiber PtXC. Indeed, the cooperator R×X S→X×Xinthefibermustbeoftheformφ(xRySz)=(x,p(xRySz)), with p(xRxSy) = y and p(xRySy) = x. The map p: R×X S → X, with the two equations, is characteristic of the fact that R and S centralize each other (see [13], and also [30] and [33]); it is called the connector between the relations R and S. It is well known that, in the category Gp of groups, two equivalence relations R and S on a group G centralize each other if and only if the nor- mal subgroups 1R and 1S given by the equivalence classes of the unit element commute inside the group G.
In the category Mon, for any monoid B the fiber PtBMon is SPtBMon- unital. So we can keep the same definition, provided that one of the relations, let us choose S, is a Schreier reflexive relation:
Definition 4.3.1. Given a reflexive relation R and a Schreier reflexive relation S on the monoid B, we say that R and S centralize each other when there
","4.3. Centrality for Schreier relations 45 is a (necessarily unique) monoid homomorphism p: R ×B S → B such that
p(xRxSy) = y and p(xRySy) = x. We denote this situation by [R, S] = 0. Example 4.3.2. Given the order ON on N, we have [ON,ON] = 0 introducing
p(x ≤ y ≤ z) = z − y + x.
Suppose S is a Schreier reflexive relation. Given any ySz, we shall denote
by 1Sq(ySz) the value of the associated Schreier retraction S → K[d0].
Proposition 4.3.3. The reflexive relation R and the Schreier reflexive relation S centralize each other in Mon if and only if, for any 1St ∈ K[d0] and any xRy, we have q(yS(y · t)) · x = x · t. In this circumstance we have p(xRySz) = q(ySz)·x. When [R, S] = 0, we have necessarily xSp(xRySz) and p(xRySz)Rz.
Proof. The condition of the existence of p and its definition is a straighforward consequence of Proposition 2.4.3. We check p(xRySy) = q(ySy) · x = 1 · x = x and p(xRxSz) = q(xSz) · x = z.
We know that, in a Mal’tsev category, two reflexive relations (i.e. two equivalence relations) R and S on X centralize each other as soon as R ∩ S = ∆X. In Mon, we have as well:
Proposition 4.3.4. The equivalence relation R and the Schreier equivalence re- lation S on the monoid X centralize each other in Mon as soon as R∩S = ∆X.
Proof. Suppose that 1St and xRy. From the first point we get xS(x·t); for the same reason we have xS(q(yS(y · t)) · x) and since S is an equivalence relation we get (x·t)S(q(yS(y·t))·x). On the other hand, from xRy, we get (x·t)R(y·t) and (q(yS(y · t)) · x)R(q(yS(y · t)) · y) = (q(yS(y · t)) · x)R(y · t); since R is an equivalence relation we get (x · t)R(q(yS(y · t)) · x). Now since R ∩ S = ∆X , we have q(yS(y · t)) · x = x · t and [R, S] = 0.
Given a relation R, we denote by Rop the relation obtained from R by twisting d0 and d1. In other terms, xRopy if and only if yRx.
Proposition 4.3.5. Suppose both Rop and S are Schreier reflexive relations on a monoid X. Then the reflexive relations R and S centralize each other if and only if the submonoids dR0 kdR1 : K[dR1 ] X and dS1 kdS0 : K[dS0 ] X commute in X. Suppose that both R and S are Schreier equivalence relations, then R and S centralize each other if and only if the equivalence classes 1R and 1S commute in X.
Proof. It is a straighforward consequence of Theorem 2.4.6.
Proposition 4.3.6. Let (A,B,f,s) be a Schreier split epimorphism such that R[f] is a Schreier equivalence relation. We have [R[f],R[f]] = 0 if and only if its kernel K[f] is an abelian group.
","46 Chapter 4. Mal’tsev aspects of Mon
Proof. Since R[f] is a Schreier equivalence relation, K[f] is a group. By the reflection of commutativity (Theorem 2.4.6), we have [R[f],R[f]] = 0 if and only if K[f] is commutative.
The following result is analogous to a known one in the context of Mal’tsev categories (see [17]):
Proposition 4.3.7. Consider a Schreier reflexive graph such that K[d0] is a group:
d0 // X1oos0 //X0
d1
The following conditions are equivalent:
1) this graph is underlying a Schreier category;
2) this graph is underlying a Schreier internal groupoid;
3) the kernel equivalence relations R[d0] and R[d1] centralize each other; 4) the kernels K[d0] and K[d1] commute in X1.
Proof. Since the split epimorphism (X1, X0, d0, s0) is a Schreier one and K[d0] is a group, the kernel equivalence relation R[d0] is a Schreier one, and the fact that any other equivalence relation on X1 centralizes with it makes sense.
1) ⇒ 2) Suppose the graph is underlying a Schreier internal category. Since K[d0] is a group, it is a Schreier internal groupoid.
2) ⇒ 3) Suppose the graph is underlying a Schreier internal groupoid. Let us first observe that the split epimorphism (X1, X0, d1, s0) is isomorphic to (X1, X0, d0, s0). The isomorphism is induced by the isomorphism γ : X1 → X1 which associates with any element of X1 (that is, with any arrow of the in- ternal groupoid) its inverse with respect to the composition of arrows in the internal groupoid. Then (X1,X0,d1,s0), being isomorphic to a Schreier split epimorphism, is a Schreier one, too. This implies that K[d1] is a group; accord- ingly R[d1] is a Schreier equivalence relation. On the other hand, an internal groupoid being an internal category, the kernels K[d0] and K[d1] commute in X1, and consequently, according to the reflection of commutativity by the ker- nel functor(Theorem 2.4.6), the kernel equivalence relations R[d0] and R[d1] centralize each other.
3) ⇒ 4) Suppose that the kernel equivalence relations R[d0] and R[d1] central- ize each other. The preservation of commutativity by the kernel functor implies that the kernels K[d0] and K[d1] commute in X1.
4) ⇒ 1) Suppose that the kernels K[d0] and K[d1] commute in X1. First notice that when K[d0] is a group, any image of an element in K[d0] is invertible. To prove that the graph is underlying an internal category, we have to show that, for any pair (σ, α) ∈ X1 × K[d0], we have:
q(s0d1(σ) · α) · σ = σ · α
","4.4. Double centralizing relation 47
We have that:
q(s0d1(σ) · α) · σ = q(s0d1(σ) · α) · q(σ) · s0d0(σ) =
= q(s0d1(σ) · α) · q(σ) · s0d1q(σ)−1 · s0d1q(σ) · s0d0(σ) = (∗).
Now q(σ) · s0d1q(σ)−1 is in K[d1] and hence it commutes with q(s0d1(σ) · α),
while
s0d1q(σ) · s0d0(σ) = s0d1q(σ) · s0d1s0d0(σ) = s0d1(q(σ) · s0d0(σ)) = s0d1(σ).
Then we get
(∗) = q(σ) · s0d1q(σ)−1 · q(s0d1(σ) · α) · s0d1(σ) = q(σ)·s0d1q(σ)−1 ·s0d1(σ)·α=q(σ)·s0d0(σ)·α=σ·α.
4.4 Double centralizing relation
We shall show in the next chapter that any Schreier equivalence relation has a centralizer; we need for that the following precisions (we refer to [13] for a more detailed account). According to the end of Proposition 4.3.3, when we have [R,S] = 0, we necessarily get xSp(xRySz) and p(xRySz)Rz. In set theoretical terms, this means that, with any triple xRySz, we can associate a square of related elements:
x S // p(x,y,z) R // R
y z.
S
More acutely, this says that any connected pair of reflexive relations (R,S) on the monoid X produces the following diagram of reflexive relations in Mon:
R ×X S oo p1 //// S OOR OO
(d0 p0,p) (p,dS1p1) dS0 dS1
(4.4.1)
p0
dR1 //
R oo
// X
dR0
It is called the centralizing double relation associated with the connector and it characterizes the fact that [R,S] = 0. When R is an equivalence relation, the
","48 Chapter 4. Mal’tsev aspects of Mon upper row becomes an equivalence relation, and, moreover, the two commuta-
tive squares in the diagram below are pullbacks:
R× Soo p1 //S
// OO dR1 //
OOX
(dR0 p0,p) dS0
p0
When R and S are equivalence relations, all the reflexive relations in this di- agram are equivalence relations, and, moreover, the four commutative squares in diagram 4.4.1 are pullbacks.
When G is a group and (R,S) is a pair of equivalence relations on G centralizing each other, the square of related elements is the following one:
x S //z·y−1·x R // R
y z.
S
When M is a monoid and (R, S) is a pair of reflexive relations on M centralizing each other, with S a Schreier one, we noticed that the square of related elements is the following one:
x S // q(ySz)·x R // R
y z.
S
R oo
// X
dR0
","Chapter 5
Schreier and homogeneous representability
5.1 The Schreier split extension classifier
This section is devoted to showing that, for any monoid X, the monoid End(X) of endomorphisms of X has a universal property similar to the automorphisms group Aut(G) in the category of groups [5, 6, 3], namely that it allows to clas- sify the Schreier split epimorphisms.
Given a monoid X, we denote by Hol(X) the monoid whose underlying set is X × End(X) and whose monoid operation is the following:
(x1, γ1) · (x2, γ2) = (x1 · γ1(x2), γ1γ2)
where the operation in the second component is the usual composition of ho- momorphisms. It is immediate to prove that this operation is associative and that (1,1X) is the identity for it. It is also easy to see that, in this way, we obtain a split epimorphism:
Hol(X) oo ρX // End(X), (5.1.1) θX
where ρX (γ) = (1, γ) and θX is the projection. The symbol Hol stands for holo- morph, since the construction above is analogous to the one of the holomorph in group theory.
Lemma 5.1.1. The split epimorphism (5.1.1) is a Schreier one. Its kernel is, up to isomorphism, the monoid X itself.
49
","50 Chapter 5. Schreier and homogeneous representability
Proof. The kernel of θX is given, up to isomorphism, by the map ιX : X Hol(X) defined by ι(x) = (x,1X). We can define a map q: Hol(X) → X by q(x, γ) = x and then we have:
and
(x, γ) = (x, 1X ) · (1, γ) = ιX q(x, γ) · ρX θX (x, γ) ιX q((x, 1X ) · (1, γ)) = ιX q(x, γ) = (x, 1X ).
Theorem 5.1.2. For any Schreier split epimorphism with kernel X: Aoo s //B
f
there exists a unique morphism φ: B → End(X) such that the following dia- gram is commutative and its right hand side part is a pullback of split epimor- phisms:
X oo qf // A oo s // B kf
X
ιX
// Hol(X)
// End(X).
φ¯ φ ooq ooρX
The explicit definition of φ is φ(b)(x) = qf (s(b) · x). The pair (k, s) being jointly strongly epimorphic, the homomorphism φ¯ is completely determined by the commutativity of this diagram, and we have: φ¯(a) = (qf (a), φf (a)).
Proof. We start by observing that, for any b ∈ B, φ(b) ∈ End(X). Indeed, for any b ∈ B and any x1,x2 ∈ X:
φ(b)(x1 · x2) = qf (s(b) · x1 · x2), φ(b)(x1) · φ(b)(x2) = qf (s(b) · x1) · qf (s(b) · x2)
and the two expressions are equal, because:
qf(s(b)·x1 ·x2) = qf(s(b)·x1)·qf(sf(s(b)·x1)·x2) = qf(s(b)·x1)·qf(s(b)·x2).
Moreover, φ is a monoid homomorphism. Indeed, for any b1,b2 ∈ B and any x ∈ X:
φ(b1b2)(x) = qf (s(b1b2) · x) = qf (s(b1) · s(b2) · x), which, thanks to Proposition 2.1.5, is equal to
qf (s(b1)) · qf (sfs(b1) · qf (s(b2) · x)) = 1 · qf (s(b1) · qf (s(b2) · x)) =
θX
","5.1. The Schreier split extension classifier 51 = qf (s(b1) · qf (s(b2) · x)),
while
φ(b1)(φ(b2)(x)) = φ(b1)(qf (s(b2) · x)) = qf (s(b1) · qf (s(b2) · x)),
and the two expressions are equal.
Also we have to check that φ¯ is a monoid homomorphism, which is equiva-
lent to: qf (a·a′) = qf (a)·qf (sf(a)·qf (a′)), and this is true thanks to Proposition 2.1.5 (e). Moreover, the fact that the square θXφ¯ = φf is a pullback comes from Theorem 2.3.7.
It remains to prove the uniqueness of the pair (φ, φ¯). So, suppose we have another pair (ψ, ψ¯) satisfying the same conditions:
X oo X
qf // A oo s // B kf
ψ¯ ψ ooq ooρX
ιX
// Hol(X)
// End(X)
The commutativity of the square at the level of split epimorphisms implies that ψ¯(a) = (q¯(a),ψf(a)) for some set theoretical map q¯: A → X. The commutativ- ity at the level of the sections implies that q¯(s(b)) = 1. The fact that the square is a pullback of Schreier split epimorphisms is equivalent to the fact that the pair (f,ψ¯) is jointly monomorphic, which, itself, is equivalent to the fact that the pair (f, q¯) is jointly monomorphic. And finally, we have ψ¯k = ιX if and only if we have q¯k = 1X . The fact that ψ¯ is a monoid homomorphism is equivalent to q¯(a·a′) = q¯(a)·ψ(f(a))(a′). Whence q¯(s(b)·a′) = q¯(s(b))·ψ(b)(a′) = ψ(b)(a′) which gives the definition of ψ via q¯.
Let us check now that q¯(a)·sf(a) = a. For that we check it by composition with the jointly monomorphic pair (f,q¯). This is clear for f, and we have q¯(q¯(a) · sf(a)) = q¯(q¯(a)) · q¯(sf(a)) = q¯(q¯(a)) = q¯(a). The uniqueness of the factorization associated with the Schreier split epimorphism in question implies that q¯(a) = qf (a).
The previous proposition allows us to give the following definition, which is inspired by Definition 1.1 in [3]:
Definition 5.1.3. For any monoid X, the monoid End(X) will be called the Schreier split extension classifier of X. For any Schreier split epimorphism (A,B,f,s) with kernel X, the pair (φ,φ¯) will be called its classifying map or its index.
Proposition 3.1.12 has the following immediate corollary:
Theorem 5.1.4. R[θX] is a Schreier equivalence relation if and only if X is a group.
θX
","52 Chapter 5. Schreier and homogeneous representability 5.2 Semidirect products
In analogy with the classical case of groups, given two monoids B and X, we will call action of B on X a morphism φ: B → End(X). If X is a commutative monoid, we will say that φ gives X a B-module structure. Given two B-modules φ: B → End(X) and ψ: B → End(X′), a B-module homomorphism is a monoid homomorphism h: X → X′ such that h(φ(b)(x)) = ψ(b)(h(x)) for all b ∈ B and all x ∈ X.
The following classical construction (see Section V4 in [19], or Section 1.2.2 in [32]) gives then a converse of Theorem 5.1.2:
Definition 5.2.1. Given an action φ: B → End(X) of a monoid B on a monoid X, the semidirect product of B and X w.r.t φ is the monoid X φ B whose underlying set is the cartesian product X × B, and whose operation is defined as follows:
(x1, b1) · (x2, b2) = (x1 · φ(b1)(x2), b1 · b2)
It is easy to see that the operation defined above gives X φ B a monoid
structure, and that the following diagram is a split exact sequence in Mon: X ⟨1,0⟩//X φBoo⟨0,1⟩ //B.
πB
Actually it is a Schreier split sequence; the Schreier retraction q is nothing but
theprojectionπX onX;indeed,foranyx∈X andanyb∈B: (x, b) = (x, 1) · (1, b) = ⟨1, 0⟩πX (x, b) · ⟨0, 1⟩πB (x, b),
and
This fact, together with Theorem 5.1.2 gives the following:
Proposition 5.2.2. Given two monoids B and X, there is a natural bijection between the set of Schreier split epimorphisms with codomain B and kernel X and the set of actions φ: B → End(X) of B on X.
Let us observe that, given an action φ: B → End(X) of B on X, the construction of the semidirect product X φ B can be obtained by building explicitly the pullback of the morphism θX : Hol(X) → End(X) along φ.
5.3 The homogeneous split extension classifier
We are now going to show that the homogeneous split epimorphisms also have their classifier, given by the group Aut(X) of automorphisms of a monoid X.
πX ((x, 1) · (1, b)) = πX (x, b) = x.
","5.3. The homogeneous split extension classifier 53 For that, let us consider the following pullback of split epimorphisms, where i
is the inclusion Aut(X) → End(X):
X // ¯ιX
// Hol(X) oo ρ¯X // Aut(X) θ¯X
//
Since Aut(X) is a group, the split epimorphism is homogeneous.
X
ιX
Hol(X) // End(X) θX
¯i i oo ρX
//
The upper split epimorphism is a Schreier one, because so is the lower one.
Theorem 5.3.1. For any homogeneous split epimorphism with kernel X: Aoo s //B
f
there exists a unique morphism φ: B → Aut(X) such that the following diagram is commutative and its right hand side part is a pullback of split epimorphisms:
X oo X
qf // A oo s // B kf
φ¯ φ // ooρ¯X
¯ιX
Hol(X)
θ¯X
// Aut(X).
The explicit definition of φ is φ(b)(x) = qf (s(b) · x), where qf is the Schreier retraction. The pair (k, s) being jointly strongly epimorphic, the homomorphism φ¯ is completely determined by the commutativity of this diagram and we have: φ¯(a) = (φf (a), qf (a)).
Proof. A homogeneous split epimorphism being a Schreier one, it is enough to prove that a Schreier split epimorphism (A,B,f,s) is a homogeneous one if and only if its classifying homomorphism φ: B → End(X) factors through Aut(X). If φ factors through Aut(X), as a pullback of a homogeneous split epimorphism, (A, B, f, s) is homogeneous. Conversely, suppose that (A, B, f, s) is homogeneous. Let us show that any endomorphism φ(b) is an automorphism. Suppose qf (s(b) · x) = qf (s(b) · x′), then we have
s(b)·x = qf (s(b)·x)·sf(s(b)·x) = qf (s(b)·x)·s(b) = qf (s(b)·x′)·s(b) = s(b)·x′.
Since the split epimorphism is left homogeneous, we have x = x′ and φ(b) is injective. Let z be in X. Since the split epimorphism is left homogeneous, there existst∈X suchthats(b)·t=z·s(b).Then
φ(b)(t) = qf (s(b) · t) = qf (z · s(b)) = z
since z ∈ X = K[f], and consequently φ(b) is surjective.
","54 Chapter 5. Schreier and homogeneous representability Definition 5.3.2. For any monoid X, the monoid Aut(X) will be called the
homogeneous split extension classifier of X.
Using again the construction of the semidirect product, we get a converse
of Theorem 5.3.1, in analogy with Proposition 5.2.2:
Proposition 5.3.3. Given two monoids B and X, there is a natural bijection between the set of homogeneous split epimorphisms with codomain B and kernel X and the set of actions φ: B → Aut(X) of B on X.
5.4 Centralizers of Schreier reflexive relations
d0 //
Let S oo s0 // M be a Schreier reflexive relation on the monoid M. Denote by
d1
k: X S the kernel of d0. Take the index (φ,φ¯) of the induced Schreier split extension (S, M, d0, s0, k) and complete the diagram with the kernel equivalence relations:
R[φ¯] oo OO
dS0 dS1
// φ¯ S
// Hol(X) OO
// OO
R(d0) R(s0) d0 s0 d1 dM0//
θX ρX
// M φ // End(X)
R[φ] oo
Since the right hand side square is a pullback, so are the two left hand side
dM1
ones, and in particular the following one:
R[φ¯] dS1 R(d0 )
R[φ]
dM1
which shows that R[φ¯] actually is the pullback R[φ] ×M S and d1dS0 is the
connector making the reflexive relations R[φ] and S centralize each other.
Proposition 5.4.1. The equivalence relation R[φ] is the largest equivalence re- lation on M centralizing the reflexive relation S; in other words it is the cen- tralizer of the reflexive relation S.
Proof. We just observed that we have [R[φ], S] = 0. Suppose now we have an equivalence relation R on M such that [R,S] = 0. Consider the associated
//S //
d0 M
","5.5. The case of commutative monoids 55 double centralizing relation (lower left hand side part of the diagram below),
see Section 4.4:
XXX
σ0k k ιX
p1 // φ¯
R ×M S oo σ // S // Hol(X)
p0
OO 0 OO (dR0 p0,p)
(p,dS1 p1) dS0 dS1 θX dR1 //
R oo
OO
ρX
dR0
// M φ // End(X)
All the commutative squares of the lower part are pullbacks, again see Section 4.4; accordingly both leftward whole lower rectangles are pullbacks. We are going to show that the two maps φdR0 and φdR1 classify the same split extension (R×M S, R, p0, s0, σ0k) with kernel X. According to the property of the Schreier split extension classifier, they will be equal, and consequently we shall get a factorization, which means R ⊂ R[φ]. So let us denote by σ0 : S R ×M S the above horizontal map. Then necessarily σ0k is a kernel of p0 : R ×M S → R, so that we get p1σ0k = k and (p,dS1 p1)σ0k = k since σ0 is a section of any map of the pair ((p, dS1 p1), p1). So, both φdR0 and φdR1 classify the split extension with
kernel X in question.
5.5 The case of commutative monoids
Proposition 5.5.1. Given a Schreier split epimorphism (A, B, f, s), whose cor- responding split sequence is the following:
K[f]oo q //Aoo s //B, kf
A is a commutative monoid if and only if K[f] and B are commutative and A is isomorphic to the direct product K[f] × B.
Proof. Obviously, if K[f] and B are commutative, then also K[f]×B is. Hence, if A is isomorphic to K[f] × B, it is commutative.
Conversely, if A is commutative, then also K[f] and B are, since they are submonoids of A. It remains to prove that A is isomorphic to K[f] × B. For this, consider the map ψ: A → K[f] × B sending an element a ∈ A to the pair (q(a), f(a)). It is a bijection, whose inverse is the map φ: K[f] × B → A
","56 Chapter 5. Schreier and homogeneous representability sending a pair (x, b) to the element x · s(b). Indeed:
φψ(a) = φ(q(a), f (a)) = q(a) · sf (a) = a ψφ(x, b) = ψ(x · s(b)) = (q(x · s(b)), b) = (x, b),
and
where the last equality comes from Proposition 2.1.4. Finally, φ (and hence ψ)
is a homomorphism; indeed, using the commutativity of A, we have: φ((x1,b1)·(x2,b2))=φ(x1 ·x2,b1 ·b2)=(x1 ·x2)·s(b1 ·b2)=
=x1 ·x2 ·s(b1)·s(b2)=x1 ·s(b1)·x2 ·s(b2)=φ(x1,b1)·φ(x2,b2),
and obviously φ(1, 1) = 1.
Corollary 5.5.2. Given a Schreier split epimorphism (A, B, f, s), if A is a com- mutative monoid, then the Schreier retraction q is a homomorphism of monoids (and not only a set-theoretical map).
Proof. Consider the following commutative diagram:
K[f] oo q // A oo s k OO f
ψφ
ooπK[f] oo⟨0,1⟩
// B // B,
Proposition 5.5.1 implies that, in the category CMon of commutative monoids, the unique Schreier split epimorphisms are the direct product projec- tions (which are also homogeneous). Then we have the following
Proposition 5.5.3. In the category CMon, the trivial monoid 0 acts as a Schreier (and homogeneous) split extension classifier for every commutative monoid X.
Proof. Consider the following diagram, where the upper row is a Schreier split sequence in CMon:
Xoo q //Aoo s //B kf
q
1X
// K[f] × B
Then q = πK[f]ψ is a homomorphism.
K[f]
where φ and ψ are the morphisms defined in the proof of Proposition 5.5.1.
⟨1,0⟩
πB
oo 1X oo
// X // 0;
X
since A is isomorphic to X × B, the square on the right is a pullback.
","Chapter 6
Semirings
In this chapter we will show that many of the results presented above in the category Mon of monoids are still valid in the category SRng of semirings. We recall that a semiring (A,+,·,0) is a commutative monoid (A,+,0) endowed with a binary operation ·: A × A → A which is associative and distributive w.r.t. +. We have then a forgetful functor
U : SRng → CMon
associating with any semiring (A, +, ·, 0) the commutative monoid (A, +, 0).
Since we do not require a unit element for the second operation (that will be called multiplication), the category SRng of semirings is pointed, the zero object being the semiring {0}; the category SRng is also unital (this can be seen in the same way as for the category Mon of monoids). Concerning the commutativity (or cooperation) of two morphisms with the same codomain (as in Definition 1.2.3), we have the following characterization:
Proposition 6.0.4. In SRng, two morphisms f and g, as in Definition 1.2.3, cooperate if and only if
f(x)g(y) = g(y)f(x) = 0 for all x ∈ X, y ∈ Y. (6.0.1) Proof. If Condition 6.0.1 is true, we can define the cooperator as
φ(x, y) = f (x) + g(y).
It is easy to show that φ is a morphism of semirings and φ⟨1, 0⟩ = f, φ⟨0, 1⟩ = g.
Conversely, if a cooperator φ exists, then we have
0 = φ(0, 0) = φ((x, 0)(0, y)) = φ(x, 0)φ(0, y) = f (x)g(y),
57
","58 Chapter 6. Semirings and
0 = φ(0, 0) = φ((0, y)(x, 0)) = φ(0, y)φ(x, 0) = g(y)f (x) foranyx∈X andy∈Y,andhencewegetCondition6.0.1.
Given a semiring M, the identity 1M cooperates with itself if and only if x · y = 0 for all x, y ∈ M .
Definition 6.0.5. We shall call trivial a semiring with a trivial multiplication.
Although not very interesting by themselves, we shall see that the trivial semirings will allow to characterize some specific properties inside SRng.
Since the underlying additive structure of a semiring is a commutative monoid, the notions of right homogeneous and left homogeneous split epimor- phism coincide at this level. This allows us to give the following:
Definition 6.0.6. A split epimorphism (A,B,f,s) of semirings is said to be homogeneous when, for any element b ∈ B, the map μb : K[f] → f−1(b) defined by μb(k) = k + s(b), is bijective.
Definition 6.0.7. A split epimorphism (A,B,f,s) of semirings is said to be a Schreier split epimorphism if, for every a ∈ A, there exists a unique x ∈ K[f] such that a = x + sf(a).
Lemma 6.0.8. A split epimorphism (A, B, f, s) of semirings is a Schreier (resp. homogeneous) one if and only if the split epimorphism (UA,UB,Uf,Us) of monoids is. In other words the functor U : SRng → CMon preserves and re- flects the Schreier split epimorphisms.
Thanks to the lemma above, Propositions 2.1.3 and 2.1.4 have the follow- ing immediate consequences:
Corollary 6.0.9. A split epimorphism (A,B,f,s) of semirings is homogeneous if and only if it is a Schreier split epimorphism.
Corollary 6.0.10. A split epimorphism (A,B,f,s) is a Schreier split epimor- phism if and only if there exists a set-theoretical map
such that
q: A K[f]
q(a) + sf(a) = a q(x + s(b)) = x
for every a∈A, x∈K[f] and b∈B.
","6.1. Properties of Schreier split epimorphisms of semirings 59
Thanks to Corollary 5.5.2, we have that the map q described in the corol- lary above is forced to be a morphism of monoids. However, it does not preserve the multiplication, in general, so it is not a morphism of semirings. The map q has the following properties:
Proposition 6.0.11. Given a Schreier split epimorphism (A,B,f,s) of semir- ings, we have:
(a) qk = 1K[f];
(b) qs = 0;
(c) q(0) = 0;
(d) if b ∈ B and x ∈ K[f], then q(s(b)·x) = s(b)·x and q(x·s(b)) = x·s(b); (e) for every a,a′ ∈A q(a·a′)=q(a)·q(a′)+sf(a)·q(a′)+q(a)·sf(a′).
Proof. The properties from (a) to (c) are immediate consequences of Lemma 6.0.8 and Proposition 2.1.5. Concerning (d), it suffices to observe that s(b) · x and x · s(b) belong to the kernel of f, so the thesis follows from (a). Finally, concerning (e), we have that q(a · a′) is the unique element of K[f] such that
a·a′ =q(a·a′)+sf(a·a′), so it suffices to prove that
q(a) · q(a′) + sf(a) · q(a′) + q(a) · sf(a′) + sf(a · a′) = a · a′. In fact we have:
q(a)·q(a′)+sf(a)·q(a′)+q(a)·sf(a′)+sf(a·a′) =
= q(a) · q(a′) + sf(a) · q(a′) + q(a) · sf(a′) + sf(a) · sf(a′) =
= (q(a) + sf(a)) · (q(a′) + sf(a′)) = a · a′.
6.1 Properties of Schreier split epimorphisms of semir- ings
All the following results, already proved for Schreier split epimorphisms of monoids, are still valid for semirings, thanks to Lemma 6.0.8.
Lemma 6.1.1. A Schreier split epimorphism of semirings is a strongly split epimorphism in the category SRng.
","60 Chapter 6. Semirings Lemma 6.1.2. Given a Schreier split epimorphism (A, B, f, s) of semirings, the
following diagram
ooq oos
K[f] // A // B,
kf
makes s: B → A the kernel of q in the category of pointed sets. Moreover, since q is a morphism of monoids, s is its kernel in the category of commutative monoids, too.
Proposition 6.1.3. Given a direct product diagram in semirings
ooπX ⟨1,0⟩
// X × B oo
πB // ⟨0,1⟩
X
the canonical split epimorphism (X × B, B, πB , ⟨0, 1⟩) is a Schreier split epi-
morphism.
Corollary 6.1.4. The terminal split epimorphism
X oo // 1 is a Schreier split epimorphism.
Corollary 6.1.5. The identity split epimorphism
1X
B,
//
and more generally any isomorphism, is a Schreier split epimorphism.
X,
split epimorphism (A, B, f, s) is a Schreier split epimorphism.
Proposition 6.1.7. Schreier split epimorphisms are stable under products, i.e.
the product of two Schreier split epimorphisms is a Schreier split epimorphism.
Proposition 6.1.8. Schreier split epimorphisms are stable under pullbacks along any morphism.
Theorem 6.1.9. Consider a commutative diagram
Xk′ //Coof′ //D s′
uv
ooq f//
X oo
Proposition 6.1.6. If B is a ring (i.e. (B, +, 0) is an abelian group), then every
X
// A oo
s
B
k
","6.2. The fibration of Schreier points in semirings 61
where the lower row is a Schreier split sequence of semirings, while the top row is any split epimorphism with the same kernel X. The following conditions are equivalent:
(a) the pair (u,f′) is jointly monomorphic;
(b) the commutative square fu = vf′ is a pullback square;
(c) the upper row (with the map q′ = qu) is a Schreier split epimorphism.
Proposition 6.1.10. Consider the following commutative diagram in SRng, where the two rows are Schreier split sequences:
Y
oo q′ l
// C
oo β α
// // D
hgt oo q oo s
if t and h are;
(ii) g is a monomorphism if and only if t and h are.
Proposition 6.1.11. Consider a commutative diagram in SRng of split se- quences of the form
f
// E // // B. kf
X
(i) g is a regular epimorphism (i.e. a surjective homomorphism) if and only
oog oor
// // B // // B,
if and only if the lower split sequence is a Schreier one.
6.2 The fibration of Schreier points in semirings
Let us consider the category Pt(SRng) of split epimorphisms in SRng and its full subcategory SP t(SRng) whose objects are the Schreier split epimorphisms. As for monoids, we have that, according to Proposition 6.1.8, the restriction ¶S of the fibration ¶: Pt(SRng) → SRng to the full subcategory SPt(SRng) is still a fibration, which we shall call the fibration of Schreier points. The following results follow then immediately from the analogous ones in the category of monoids:
X
// A φ
k
ooq oos
// E
where the upper split sequence is Schreier. The morphism φ is an isomorphism
X
p
l
","62 Chapter 6. Semirings Proposition 6.2.1. The change-of-base functors of the fibration ¶S are conser-
vative (i.e. they reflect isomorphisms).
Theorem 6.2.2. Given any semiring B, the fiber PtB(SRng) is SPtB(SRng)- unital.
Proposition 6.2.3. For any semiring B, the fiber SPtB(SRng) is closed under finite limits. Hence, the fiber SPtB(SRng) is a unital category.
As for the case of monoids, according to Section 1.3 we can define the commutativity of two morphisms (m,n) in PtB(SRng) provided that one of the domains of the pair (m, n) is a Schreier split epimorphism.
Proposition 6.2.4. Let us denote by (A, B, f, s) the Schreier split epimorphism which is the domain of m and by (C, B, g, t) the domain of n, as in the diagram below. The cooperator φ: A×B C → X of the pair (m,n) in PtB(SRng), when it exists, is necessarily (being unique) defined by φ(a, c) = mqf (a) + n(c). It is effectively a cooperator if and only if we have mqf (sg(c) · α) = n(c) · m(α) and mqf(α·sg(c))=m(α)·n(c) for α∈K[f] and c∈C.
A ×B C
φ
;;
cc
eA
eC
m // oo n AXC cc OO ;; fg
s
Proof. If the cooperator φ exists, then it is the cooperator of the two points (UA, UB, Uf, Us) and (UC, UB, Ug, Ut) of (additive) monoids. Hence, thanks to Proposition 2.4.3, we already know that the cooperator, if it exists, is given by φ(a, c) = mqf (a) + n(c). Moreover, since all the monoids involved are com- mutative, φ is always a morphism of (additive) monoids. It remains to prove that φ is a morphism of semirings if and only if
mqf (sg(c) · α) = n(c) · m(α) and mqf (α · sg(c)) = m(α) · n(c) (6.2.2) for any α ∈ K[f] and c ∈ C. We have that
φ((a1,c1)·(a2,c2))=φ(a1 ·a2,c1 ·c2)=mqf(a1 ·a2)+n(c1)·n(c2)=
= mqf (a1)·mqf (a2)+mqf (sf(a1)·qf (a2))+mqf (qf (a1)·sf(a2))+n(c1)·n(c2) = = mqf (a1)·mqf (a2)+mqf (sg(c1)·qf (a2))+mqf (qf (a1)·sg(c2))+n(c1)·n(c2), while
## {{ B
t
φ(a1, c1) · φ(a2, c2) = (mqf (a1) + n(c1)) · (mqf (a2) + n(c2)) =
","6.2. The fibration of Schreier points in semirings 63 = mqf (a1) · mqf (a2) + mqf (a1) · n(c2) + n(c1) · mqf (a2) + n(c1) · n(c2),
and it is easy to see that these two expressions are equal if and only if the conditions 6.2.2 are satisfied.
Theorem 6.2.5. The kernel functor reflects commutativity. Given any pair of maps in PtB(SRng) with their two domains in SPtB(SRng):
l
v
functor cooperate in SRng.
Proof. As in the case of monoids (see Theorem 2.4.6), since the kernel functor is left exact, it preserves the cooperating pairs. Conversely, suppose that the following pair cooperates in SRng:
K[w] K(l) // K[f] oo K(n) K[w′]
Let us denote by qw : U → K[w] (resp. qw′ : U′ → K[w′]) the Schreier retraction of the kernel kw (resp. kw′ ). Thanks to Theorem 2.4.6, we already know that the unique possible cooperator φ: U ×B U′ → A is given by φ(u,u′) = lqw(u)+ n(u′), and that it is a morphism of (additive) monoids. We only have to prove that it also preserves multiplication. We have that
φ((u1, u′1) · (u2, u′2)) = φ(u1 · u2, u′1 · u′2) = lqw(u1 · u2) + n(u′1) · n(u′2) =
= lqw(u1)·lqw(u2)+lqw(vw(u1)·qw(u2))+lqw(qw(u1)·vw(u2))+n(u′1)·n(u′2),
while
φ(u1, u′1) · φ(u2, u′2) = (lqw(u1) + n(u′1))(lqw(u2) + n(u′2)) =
= lqw(u1) · lqw(u2) + lqw(u1) · n(u′2) + n(u′1) · lqw(u2) + n(u′1) · n(u′2). So we have to show that
(1) lqw(vw(u1) · qw(u2)) = n(u′1) · lqw(u2), and
(2) lqw(qw(u1) · vw(u2)) = lqw(u1) · n(u′2).
Since (u1, u′1), (u2, u′2) ∈ U ×B U′, we have that w(u1) = w′(u′1) and w(u2) = w′(u′2), hence conditions (1) and (2) become
U
ff
// OOA oo
w v′
n 88 U′ && xx w′
sf
B
l and n cooperate in the fiber PtB(SRng) if and only their images by the kernel
","64 Chapter 6.
(1′) lqw(vw′(u′1) · qw(u2)) = n(u′1) · lqw(u2), and
(2′) lqw(qw(u1) · vw′(u′2)) = lqw(u1) · n(u′2). We have that
n(u′1) · lqw(u2) = (nqw′ (u′1) + nv′w′(u′1)) · lqw(u2) = = nqw′ (u′1) · lqw(u2) + nv′w′(u′1) · lqw(u2) = (∗),
Semirings
but, since qw′ (u′1) ∈ K[w′], qw(u2) ∈ K[w] and K[w′] and K[w] commute, we have that nqw′ (u′1) · lqw(u2) = 0 and hence
(∗) = nv′w′(u′1) · lqw(u2) = lvw′(u′1)) · lqw(u2).
Then Condition (1′) is verified. Indeed, observe that vw′(u′1) · qw(u2) ∈ K[w],
and then
lvw′(u′1) · lqw(u2) = l(vw′(u′1) · qw(u2)) = lqw(vw′(u′1) · qw(u2)).
Condition (2′) can be proved similarly. We noticed that the fiber SPtB(SRng) is unital. Accordingly, there is,
inside this fiber, a natural notion of internal commutative object.
Corollary 6.2.6. A Schreier split epimorphism Aoo s ////B,
f
seen as an object in the fiber PtB(SRng) is commutative if and only if the kernel K[f] of f is a trivial ring. Accordingly any Schreier split epimorphism has at most one (commutative) monoid structure in this fiber.
Accordingly, the previous condition is the characteristic condition which makes the application m : R[f] → A defined by m(a, a′) = q(a) + a′ a semiring homomorphism; it is the binary operation which gives the commutative monoid structure structure inside the fiber PtB(SRng).
6.3 Schreier internal structures in semirings
The following definition is inspired by the analogous one (Definition 3.0.10) in the category of monoids.
","6.3. Schreier internal structures in semirings 65 Definition 6.3.1. An internal reflexive graph in the category of semirings
d0 // X1oos0 //X0
d1
is a Schreier reflexive graph if the split epimorphism (d0 , s0 ) is a Schreier one.
The definitions of Schreier reflexive relation, Schreier internal category and Schreier internal groupoid are analogous.
Thanks to Lemma 6.0.8, the following results are immediate consequences of the analogous ones for monoids.
Example 6.3.2. For every semiring X, the discrete internal equivalence relation: 1X //
Xoo1X //X 1X
is a Schreier internal equivalence relation.
Example 6.3.3. The internal order in SRng given by the usual order between natural numbers:
p0 // ON oo s0 // N,
p1
where
is a Schreier order relation.
ON = {(x, y) ∈ N × N | x ≤ y},
Proposition 6.3.4. Any Schreier reflexive relation is transitive. It is an equiva- lence relation if and only if K[d0] is a ring (i.e. if and only if (K[d0],+,0) is an abelian group).
Proposition 6.3.5. Given a semiring X, the indiscrete equivalence relation ∇X given by:
p1
Corollary 6.3.6. Given any semiring B, the following conditions are equivalent: (a) the semiring B is a ring
(b) any split epimorphism with codomain B is a Schreier split epimorphism.
//
is a Schreier equivalence relation if and only if X is a ring.
p0 X×Xoo s0 //X
","66 Chapter 6. Semirings
Proposition 6.3.7. Given a split epimorphism (X, Y, f, s), its kernel equivalence relation R[f] is a Schreier one if and only if the split epimorphism is itself a Schreier one and the kernel K[f] is a ring.
Proposition 6.3.8. A Schreier reflexive graph in SRng:
d0 // X1oos0 //X0
d1
with q: X1 → K[d0] its associated Schreier retraction, has at most one structure of internal category. When it is the case, the composition of a composable pair (τ,σ) is given by: d1(τ,σ) = q(τ) + σ. This map is a morphism of semirings (and hence it is the composition of an internal category) if and only if, for any α ∈ K[d0] and any σ ∈ X1 the following conditions hold:
(1) q(α·s0d1(σ))=α·σ,
(2) q(s0d1(σ)·α)=σ·α.
In particular, these conditions imply that K[d0] and K[d1] commute in X1.
Proof. The fact that the unique possible composition is given by d1(τ,σ) = q(τ) + σ comes from Proposition 3.2.3, through Lemma 6.0.8. It is easy to check that such composition becomes a homomorphism of semirings if and only if conditions (1) and (2) are satisfied. Moreover, when σ is in K[d1], we get
α · σ = q(α · s0d1(σ)) = q(α · 0) = 0 σ · α = q(s0d1(σ) · α) = q(0 · α) = 0,
and
and hence K[d0] and K[d1] commute in X1.
Proposition 6.3.9. Given a Schreier internal category X1, the following condi- tions are equivalent:
(a) X1 is a Schreier groupoid;
(b) R[d0] is a Schreier equivalence relation;
(c) K[d0] is a ring.
d0 //
Proposition 6.3.10. Let X1 oo s0 // X0 be a Schreier reflexive graph such that
d1
K[d0] is a ring. The following conditions are equivalent:
(a) the kernels of d0 and d1 cooperate in SRng; (b) the reflexive graph is an internal category;
","6.3. Schreier internal structures in semirings 67 (c) the reflexive graph is an internal groupoid.
Proof. We already proved (see Proposition 6.3.8) that, if a Schreier reflex- ive graph is an internal category, then the kernels of the domain and of the codomain cooperate. Hence (b) implies (a).
Conversely, suppose that we have a Schreier reflexive graph
d0 // K[d0] // X1 oo s0 // X0,
ooq k
d1
with the Schreier split epimorphism associated with the codomain d1 given by
oo t oo s0 K[d1] // X1
// X0,
such that k and l cooperate (we recall that (d1,s0) is a Schreier split epimor-
phism thanks to Proposition 3.1.13 and Lemma 6.0.8). This means that x·y=y·x=0 forall x∈K[d0], y∈K[d1].
We already know from Proposition 3.3.5 that the unique possible multiplication is the map defined by
m(a′, a) = q(a′) + a,
and that it is a morphism of monoids (since the additive monoids involved are all commutative). We only have to show that it is a morphism of semirings, i.e. that m(a′1 · a′2, a1, a2) = m(a′1, a1) · m(a′2, a2). We have that:
m(a′1 ·a′2,a1,a2)=q(a′1 ·a′2)+a1 ·a2 =
= q(a′1) · q(a′2) + q(s0d0(a′1) · q(a′2)) + q(q(a′1) · s0d0(a′2)) + a1 · a2,
while
m(a′1, a1) · m(a′2, a2) = (q(a′1) + a1) · (q(a′2) + a2) = =q(a′1)·q(a′2)+q(a′1)·a2 +a1 ·q(a′2)+a1 ·a2,
so it suffices to show that
q(s0d0(a′1) · q(a′2)) = a1 · q(a′2)
and
q(q(a′1) · s0d0(a′2)) = q(a′1) · a2.
a1 · q(a′2) = (t(a1) + s0d1(a1)) · q(a′2) = t(a1) · q(a′2) + s0d1(a1) · q(a′2) =
We have:
l d1
","68 Chapter 6. Semirings = 0 + s0d1(a1) · q(a′2) = q(s0d1(a1) · q(a′2)),
where the last equality holds because s0d1(a1) · q(a′2) ∈ K[d0]. Similarly, we have:
q(a′1) · a2 = q(a′1) · (t(a2) + s0d1(a2)) = q(a′1) · t(a2) + q(a′1) · s0d1(a2) = = 0 + q(a′1) · s0d1(a2) = q(q(a′1) · s0d1(a2)).
The equivalence between (b) and (c) comes from Proposition 6.3.9. Proposition 6.3.11. Consider the following commutative diagram
d′0 //
′′ X1′oos0 X
// 0 d′1
uv d0 //
X, X1 oo s // 0
0
d1
where the two rows are Schreier reflexive graphs, u is a regular epimorphism, and the kernels of d0 and d′0 are rings. If the upper row is an internal groupoid, then so is the lower one.
Proof. Thanks to Proposition 6.3.10, we only have to show that k: X → A and l: Y → A cooperate, where k is the kernel of d0 and l is the kernel of
k′ : K[d′0] → X1′ and l′ : K[d′1] → X1′ cooperate, where k′ is the kernel of d′0 and l′ is the kernel of d′1. This means that
x′·y′ =y′·x′ =0 forall x′ ∈K[d′0], y′ ∈K[d′1].
Consider the following diagrams, where i and j are induced by u via the uni-
versal property of the kernels:
d′0 // ′′
d1. Since X1′ oo s0 // X0 is a Schreier internal groupoid, we already know that d′1
s′ ′ k′ //′oo0
′
s′
′ l′ //′oo0 ′
K[d0] X1
d′0
// X0
K[d1] X1
// X0
d′1 iuvjuv
// oo s0
K[d0] k X1
d0
// X0,
K[d1]
// oo s0
l
X1
// X0.
Since u is a regular epimorphism, i.e. it is surjective, also i is. Indeed, there exists a set-theoretical map r : X1 → X1′ such that ur = 1X1 . Then the map
d1
","6.4. Mal’tsev aspects of SRng related to Schreier split epimorphisms 69 q′rk: K[d0] → K[d′0] (where q′ is the map given by the Schreier condition for
the split epimorphism d′0) is such that
iq′rk = qurk = qk = 1K[d0],
where q is the map given by the Schreier condition for the split epimorphism d0, and this proves that i is surjective. Similarly we prove that j is surjective, too. Now, given x ∈ K[d0] and y ∈ K[d1], there exist x′ ∈ K[d′0] and y′ ∈ K[d′1] such that i(x′) = x and j(y′) = y. Then we have:
x·y=i(x′)·j(y′)=u(x′ ·y′))=u(0)=0 y·x=j(y′)·i(x′)=u(y′ ·x′)=u(0)=0,
and
and this proves that k and l cooperate.
6.4 Mal’tsev aspects of SRng related to Schreier split epimorphisms
As for the category Mon of monoids, we can recover in the category SRng of semirings some partial aspects of Mal’tsev categories related to the Schreier split epimorphisms. The observations about Schreier reflexive graphs, categories and groupoids are the same.
In the same way, since the fiber PtBSRng is SPtBSRng-unital, we can introduce the following:
Definition 6.4.1. Given a reflexive relation R and a Schreier reflexive relation S on the semiring B, we say that R and S centralize each other, when there is a (necessarily unique) semiring homomorphism p: R ×B S → B such that p(xRxSy) = y and p(xRySy) = x. We denote this situation by [R, S] = 0.
Proposition 6.4.2. The reflexive relation R and the Schreier reflexive relation S centralize each other in SRng if and only if, for any 1St ∈ K[d0] and any xRy, we have q(yS(y·t)) = x·t and q(yS(t·y)) = t·x. In this circumstance we have p(xRySz) = x + q(ySz). When [R, S] = 0, we have necessarily xSp(xRySz) and p(xRySz)Rz.
Proof. The definition of p and the last assertion are a consequence of Proposi- tion 4.3.3. The condition on p is a straighforward consequence of Proposition 6.2.4.
We get also:
","70 Chapter 6. Semirings Proposition 6.4.3. The equivalence relation R and the Schreier equivalence rela-
tion S on the semiring M centralize each other in SRng as soon as R∩S = ∆X .
Proposition 6.4.4. Suppose both Rop and S are Schreier reflexive relations on a semiring M. Then the reflexive relations R and S centralize each other if and only if the subobjects dR0 kdR1 : K[dR1 ] X and dS1 kdS0 : K[dS0 ] X cooperate in the category SRng.
Proof. It is a straighforward consequence of Theorem 6.2.5.
Proposition 6.4.5. Let (A,B,f,s) be a Schreier split epimorphism such that R[f] is a Schreier equivalence relation. We have [R[f],R[f]] = 0 if and only if its kernel K[f] is a trivial ring.
Proof. Since R[f] is a Schreier equivalence relation, K[f] is a ring. By the reflection of commutativity, we have [R[f],R[f]] = 0 if and only if the ring K[f] is trivial.
When we have [R,S] = 0, the same diagram in the category SRng as the one described in Section 4.4 produces the associated double centralizing relation.
6.5 Schreier accessibility
We showed that in the category Mon there are classifiers for Schreier split extensions with a given kernel X. Some classification process for Schreier split extensions with a given kernel X in SRng does exist, but it is not so simple. Actually it is analogous to the one given for rings in [15] and related to the accessibility property. Again it will allow us to produce centralizers for Schreier reflexive relations.
Given two Schreier split epimorphisms of semirings (which, in particular, are split extensions, as observed in Lemma 1.2.8) (B,A,p,s) and (D,C,r,t) with the same kernel X, a morphism between them
(g,f) : (B,A,p,s) −→ (D,C,r,t) is a pair (g,f) of morphisms:
0 // X oo q // A oo p // B // 0
s
1X f g
(6.5.3)
(6.5.4)
k
// oo q′ r //
//
0 X //Coo D 0
lt
","6.5. Schreier accessibility 71
such that l = fk, rf = gp and fs = tg.
Schreier split extensions with fixed kernel X form a category, which we
will denote by SSplExtSRng(X), or simply by SSplExt(X).
Remark. Since the pair (k, s) is jointly strongly epimorphic, the morphism
f in (6.5.4) is uniquely determined by g.
Definition 6.5.1. An object in SSplExt(X) is said to be faithful if any object
in SSplExt(X) admits at most one morphism into it.
Definition 6.5.2. An object in SSplExt(X) is said to be accessible if it admits a morphism into a faithful object. We call that unique possible morphism an index of the Schreier split extension in question.
Proposition 6.5.3. Given a Schreier split extension
// oo q′ r // //
0 X //Coo D 0 lt
in SRng, the following conditions are equivalent: (i) the split extension is faithful;
(ii) if d1,d2 ∈ D are such that t(d1) · l(x) = t(d2) · l(x) and l(x) · t(d1) = l(x)·t(d2) for every x∈X, then d1 =d2.
Proof. Let us first prove that (ii) implies (i). Given another Schreier split ex- tension
0 //Xoo q //Aoo p //B //0,
k
s
suppose we have two morphisms (f,g) and (f′,g′) of Schreier split extensions, as in the following commutative diagram
0 // X oo q // A oo p // B // 0
s
k
1X ff′gg′
// oo q′ r //
0 X //Coo D 0.
lt
It suffices to prove that g = g′. For any b ∈ B and any x ∈ X, we have that s(b) · k(x) ∈ k(X), hence there exists y ∈ X such that s(b) · k(x) = k(y). So we have that
tg(b) · l(x) = fs(b) · fk(x) = f(s(b) · k(x)) = fk(y) = l(y) = f′k(y) = f′(s(b) · k(x)) = f′s(b) · f′k(x) = tg′(b) · l(x),
//
","72
and, similarly,
Chapter 6.
Semirings
l(x) · tg(b) = l(x) = tg′(b). Conversely, suppose that the Schreier split extension
// oo q′ r // //
0 X //Coo D 0
lt
Condition (ii) then implies that g(b) = g′(b) for every b ∈ B.
is faithful. Fixed an element d ∈ D, we can construct another Schreier split extension
0 //Xoo q //Aoo p //B //0,
k
s
with a morphism (f,g) into the faithful one in the following way. Let B = F({z}) be the free semiring on a singleton {z} (which is isomorphic to the semiring N of natural numbers, with the usual sum and multiplication). Let g: B → D be the unique morphism such that g(z) = d. Let A be the commu- tative monoid X × B, with the multiplication defined by
(x1, b1) · (x2, b2) = (x1 · x2 + q′(tg(b1) · l(x2)) + q′(l(x1) · tg(b2)), b1 · b2).
It is easy to see (and it follows from the results in [26]) that A is a semiring
and the following is a Schreier split extension:
0 //Xoo q //Aoo p //B //0,
s
where q = πX, p = πB, k = ⟨1,0⟩ and s = ⟨0,1⟩. If we define f: A → C by
putting
we obtain the following commutative diagram:
0 // X oo q // A oo p // B // 0
s
1X f g
k
f (x, b) = l(x) + tg(b),
k
// oo q′ r //
0 X //Coo D 0.
lt
We still have to prove that f is a morphism of semirings: f((x1,b1)+(x2,b2))=f(x1 +x2,b1 +b2)=
//
=l(x1 +x2)+tg(b1 +b2)=l(x1)+l(x2)+tg(b1)+tg(b2)=
","6.5. Schreier accessibility 73 = l(x1) + tg(b1) + l(x2) + tg(b2) = f(x1, b1) + f(x2, b2),
and, moreover
f((x1,b1)·(x2,b2)=f(x1 ·x2 +q′(tg(b1)·l(x2))+q′(l(x1)·tg(b2)),b1 ·b2)=
= l(x1 · x2) + lq′(tg(b1) · l(x2)) + lq′(l(x1) · tg(b2)) + tg(b1 · b2) = = l(x1) · l(x2) + tg(b1) · l(x2) + l(x1) · tg(b2) + tg(b1) · tg(b2) = = (l(x1) + tg(b1)) · (l(x2) + tg(b2)) = f(x1, b1) · f(x2, b2),
where we are using the fact that, by Proposition 6.0.11 (d), lq′(tg(b1)·l(x2)) = tg(b1) · l(x2) and lq′(l(x1) · tg(b2)) = l(x1) · tg(b2), since tg(b1) · l(x2) and l(x1) · tg(b2) belong to l(X). Hence f is a morphism of semirings.
Given another element d′ ∈ D, we can repeat the same construction as above, obtaining a Schreier split extension
0 //Xooq¯//A′oop′ //B′ //0, k′ s′
and a morphism of Schreier split extensions
0 //Xooq¯//A′oop′//B′ //0 k′ s′
1X f′ g′
// oo q′ r //
0 X //Coo D 0,
//
whereB′ =B,k=k′,p=p′,s=s′ andq¯=q,andmoreoverA′ =Aasa
monoid.
Suppose now that d and d′ are such that t(d) · l(x) = t(d′) · l(x) and l(x) · t(d) = l(x) · t(d′) for every x ∈ X. If we prove that, under this hypothesis, the multiplications defined on A and A′ are equal, then, by the faithfulness of the split extension
// oo q′ r // //
0 X //Coo D 0,
lt
we can conclude that g = g′, and hence d = d′. In order to do this, it suffices toprovethat,foreveryx∈X andeveryb∈B
tg(b) · l(x) = tg′(b) · l(x)
lt
","74
and
Chapter 6. Semirings
l(x) · tg(b) = l(x) · tg′(b).
By hypothesis, these conditions are satisfied when b = z (because in this case
g(z) = d and g′(z) = d′). Then, it suffices to prove that the set Z of all ele- ments b ∈ B such that the conditions above are satisfied for any x ∈ X is a subsemiring of B, because in this case it coincides with B, since B is generated by z, and z ∈ Z.
So,letb1,b2 ∈Z.Then
tg(b1 + b2) · l(x) = (tg(b1) + tg(b2)) · l(x) = tg(b1) · l(x) + tg(b2) · l(x) =
= tg′(b1) · l(x) + tg′(b2) · l(x) = (tg′(b1) + tg′(b2)) · l(x) = tg′(b1 + b2) · l(x), and, similarly, l(x) · tg(b1 + b2) = l(x) · tg′(b1 + b2), so b1 + b2 ∈ Z. Obviously
0∈Z,soitremainstoprovethatb1·b2 ∈Z:
tg(b1 · b2) · l(x) = (tg(b1) · tg(b2)) · l(x) = tg(b1) · (tg(b2) · l(x)) =
= tg(b1) · (tg′(b2) · l(x)) = tg′(b1) · (tg′(b2) · l(x)) = tg′(b1 · b2) · l(x), where the first equality in the last row holds because tg′(b2) · l(x) ∈ l(X). This
concludes the proof. Theorem 6.5.4. For any semiring X, every object in SSplExt(X) is accessible. Proof. Given a Schreier split extension
0 //Xoo q //Aoo p //B //0,
we have to build a faithful one
k
s
// oo q′ r //
0 X //Coo D 0
lt
and a morphism of Schreier split extensions into it, the so-called index of the Schreier split extension in question. Let us define the following relation R on B:
bRb′ ⇐⇒ s(b)·k(x) = s(b′)·k(x) and k(x)·s(b) = k(x)·s(b′) for all x ∈ X.
It is immediate to see that R is an equivalence relation. It is a congruence on B, indeed, if b1Rb′1 and b2Rb′2, then (b1 + b2)R(b′1 + b′2) and (b1 · b2)R(b′1 · b′2), because
s(b1+b2)·k(x)=s(b1)·k(x)+s(b2)·k(x)=s(b′1)·k(x)+s(b′2)·k(x)=s(b′1+b′2)·k(x),
//
","6.5. Schreier accessibility 75
s(b1 · b2) · k(x) = s(b1) · (s(b2) · k(x)) = s(b1) · (s(b′2) · k(x)) = = s(b′1) · (s(b′2) · k(x)) = s(b′1 · b′2) · k(x),
and similarly for the other two equalities.
Then we can define D as the quotient B of B w.r.t. the congruence R,
R
and a morphism g: B → D, which is the canonical projection. Moreover, we
can construct a semiring C in the following way: as a monoid, it is the direct
product of X and B , and the multiplication is defined by R
(x1, [b1]) · (x2, [b2]) = (x1 · x2 + q(s(b1) · k(x2)) + q(k(x1) · s(b2)), [b1 · b2]).
It is easy to see that this definition does not depend on the choice of the representatives of the classes [b1] and [b2], and that in this way we obtain a semiring and a Schreier split extension
// oo q′ r // //
0 X //Coo D 0,
lt
where r = πD, q′ = πX, l = ⟨1,0⟩ and t = ⟨0,1⟩. It follows from Proposition 6.5.3 that it is a faithful Schreier split extension. It remains to build a morphism f : A → C making the following diagram commutative:
0 // X oo q // A oo p // B
s
1X f g
// oo q′ r //
0 X //Coo D 0
lt
in order to complete the index of our given Schreier split extension. We define f by putting f(a) = (q(a),[p(a)]). It clearly makes the diagram commutative, so we only need to prove that it is a morphism of semirings:
f(a1 +a2)=(q(a1 +a2),[p(a1 +a2)])=(q(a1)+q(a2),[p(a1)]+[p(a2)])= = (q(a1), [p(a1)]) + (q(a2), [p(a2)]) = f(a1) + f(a2),
indeed, since the upper Schreier split extension is a Schreier split epimorphism in the category of commutative monoids, the map q is a morphism of monoids. Moreover, A is isomorphic, as a monoid, to the direct product X × B (thanks to Proposition 5.5.1). As shown in [26], the multiplication in X × B is defined by
(x1, b1) · (x2, b2) = (x1 · x2 + q(s(b1) · k(x2)) + q(k(x1) · s(b2)), b1 · b2).
This immediately implies that f preserves the multiplication, and hence it is a morphism of semirings. This concludes the proof.
k
// 0 //
","76 Chapter 6. Semirings Inspired by the definition of an action accessible category given in [15],
we can say that the category SRng of semirings is Schreier accessible.
6.6 Centralizers of Schreier reflexive relations
As in the case of the category Mon of monoids, we shall show here that any Schreier reflexive relation in SRng has a centralizer.
d0 //
Let S oo s0 // M be a Schreier reflexive relation on a semiring M. De-
d1
note by k: X S the kernel of d0. We can take the index (φ,φ¯) of the in- duced Schreier split extension (S, M, d0, s0, k) towards a faithful split extension (S(X),M(X),θX,ρX,ιX) and complete the diagram by the kernel equivalence relations:
R [ φ¯ ] o o OO
dS0 dS1
// φ¯ S
/ / S ( X ) OO
// OO
R(d0) R(s0) d0 s0 d1 dM0//
θX ρX //M φ //M(X)
R[φ]oo
Since the right hand side square is a pullback, so are the two left hand side
dM1 ones, and in particular the following one:
R[φ¯] dS1 R(d0 )
R[φ]
dM1
which shows that R[φ¯] is the pullback R[φ] ×M S and d1dS0 is the connector
making the reflexive relations R[φ] and S centralize each other.
Proposition 6.6.1. The equivalence relation R[φ] is the largest equivalence re- lation on M commuting with the reflexive relation S; in other words it is the centralizer of the reflexive relation S in SRng.
Proof. We just observed that we have [R[φ], S] = 0. Suppose now we have an equivalence relation R on M such that [R,S] = 0. Consider the associated double centralizing relation (lower left hand side part of the diagram below),
//S //
d0 M
","6.7. Semidirect products in SRng 77 see Section 4.4:
XXX
σ0k k ιX
p1 // φ¯
R×MSoo σ
OO 0
//S OO
//S(X) OO
p0
dS1 θX
R oo
ρX // M φ // M(X)
(dR0 p0,p) (p,dS1 p1)
dS0
dR1 //
dR0
All the commutative squares of the lower part are pullbacks, again see Section 4.4; accordingly both leftward whole lower rectangles are pullbacks. We are going to show that the two maps φdR0 and φdR1 are an index for the same split extension (R ×M S, R, p0, s0, σ0k) with kernel X. According to the property of the faithful split extensions, they will be equal, and consequently we shall get a factorization, which means R ⊂ R[φ]. So, let us denote by σ0 : S R ×M S the above horizontal map. Then necessarily σ0k is a kernel of p0 : R ×M S → R; so that we get p1σ0k = k and (p,dS1p1)σ0k = k since σ0 is a section of any map of the pair ((p, dS1 p1), p1). So, both φdR0 and φdR1 are an index of the split extension with kernel X in question.
6.7 Semidirect products in SRng
In analogy with Section 5.2, we now describe the explicit form of Schreier split epimorphisms of semirings by means of a semidirect product construction. In the case of semirings it is not possible to define an action of a semiring B on a semiring X simply as a morphism from B to a semiring depending only on X. We consider then the following
Definition 6.7.1. Given two semirings B and X, an action of B on X is a pair of bilinear maps (a left action and a right action)
αl : B × X → X and αr : X × B → X satisfying the following properties:
(i) αl(b,x1 ·x2)=αl(b,x1)·x2; (ii) αr(x1 ·x2,b)=x1 ·αr(x2,b);
(iii) αl(b1 · b2, x) = αl(b1, αl(b2, x)); (iv) αr(x,b1 ·b2)=αr(αr(x,b1),b2);
","78 Chapter 6. Semirings
(v) x1 ·αl(b,x2)=αr(x1,b)·x2;
(vi) αr(αl(b1,x),b2)=αl(b1,αr(x,b2))
for any b,b1,b2 ∈ B and any x,x1,x2 ∈ X.
In the definition above, by bilinear we mean, as usual, that, for any ¯b ∈ B and any x¯ ∈ X, the restrictions of αl to {¯b}×X and B×{x¯} and of αr to {x¯} × B and X × {¯b} are linear. We have the following
Proposition 6.7.2. Given two semirings B and X, there is a natural bijection between the set of actions of B on X and the set of isomorphic classes of Schreier split epimorphisms with codomain B and kernel X.
Proof. We only give a sketch of the proof; we refer to [26] for more details. Given a Schreier split epimorphism
Xoo q //Aoo s //B kf
we define an action of B on X by putting
αl(b, x) = q(s(b) · x), αr(x, b) = q(x · s(b)).
Conversely, given an action α = (αl,αr), we define a semiring X α B, the semidirect product of B and X w.r.t. α, whose underlying set is the cartesian product X × B, the sum is componentwise and the multiplication is given by:
(x1,b1)·(x2,b2)=(x1 ·x2 +αl(b1,x2)+αr(x1,b2),b1 ·b2), and a Schreier split epimorphism
Xoo πX //X αBoo⟨0,1⟩ //B. ⟨1,0⟩ πB
When X is a trivial semiring, an action of B on X will be called a B- bimodule structure. Given two B-bimodules X and X′ (with actions α and α′, respectively), a semiring homomorphism h: X → X′ is a B-bimodule homo- morphism if
hαl (b, x) = αl′ (b, h(x)) and hαr (x, b) = αr′ (h(x), b) forallx∈X andb∈B.
","Chapter 7
Special Schreier and special homogeneous surjections
Coming back to the category Mon of monoids, in this chapter we will study a particular class of extensions with abelian kernel, obtaining results that are similar to some classical ones known for groups. For this purpose, we will make use of the categorical approach described in [14]. In particular, we shall show in this chapter that the above mentioned class of extensions with abelian kernel has a canonical abelian group structure, on the model of what happens in the category Gp of groups (see, for example, [24] for the case of groups). In order to do that, we shall use the general categorical process developed in [9, 11].
7.1 Special Schreier surjections in Mon
Let f : X Y be a surjective homomorphism in M on. It is now rather natural
to introduce the following:
Definition 7.1.1. The surjective homomorphism f is said special Schreier when its kernel equivalence relation R[f] is a Schreier one, and special homogeneous when the equivalence relation R[f] is homogeneous.
In this case the kernel K[f] of f, being isomorphic to the kernel of the projec- tion p0 : R[f] → X, is necessarily a group (because R[f] is a Schreier internal groupoid, see Proposition 3.3.2). This rather awkward terminology is compelled by the following observations:
1) a Schreier split epimorphism is not necessarily a special Schreier surjection; 2) by Proposition 3.1.12, a special Schreier surjection which is split is a Schreier split epimorphism.
79
","80 Chapter 7. Special Schreier and special homogeneous surjections Example 7.1.2. According to Example 3.1.9, the morphism abs: Z∗ → N∗ is a
special Schreier surjection (actually a special Schreier split epimorphism).
Similarly to Schreier split epimorphisms, we get:
Proposition 7.1.3. Given any special Schreier (a fortiori special homogeneous) surjection f : X Y , the following sequence is exact:
K[f]//k //X f////Y, namely f is the cokernel of its kernel k.
Proof. Suppose we have a map h:X → Z such that hk = 0. Let us show that h coequalizes the kernel equivalence relation R[f]. For that, consider the following diagram:
R[f] oo s0 //oo X ;; p0
p1 f
(0,k)
;; K[f] // //X //// Y
kf
The map (0,k) is the kernel of p0. Accordingly, since R[f] is a Schreier equiva- lence relation, the pair (s0, (0, k)) is jointly strongly epimorphic. The equality hp0 = hp1 is checked by composition with this pair (since hk = 0). Since f is the quotient of R[f], we get a unique factorization h¯: Y → Z such that h = h¯f.
Proposition 7.1.4. The special Schreier (resp. special homogeneous) surjections are stable under products and pullbacks.
Proof. The first assertion comes from the stability of the Schreier (resp. homo- geneous) equivalence relations under products. On the other hand, in Mon as in any variety of algebras, the surjective homomorphisms are stable under pull- backs. Then consider the following diagram, where the right hand side square is a pullback:
p0 // f R[f]oo s0 //X ////Y
p1 R(x)
xy p′0//
R[f′]oo s′0 //X′ ////Y′.
p′ 1
f′
Then any of the left hand side ones are pullbacks (see the Appendix). Therefore, when R[f′] is a Schreier (resp. homogeneous) equivalence relation, so is R[f].
","7.2. Special Schreier short five lemma 81 Proposition 7.1.5. Given any pullback in Mon with h a surjective homomor-
phism:
X f ////Y
g X′
h // // Y ′
f′
if f is a special Schreier (resp. special homogeneous) surjection, so is f′.
Proof. First, since the surjections in Mon are stable under pullbacks, we have that g is surjective. Let us then complete the previous diagram with the kernel equivalence relations, which produce the left hand side pullbacks:
// // Y
h
Suppose f is a special Schreier (resp. homogeneous) surjection. According to Corollary 2.3.6, since g is a surjective homomorphism and (p0,s0) is a Schreier (resp. homogeneous) split epimorphism, so is (p′0, s′0) and f′ is a special Schreier (resp. homogeneous) surjection.
7.2 Special Schreier short five lemma
We will call special Schreier extension an exact sequence K[f]//k //Xf////Y
such that f is a special Schreier surjection. The exact sequence will be called special homogeneous when f is special homogeneous.
The name Schreier extension was introduced, with a different meaning, by R´edei in [31] in the context of semigroups, semirings and semimodules over a semiring, and later developed by Patchkoria in [27, 29]. This is another reason why we prefer to use the name special Schreier extension. The study of the relationships between these two different concepts will be material for a future work.
p0 R[f]oo s0 // X
p1 R ( g ) p ′0
// f g
// ′
R[f]s0 //X ′ Y.
′ oo ′
p′ f
1
//// ′
","82 Chapter 7. Special Schreier and special homogeneous surjections Proposition 7.2.1. Consider a commutative diagram of special Schreier (resp.
homogeneous) extensions:
k′
The map u is an isomorphism if and only if K(u) is an isomorphism.
Proof. The fact that K(u) is an isomorphism as soon as so is u holds in any pointed category with pullbacks. For the converse, complete the diagram with the kernel equivalence relations:
K[f] (0,k) //R[f]oo p0 ////X f ////Y p1
K[f]//k //Xf////Y
K(u)
u
K[f′] // //X′ ////Y
f′
u
′ // ′oop′0// ′ ////
K(u) R(u)
K[f] ′ R[f] //X ′ Y (0,k ) p′ f
1
The two horizontal left hand side maps are the kernels of p0 and p′0, respectively. According to Theorem 2.3.7, since K(u) is an isomorphism and the split epi- morphisms (p0, s0) and (p′0, s′0) are Schreier ones, the associated middle square is a pullback. Thanks to the Barr-Kock Theorem (see Proposition 9.2.2 in the Appendix), the right hand side square is a pullback and consequently u is an isomorphism.
More precisely, we have
Proposition 7.2.2. Consider a commutative diagram of special Schreier exten- sions:
K[f]//k //Xf////Y
K(u)
u
K[f′] // //X′ ////Y
f′
k′
The map u is a monomorphism (resp. a surjective homomorphism) if and only
if so is K(u).
Proof. If u is a monomorphism, then k′K(u) = uk also is, and this implies that K(u) is a monomorphism. The fact that K(u) is a surjective homomorphism as soon as so is u holds in any pointed variety of universal algebra. As for the
","7.3. Special Schreier extensions with abelian kernel 83 converse:
1) Suppose that K(u) is a monomorphism and that u(x1) = u(x2). Then f(x1) = f′u(x1) = f′u(x2) = f(x2),
which means that x1R[f]x2. Since R[f] is a Schreier equivalence relation, de- noting by q its associated Schreier retraction, we get q(x1R[f]x2) · x1 = x2 thanks to Lemma 3.1.4. Then
uq(x1R[f]x2) · u(x1) = u(x2) = u(x1),
and the uniqueness condition for the equivalence relation R[f′] (which is also a Schreier one) implies that uq(x1R[f]x2) = 1, and since K(u) (which is nothing but the restriction of u to the kernels) is a monomorphism, we have q(x1R[f]x2) = 1. Since the kernel of q, in the category of pointed sets, is s0 : X → R[f], we have that the pair (x1, x2) belongs to the image of s0, which means that x1 = x2.
2) Suppose that K(u) is a surjective homomorphism. Let x′ be in X′. There exists x ∈ X such that f(x) = f′(x′); so we get f′u(x) = f(x) = f′(x′), which means that u(x)R[f′]x′. Denoting by q′ the Schreier retraction of the Schreier equivalence relation R[f′], we obtain f′(x′) = q′(u(x)R[f′]x′) · u(x), again thanks to Lemma 3.1.4. But q′(u(x)R[f′]x′) belongs to K[f′] and so, since K(u) is surjective, there exists t ∈ K[f] such that q′(u(x)R[f′]x′) = u(t). Then we have f′(x′) = u(t) · u(x) = u(t · x).
7.3 Special Schreier extensions with abelian kernel
Definition 7.3.1. A special Schreier (resp. homogeneous) extension with abelian kernel is an exact sequence where f is a special Schreier (resp. special homoge- neous) surjection and K[f] is an abelian group A:
A=K[f]//k //Xf////Y
Thanks to Proposition 3.1.12, we have that, when a special Schreier surjection is split, it is a Schreier split epimorphism. According to the property of the Schreier split extension classifier, the split special Schreier surjections with kernel the abelian group A are in one-to-one correspondence with the structures of Y -modules: Y → End(A).
The kernel K[f] is abelian if and only if the kernel inclusion kf : K[f] X cooperates with itself, and, according to Proposition 4.3.6, this happens if and only if the Schreier equivalence relation R[f] is such that [R[f],R[f]] = 0. Consider then the associated double centralizing relation and its levelwise
","84 Chapter 7. Special Schreier and special homogeneous surjections horizontal quotients:
R[f]×X R[f]oo p1 ////R[f] f¯ ////Q
OO (d0p0,p) OO (p,d1p1) d0 s0 d1
OO ψ σ
p0
R[f] //XfY
oo d1 // d0
¯¯
(ψ,σ) such that ψf = fd0 = fd1 and σf = fs0. Moreover, since any of the
left hand side commutative squares is a pullback, by the Barr-Kock Theorem (Theorem 9.2.2) the right hand side one is a pullback as well.
Proposition 7.3.2. Let (f,k) be a special Schreier (resp. homogeneous) exten- sion with abelian kernel A. The split epimorphism (ψ,σ) is a Schreier (resp. homogeneous) one and its kernel is isomorphic to the abelian group A. Accord- ingly R[ψ] is a Schreier (resp. homogeneous) equivalence relation.
Proof. The right hand side square being a pullback, the kernel of ψ is iso- morphic to the kernel of d0, which is itself isomorphic to the kernel A of f. Moreover, since (d0,s0) is a Schreier (resp. homogeneous) split epimorphism, so is the split epimorphism (ψ, σ), by Corollary 2.3.6.
Definition 7.3.3. The split epimorphism (ψ, σ) in the proposition above is called the direction of the special Schreier (resp. homogeneous) extension (f,k).
The name direction was first used, in a more general context, in [9], being in- spired by the fact that the direction of an affine space is the associated vector space (both affine spaces and extensions are particular cases of the general con- text studied in [9]).
We have the immediate corollary
Corollary 7.3.4. Given any special Schreier extension with abelian kernel A, it is homogeneous if and only if so is its direction.
The direction of a Schreier extension with abelian kernel has an alternative construction, giving more directly the structure of Y -module on A. It is based on the following observation:
Proposition 7.3.5. Given a special Schreier extension with abelian kernel A, the classifying map φ: X → End(A) of the Schreier split epimorphism (d0,s0) coequalizes the equivalence relation R[f].
////
The universal property of the quotients produces a unique split epimorphism
","7.3. Special Schreier extensions with abelian kernel 85
Proof. Let q be the Schreier retraction of the Schreier split epimorphism (d0,s0). Since [R[f], R[f]] = 0, for any xR[f]x′ and any t ∈ A we have q(x′R[f](x′ · t)) · x = x · t thanks to Proposition 4.3.3. Hence
φ(x′)(t) = q(x′R[f](x′ · t)) and φ(x)(t) = q(xR[f](x · t))
(see the construction of φ given in Theorem 5.1.2). If we have (x,x′) ∈ R[f], then we get q(x′R[f](x′ · t)) · x = x · t. On the other hand, by the uniqueness in the Schreier condition, q(xR[f](x · t)) is the only element in A such that q(xR[f](x · t)) · x = x · t. Accordingly φ(x′)(t) = φ(x)(t), and consequently φ(x) = φ(x′).
Since f is the quotient of R[f], according to the previous proposition there is a factorization φ˜: Y → End(A) such that φ = φ˜f. We have φ˜(y)(a) = q(xR[f](x · a)) for any x such that f(x) = y. Consider now the following diagram:
R[f] ×X R[f] // R[f]
Q
Hol(A) OO
φ¯ oop1//f¯//// //
OO (d0 p0 ,p)
d0
OO //
// X
d1 (1) ψ ////
OO Y
φˇ
(2) θA
p0 R[f]
σ
ρA End(A)
(p,d1p1)
oo d1
// f φ˜ GG
d0
Let us check now that φ¯ coequalizes the upper horizontal relation. Given any triple xR[f]x′R[f]x′′ belonging to R[f]×XR[f], we have (recalling the definition of φ¯ given in Theorem 5.1.2) that
φ¯p1(xR[f]x′R[f]x′′) = φ¯(x′, x′′) = (q(x′R[f]x′′), φ(x′)). On the other hand, we have that:
φ¯(d0p0,p)(xR[f]x′R[f]x′′) = φ¯(x,p(xR[f]x′R[f]x′′)) = = (q(xR[f]p(xR[f]x′R[f]x′′)), φ(x)).
According to the explicit expression of p given in Proposition 4.3.3, this is equal to
(q(xR[f](q(x′R[f]x′′) · x), φ(x))) = (q(x′R[f]x′′), φ(x)),
where the last equality comes from Lemma 3.1.4. Observing that φ(x) = φ(x′), because (x,x′) is in R[f], we conclude the proof that φ¯ coequalizes the upper
φ
","86 Chapter 7. Special Schreier and special homogeneous surjections
horizontal relation. Since f¯ is the quotient of this relation, we get a unique factorization φˇ: Q → Hol(A) such that φˇf¯= φ¯.
Since the square (1) and the rectangle (1) + (2) are pullbacks, the square
(2) is also a pullback. According to the results of Section 5.2, we have that
the monoid Q is isomorphic to the semidirect product monoid A φ˜ Y , with
the operation given by (a,y) · (a′,y′) = (a · q(xR[f](x · a′)),y · y′), where x is
any element of X such that f(x) = y. Then the morphisms φˇ and f¯ can be
reinterpreted in the following way: φˇ: A φ˜ Y → Hol(A) is given by φˇ(a, y) = ¯¯′′
(a,φ˜(y))andf:R[f] A φ˜Y isgivenbyf(xR[f]x)=(q(xR[f]x),f(x)). 7.4 The direction functor in Mon
The previous construction of the direction actually gives rise to a functorial construction:
D: AbSExtY (Mon) → AbSPtY (Mon)
from the category of special Schreier surjections with codomain Y and abelian kernel to the abelian category AbSPtY (Mon) of internal abelian groups in the fiber SPtY (Mon) (which are nothing but Schreier split epimorphisms whose kernel is an abelian group). Any morphism u of special Schreier extensions:
A=K[f]//k //Xf////Y
K(u)
A′ =K[f′] //
k′
u
//X′ ////Y
f′
makes K(u) a Y -module homomorphism, and with it we associate the monoid homomorphism K(u) Y : A φ˜ Y → A′ ψ˜ Y with the respective appropriate classifying maps φ and ψ.
Proposition 7.4.1. The direction functor D preserves and reflects monomor- phisms, surjective homomorphisms and consequently isomorphisms.
Proof. It is a straightforward consequence of Proposition 7.2.2.
Before making explicit another extremely important property of the func- tor D, namely that it is cofibrant above the surjective homomorphisms, let us begin with the following immediate
Lemma 7.4.2. Let two Y -module structures φ: Y → End(A) and ψ: Y → End(A′) on the abelian groups A and A′ be given. A group homomorphism
","7.4. The direction functor in Mon 87 h: A → A′ determines a morphism in AbSPtY (Mon), i.e. a monoid homo-
morphism h 1 making the following diagram commute A φ Y h 1 // A′ ψ Y
ccpφ ιψ;; YY
ιφY
if and only if h is a Y -module homomorphism, i.e. if and only if we have:
h(φ(y)(a)) = ψ(y)(h(a)) for all (a, y) ∈ A φ Y .
Proof. Straightforward calculation.
Theorem 7.4.3. Given any special Schreier extension with abelian kernel A and direction (A φ Y, Y, pφY , ιφY ):
A=K[f]//k //Xf////Y
and any surjective homomorphism h : A A′ of Y -modules, where the struc- ture of Y -module of A′ is given by the morphism ψ : Y → End(A′), there is a special Schreier extension with abelian kernel A′ and direction given by (A′ ψ Y , Y, pψY , ιψY ), called direct image of the special Schreier extension along h, together with a monoid homomorphism h¯ which makes the following diagram commutative
A//kf //Xf////Y
##Y{{ pψY
h A′ //
h¯
//X′ ////Y
f′
Given any morphism l of special Schreier extensions:
A//kf //Xf////Y
kf′
K(l)
phism, there is a unique morphism μ of special Schreier extensions A′//kf′ //X′ f′ ////Y
l
//X′′ ////Y
f′′
with a factorization mh = K(l), where m: A′ → A′′ is a Y -module homomor-
A′′ //
kf′′
mμ
A′′ //
kf′′
//X′′ ////Y f′′
","88 Chapter 7. Special Schreier and special homogeneous surjections such that μh¯ = l. Furthermore, the surjection f′ is special homogeneous as soon
as f is.
Proof. The fact that h is a morphism of Y -modules means that ψ(y)(h(a)) = hq(xR[f](x · a)) for any x such that f(x) = y. Consider now the following pullback in Mon:
f∗ ////K[h] φY kh 1Y
R[f] f¯ A φY
It produces a relation S on X defined by xSz if and only if xR[f]z and q(xR[f]z) ∈ K[h]. This relation S is clearly reflexive. The kernel of d0 : S → X is the abelian group K[h] and the reflexive relation S is a Schreier one with the Schreier retraction qS : S → K[h] defined by qS(xSz) = q(xR[f]z). Accord- ingly S is a Schreier equivalence relation on X in Mon. Let us define X′ as the quotient monoid of this equivalence relation S:
d0// h¯
Soo s0 //X ////X′.
d1
Since the kernel of d0 : S → X is the abelian group K[h], K[h] is also the kernel of h¯. On the other hand, since S is included in R[f], h¯ coequalizes p0 and p1, hence there exists a unique surjective homomorphism f′ : X′ Y making the following lower right hand side square commutative:
K(h¯)
(0,kf ) R(h¯)
////Y
S
j
// //
K[k(h¯)] //
//K[h] //1
//R[f]oo s0 //X p1
k¯ A//
kh¯ p0// f
K[f′] // (0,kf′)
p′0// h¯ //R[f′]oo s′0 //X′
////Y.
p′ f′ 1
Completing the diagram above with the kernels of the vertical maps produces the upper horizontal kernel diagram, which shows that the kernel of K(h¯) is, up to isomorphisms, the abelian group K[h]; consequently the abelian group K[f′] is, up to isomorphisms, the abelian group A′, and makes the map k′ : A′ X′, defined by k′(a′) = h¯(a) for any a such that a′ = h(a), a kernel for f′. We have to show now that f′ is a special Schreier surjection. If it is the case, the associated Schreier retraction q′ is necessarily defined by q′(x¯R[f′]z¯) =
","7.5. Baer sums in Mon 89
hq(xR[f]z) for any x,z ∈ X such that h¯(x) = x¯ and h¯(z) = z¯. Let us check that this definition does not depend on the choice of x and z. Suppose that h¯(x) = h¯(x′) and h¯(z) = h¯(z′). Then xSx′ and zSz′, and this implies that xR[f]x′ and zR[f]z′. Then, by Proposition 3.1.5, we get
q(x′R[f]z′) · q(xR[f]x′) = q(xR[f]z′) = q(zR[f]z′) · q(xR[f]z). Hence
hq(x′R[f]z′) · hq(xR[f]x′) = hq(zR[f]z′) · hq(xR[f]z).
But xSx′ and zSz′ imply also that q(xR[f]x′) ∈ K[h] and q(zR[f]z′) ∈ K[h], and so hq(xR[f]z) = hq(x′R[f]z′).
Let us now determine the direction of the special Schreier extension
A′ // k′ //X′ f′ //// Y.
It is determined by the map φ′ : Y → End(A′) defined by φ′(y)(a′) = q′(h¯(x)R[f′](h¯(x) · h¯(a))),
where f(x) = y and h(a) = a′. Hence
φ′(y)(a′) = hq(xR[f](x · a)) = ψ(y)(h(a)) = ψ(y)(a′),
and consequently φ′ = ψ. Accordingly, the direction of this special Schreier extension is (Y ψ A′, Y, pψY , ιψY ), as desired.
Suppose now that we have the morphisms l of special Schreier extensions and m of left Y -modules such that mh = K(l). Showing that l factors through h¯ is equivalent to showing that l coequalizes the equivalence relation S. So, suppose that we have xSz. Then xR[f]z and q(xR[f]z) ∈ K(h). Since f′′l = f, we get l(x)R[f′′]l(z), hence q′′(l(x)R[f′′]l(z)) · l(x) = l(z) by Lemma 3.1.4. Moreover
q′′(l(x)R[f′′]l(z)) = lq(xR[f]z) = mhq(xR[f]z) = m(1) = 1.
So we have l(x) = l(z), and hence a factorization μ: X′ → X′′ such that μh¯ = l, so that K(μ)h = K(l) = mh. Since h is surjective, we get K(μ) = m. The last point of the proposition is just Corollary 7.3.4.
7.5 Baer sums in Mon
We shall denote by SExtφ(Y,A) (resp. HExtφ(Y,A)) the set of isomorphic classes of special Schreier (resp. homogeneous) extensions with codomain Y , abelian kernel A and direction the Schreier split epimorphism (A φY, Y, pφY , ιφY )
","90 Chapter 7. Special Schreier and special homogeneous surjections
associated with the Y -module structure φ: Y → End(A). Similarly to what happens in the category Gp of groups we are going to show that SExtφ(Y,A) (resp. HExtφ(Y,A)) is endowed with a natural structure of abelian group.
For that, given a pair of classes of special Schreier extensions with abelian kernel A and same direction (A φ Y, Y, pφY , ιφY )
A // kf //X f //// Y A // kf′ //X′ f′ //// Y
we can define the Baer sum of the two previous classes as the class of the direct image of their product in AbSExtY (Mon) (given by their pullback above Y ) along the homomorphism + : A × A → A of Y -modules given by the abelian group operation in A:
kf×kf′ f×Y f′ A×A // //X×Y X′ //// Y
+ +˜
// // ′ ////
A X⊕YX ′Y kf′ f⊕Y f
This operation gives SExtφ(Y, A) (resp. HExtφ(Y, A)) an abelian group struc- ture. The zero of this group is just the isomorphic class of the split extension corresponding to the direction (A φ Y, Y, pφY , ιφY ). The inverse of a special
Schreier (resp. homogeneous) extension in M on: A // k // X f // // Y is given
by: A // −k //X f //// Y .TheproofofthecommutativityoftheBaersumis straightforward (since the pullback is commutative up to isomorphisms), while the associativity requires long and heavy calculations, which are similar to the ones classically done for groups (see, for example, [24] for more details). However, the result follows by general categorical arguments thanks to the properties of the direction functor described in the previous section. We will not describe here the details of these arguments, that can be found in Section VI in [9] and in the paper [11].
7.6 Special Schreier surjections in SRng
In this section, and in the following ones, we shall mimic in the category SRng
what we did in the category Mon.
Definition 7.6.1. A surjective homomorphism f : X Y in SRng will be said a special Schreier one when its kernel equivalence relation R[f] is a Schreier one.
","7.7. Special Schreier extensions with trivial kernel 91 In this case the kernel K[f] of f, being isomorphic to the kernel of the projection
p0 : R[f] → X, is necessarily a ring.
Lemma 7.6.2. Given a special Schreier surjection f, the Schreier retraction qf associated with the Schreier split epimorphism d0 satisfies the following identity: given xR[f]x′ and zR[f]z′, we get
qf((x·zR[f]x′ ·z′) = qf(xR[f]x′)·qf(zR[f]z′)+qf(xR[f]x′)·z+x·qf(zR[f]z′). On the other hand, for any k ∈ K[f] and any pair xR[f]x′, we have
qf ((k · x)R[f](k · x′)) = k · qf (xR[f]x′) qf ((x · k)R[f](x′ · k)) = qf (xR[f]x′) · k.
and similarly
Proof. The first point is just a translation of the point (e) in Proposition 6.0.11.
The second one is obtained from:
k·qf (xR[f]x′)+k·x = k·(qf (xR[f]x′)+x) = k·x′ = qf ((k·x)R[f](k·x′))+k·x,
where the last two equalities come from Lemma 3.1.4. The proof of the second formula is completely analogous.
For the same reasons as in Mon, the following sequence K[f] // k //X f //// Y
is exact in SRng, namely f is the cokernel of its kernel k. In SRng, the special Schreier surjections are stable under products and pullbacks, the pullbacks along surjective homomorphisms reflect the special Schreier surjections, and we have a special Schreier short five lemma.
7.7 Special Schreier extensions with trivial kernel Definition 7.7.1. A special Schreier extension in SRng with trivial kernel is an
exact sequence
A=K[f]// k //X f ////Y
where f is a special Schreier surjection and K[f] is a trivial ring A.
Lemma 7.7.2. Let f : X Y be a special Schreier surjection with trivial kernel A. Given any pair xR[f]x′, we have a·x = a·x′ and x·a = x′ ·a for any a ∈ A. Accordingly the abelian group A is endowed with a canonical structure of Y -bimodule (see Section 6.7) by: a · y = a · x and y · a = x · a, for any x such that f(x) = y.
","92 Chapter 7. Special Schreier and special homogeneous surjections
Proof. According to Lemma 7.6.2, given any pair xR[f]x′ and a ∈ A, we have qf (a · xR[f]a · x′) = a · qf (xR[f]x′) = 0 since the ring A is trivial. Hence a·x=a·x′,sincethekernelofqf (inMon)isthemorphisms0.
Given a special Schreier surjection f, the kernel K[f] is trivial if and only if the kernel inclusion kf : K[f] X cooperates with itself, and, according to Proposition 6.4.5, this happens if and only if the Schreier equivalence relation R[f] is such that [R[f],R[f]] = 0. When it is the case, consider the associated double centralizing relation and its levelwise horizontal quotients:
R[f]×X R[f]oo p1 ////R[f] f¯ ////Q
OO (d0p0,p) OO (p,d1p1) d0 s0 d1
OO ψ σ
p0
R[f] //XfY
oo d1 // d0
¯¯
(ψ,σ) such that ψf = fd0 = fd1 and σf = fs0. Moreover, since any of the
left hand side square is a pullback, by the Barr-Kock Theorem (Theorem 9.2.2 in the Appendix) the right hand side one is a pullback as well, and the split epimorphism is a special Schreier one since so is (d0,s0).
Actually the ring Q can be described in the following way: we noticed that A is canonically endowed with a Y -bimodule structure; the split epimor- phism (ψ, σ) is nothing but the canonical split epimorphism associated with this structure. Namely, the semiring Q is the semidirect product (A Y ), where the multiplication is defined by (a,y) · (a′,y′) = (x · a′ + a · x′,y · y′) for any xandx′ suchthatf(x)=yandf(x′)=y′ (a·a′ =0,sinceAistrivial).We call this split special Schreier epimorphism the direction of the special Schreier epimorphism f.
As in Mon, the previous construction of the direction gives rise to a functorial construction:
D : T rSExtY (SRng) → AbSP tY (SRng)
from the category of special Schreier extensions with codomain Y and triv- ial kernel to the abelian category AbSPtY (SRng) of internal abelian groups in the fiber SPtY (SRng) (which are nothing but Schreier split epimorphisms whose kernel is a trivial ring). This direction functor D preserves and reflects monomorphisms, surjective homomorphims and consequently isomorphisms. Again as in Mon, the direction functor is cofibrant above the surjective homo- morphims:
////
The universal property of the quotients produces a unique split epimorphism
","7.7. Special Schreier extensions with trivial kernel 93
Theorem 7.7.3. Given any special Schreier extension with trivial kernel A en- dowed with its canonical structure of Y -bimodule, and any surjective homo- morphism h : A A′ of Y -bimodules, there is a special Schreier extension with trivial kernel A′ and direction given by the Y -bimodule structure on A′, called direct image of the special Schreier extension along h, together with a morphism h¯ which makes the following diagram commutative:
A//kf //Xf////Y
h A′ //
kf′
h¯
//X′ ////Y
f′
Given any morphism of special Schreier extensions:
A//kf //Xf////Y
K(l)
morphism, there is a unique morphism of special Schreier extensions:
l
//X′′ ////Y
f′′
with a factorization mh = K(l), where m: A′ → A′′ is a Y -bimodule homo-
A′′ //
kf′′
A′//kf′ //X′ f′ ////Y
mμ
A′′ //
kf′′
//X′′
f′′
////Y
such that μh¯ = l.
Proof. The Y -bimodule structure on A is given by a · y = qf (xR[f ](a · x)) and y·a=qf(xR[f](x·a))foranyx∈X suchthatf(x)=y.ThefactthathisaY- bimodules homomorphism means that we have hqf (xR[f ](a · x)) = h(a) · f (x) and hqf (xR[f ](x · a)) = f (x) · h(a). Consider now the following pullback in SRng:
j
S f∗//// K[h] Y kh 1Y
R[f] f¯ A Y
// //
","94 Chapter 7. Special Schreier and special homogeneous surjections
It produces a relation S on X defined by xSz if and only if xR[f]z and q(xR[f]z) ∈ K[h]. This relation S is clearly reflexive. The domain of the kernel of d0 : S → X is the ring K[h] and the reflexive relation S is a Schreier one with the Schreier retraction qS : S → K[h] defined by qS(xSz) = q(xR[f]z). Accordingly S is a Schreier equivalence relation on X in SRng. Let us define X′ as the quotient semiring of this equivalence relation S:
d0 // h¯ Soos0//X ////X′
d1
Since the kernel of d0 : S → X is the ring K[h], K[h] is also the kernel of h¯. On the other hand, since S is included in R[f], h¯ coequalizes p0 and p1, hence there exists a unique surjective homomorphism f′ : X′ Y making the following lower right hand side square commutative:
K[k(h¯)] //
//K[h] //1
// R[f] oo s0 // X p1
K(h¯)
(0,kf ) R(h¯)
// // Y
k¯
A //
kh¯
p0 // f
K[f′] // (0,kf′)
p′0 / / h¯ // R[f′] oo s′0 // X′
// // Y.
p′ f′ 1
Completing in SRng the diagram above with the kernels of the vertical maps produces the upper horizontal kernel diagram, which shows that the kernel of K(h¯) is, up to isomorphisms, the ring K[h]; consequently the ring K[f′] is, up to isomorphisms, the given ring A′, and makes the map k′ : A′ X′, defined by k′(a′) = h¯(a) for any a such that a′ = h(a), a kernel for f′. We have to show now that f′ is a special Schreier surjection. If it is the case, the associated Schreier retraction q′ is necessarily defined by q′(x¯R[f′]z¯) = hq(xR[f]z) for any x, z ∈ X such that h¯(x) = x¯ and h¯(z) = z¯. Let us check that this definition does not depend on the choice of x and z. Suppose that h¯(x) = h¯(x′) and h¯(z) = h¯(z′). Then xSx′ and zSz′, and this implies that xR[f]x′ and zR[f]z′. Then, by Proposition 3.1.5, we get
q(x′R[f]z′) + q(xR[f]x′) = q(xR[f]z′) = q(zR[f]z′) + q(xR[f]z) Hence
hq(x′R[f]z′) + hq(xR[f]x′) = hq(zR[f]z′) + hq(xR[f]z).
But xSx′ and zSz′ imply also q(xR[f]x′) ∈ K[h] and q(zR[f]z′) ∈ K[h], and so hq(xR[f]z) = hq(x′R[f]z′).
","7.8. Baer sums in SRng 95
Let us now determine the direction of the special Schreier extension
A′ // k′ //X′ f′ //// Y.
The left action of y = f(x) on a′ = h(a) is, by definition, given by the term q(h¯(x)R[f′](h¯(x) · h¯(a))) = hqf (xR[f](x · a)), which is equal to f(x) · h(a), as we noticed above; but f(x) · h(a) = y · a′. The same holds for the right action. Accordingly the Y -bimodule structure on A′ is the desired one.
Suppose now that we have the morphism l of special Schreier extensions and the homomorphism m of Y -bimodules such that mh = K(l). Showing that l factors through h¯ is equivalent to showing that l coequalizes the equivalence relation S. So, suppose that we have xSz. Then xR[f]z and q(xR[f]z) ∈ K(h). Since f′′l = f, we get l(x)R[f′′]l(z), hence q′′(l(x)R[f′′]l(z)) + l(x) = l(z) by Lemma 3.1.4. Moreover
q′′(l(x)R[f′′]l(z)) = lq(xR[f]z) = mhq(xR[f]z) = m(1) = 1.
So we have l(x) = l(z), and hence a factorization μ: X′ → X′′ such that μh¯ = l,
so that K(μ)h = K(l) = mh. Since h is surjective, we get K(μ) = m. 7.8 Baer sums in SRng
Let Y be a semiring and A be a fixed Y -bimodule. We shall denote by SExt(Y,A) the set of isomorphic classes of special Schreier extensions with trivial kernel the underlying ring A and direction the Schreier split epimorphism (A Y, Y, pY , ιY ) associated with the given Y -bimodule structure on A. Similarly to what hap- pens in the category Mon of monoids we are going to show that SExt(Y,A) is endowed with a natural structure of abelian group.
For that, given a pair of classes of special Schreier extensions with trivial kernel A and same direction (A Y, Y, pY , ιY ):
A // kf //X f //// Y A // kf′ //X′ f′ //// Y
we can define the sum of the two previous classes as the class of the direct image of their product in T rSExtY (SRng) (given by their pullback above Y ) along the Y -bimodule homomorphism + : A × A → A given by the sum in the ring A:
kf×kf′ f×Y f′ A×A // //X×Y X′ //// Y
+ +˜
// // ′ ////
A X⊕YX ′Y kf′ f⊕Y f
","96 Chapter 7. Special Schreier and special homogeneous surjections
For the same reasons as in Mon, this operation gives SExt(Y,A) an abelian group structure. The zero of this group is just the isomorphic class of the split extension corresponding to the direction (A Y, Y, pY , ιY ). The inverse of a
special Schreier (resp. homogeneous) extension in SRng: A // k // X f // // Y isgivenby: A // −k //X f //// Y .
","Chapter 8
Conclusion
We shall sum up here the rich structural equipment we made explicit on the categories Mon of monoids and SRng of semirings. We recalled that a category C is a Mal’tsev category when any fiber PtXC of the fibration of points ¶C is unital; the category C is said to be protomodular (see [4] for more details) when any change-of-base functor of this same fibration is conservative; this im- plies that the category C is a Mal’tsev one. The category Gp of groups is the paradigmatic example of a pointed protomodular category, but the category Rng of rings without unit is pointed protomodular as well.
We showed that the categories Mon of monoids and SRng of semirings are unital categories C equipped with a class S of split epimorphisms determining a subfibration ¶SC of the fibration of points ¶C, where SPtC is the full subcategory of PtC whose objects are those which are in S:
SPtC // j
//PtC ¶C
satisfying the following properties:
1) any object in SPtC is a strongly split epimorphism;
2) the fibers coincide on the terminal object: SPt1C = C = Pt1C;
3) any fiber SPtXC is stable under finite limits in the fiber PtXC;
4) the change-of-base functors of the fibration ¶SC = ¶C ◦ j are conservative; 5) any fiber P tX C is SP tX C-unital, which implies by 2) that any fiber SP tX C is unital;
6) the change of base functors αX∗ : PtXC → C with respect to ¶C along the initial maps αX : 1 → X reflect the commutativity of maps having their do-
¶S C
## }} C
97
","98 Chapter 8. Conclusion mains in SPtXC.
8.1 Partial protomodularity
It is worth introducing the following definition:
Definition 8.1.1. A finitely complete pointed category C will be said to be S- protomodular when there is a class of split epimorphisms S which determines a subfibration ¶SC of the fibration of points ¶C, where SPtC is the full subcategory of PtC whose objects are those which are in S:
oo eC
OO πC OO
oo eC
OO πC OO
SPtC // j
//PtC ¶C
satisfying the following properties:
(1) any object in SPtC is a strongly split epimorphism;
(2) SPtC is stable under finite limits in PtC (in particular, it contains the terminal object 1 1 in P tC).
So, S is a class of strongly split epimorphisms. The fact that ¶SC is a subfibration of ¶C means that this class S is stable under pullbacks. Condition (2) implies that any fiber SPtXC is stable under finite limits in the fiber PtXC and that any change-of-base functor with respect to ¶SC is left exact. The fact that SPtC contains the terminal object is equivalent to the fact that the class S contains the isomorphisms and that any fiber SPtXC is pointed. From this definition, we get immediately:
Proposition 8.1.2. Let C be an S-protomodular category. Then:
1) any fiber PtXC is SPtXC-unital, which implies that any fiber SPtXC is unital by Condition (2) above;
2) any change-of-base functor with respect to the fibration ¶SC is conservative.
Proof. 1) Consider the following left hand side downward pullback of split epimorphisms, where the lower one is in the fiber SPtBC:
¶S C
## }} C
A×B C oos // //oos
A×B C
//C (8.1.1) πAeAgt eAt
//C
A//B. K[f]kA//B. ff
","8.2. Back to C′-unital categories 99
then also the upper one is in SPtBC, since ¶SC is a subfibration of ¶C. So the split epimorphism (πC , eC ) is a strongly split epimorphism. On the other hand, the right hand side square is still a pullback, so the map eAk is the kernel of πC. Accordingly, the pair (eAk,eC) is jointly strongly epimorphic. So this is equally the case for the pair (eA,eC).
2) Since any change-of-base functor with respect to ¶SC is left exact, it is enough to prove that it is conservative on monomorphisms (see the Appendix). Let us then consider the following diagram, where all the quadrangles are pull- backs and all the split epimorphisms are in SPtC:
′ kf′ //′ x // K[f] X X
%% K(m′)
%% ¯′ kf¯′
OO OO ′
mm //¯′ x¯ //¯
K[f] X X
DD
f′ s′ f s
EE
// ′ s¯ // 1YyY
f¯′ f¯
′ s¯
Suppose moreover that m′, and consequently K(m′), are isomorphisms. Cer-
tainly x¯k ¯′ ¯ ¯
f is the kernel of f , and, since the split epimorphim (f , s¯) is a strongly split epimorphism, the pair (x¯kf¯′ , s¯) is jointly strongly epimorphic. Accordingly, since K(m′) = K(m) is an isomorphism, so is m.
8.2 Back to C′-unital categories
The C′-unital categories can appear as a particular case of the previous situa- tion. Let C be a finitely complete pointed category. Let C′ be a full subcategory of C stable under finite limits and containing the terminal object. Let us de- note by ΠC′ the class of split epimorphims which are, up to isomorphisms, the canonically split direct product projections:
πX // ⟨1X ,0⟩
where the factor Y is in C′. It is clear that this class is stable under pullbacks.
The full subcategory j : ΠC′P tC P tC whose objects are those which are in
ΠC′ is such that the functor ¶ΠC′ = ¶C ◦ j: ΠC′PtC → C is a subfibration of C
¶C. On the other hand, the category ΠC′PtC is clearly stable under products, since so is C′. The fact that the terminal object 1 belongs to C′ implies that the isomorphisms are in ΠC′.
X×Y oo
X,
","100 Chapter 8. Conclusion
Proposition 8.2.1. Let C be a finitely complete pointed category and C′ be a full subcategory of C stable under finite limits and containing the terminal object. The category C is C′-unital if and only if it is ΠC′-protomodular.
Proof. Suppose C is C′-unital. Then any split epimorphism in ΠC′ is a strongly split epimorphism. Accordingly any map between such split epimorphisms is of the following form:
X×OOY f×g//X′×OOY′
πX ⟨1X ,0⟩ πX′ ⟨1X′ ,0⟩
//
X X,
f
(8.2.2)
which implies that ΠC′PtC is stable under equalizers; being already stable under finite products, it is stable under finite limits.
Conversely, suppose C is ΠC′-protomodular. The fact that any split epi- morphism in ΠC′ is a strongly split epimorphism means exactly that C is C′ -unital.
Here the fiber above 1 of this subfibration is C′, which is different from C if C is not unital.
8.3 Left exact conservative forgetful functors
Recall that a functor U is said to be conservative when it reflects the isomor- phims. Let U : C → D be a left exact functor between pointed finitely complete categories. The forgetful functor
U : SRng → CMon
is precisely a left exact conservative functor between pointed categories. More- over we noticed that the class of Schreier split epimorphisms in SRng are the inverse image of the class of Schreier split epimorphisms in CMon. We have the following very general result:
Proposition 8.3.1. Let U : C → D be a left exact functor between pointed finitely complete categories. Suppose, in addition, that the functor U is conservative. Suppose that the category D is S-protomodular. Then, setting Σ = U−1(S), the category C is Σ-protomodular.
Proof. Since U is left exact and conservative, it is straightforward that U re- flects the strongly split epimorphisms. Accordingly, any split epimorphism in Σ is a strongly split epimorphism when D is S-protomodular. The class Σ con- tains the isomorphisms since it is the case for the class S. Finally, Σ is stable under finite limits since U is left exact and S is stable under finite limits.
","Chapter 9
Appendix
This appendix is mainly devoted to set notations and to recall some notions and results without proofs. Given a category E we shall denote by E(X, Y ) the set of morphisms between X and Y . We shall suppose here that the category E is finitely complete.
9.1 Pullbacks
Given any map f : X → Y in E, its kernel equivalence relation
p0 // R[f] oo s0 // X
p1
is given by the pullback of the map f along itself. In the set-theoretical context it is defined as the set {(x0, x1) ∈ X × X | f(x0) = f(x1)}. This means that xR[f]x′ if and only if we have f(x) = f(x′). One of the most useful results concerning pullbacks is the following one with its associated variations:
Proposition 9.1.1. Given two commutative squares where both the whole rect- angle and the right hand side square are pullbacks
• // • // • // //
then so is the left hand side square.
Whence the following corollary: 101
f
// Y.
•••
","102 Chapter 9. Corollary 9.1.2. Suppose the right hand side square is a pullback:
p1 R(x)
Appendix
p0
// f R[f]oo s0 // X // Y
//
then so is any of the left hand side commutative squares.
9.1.1 Left exact conservative functors
′oop′0
R [ f ] s ′0 // X′
// Y ′.
xy
p′ 1
f′
A functor U : C → D is said to be conservative when it reflects the isomorphims.
Proposition 9.1.3. Suppose that U : C → D is a left exact functor such that, for any monomorphism m in C, if Um is an isomorphism in D then m is an isomorphism. Then U is conservative.
Proof. Given any morphism f in C, consider the kernel equivalence relation of f:
p0 // f R[f]oo s0 //X //Y.
p1
Since U is left exact, we have that UR[f] is the kernel equivalence relation of
Uf:
ooUp0 // Uf // UR[f] = R[Uf] Us0 // UX
Up1
UY.
Suppose that Uf is an isomorphism. Then Us0 is an isomorphism. Since s0 is a monomorphism, our hypothesis implies that s0 is an isomorphism. But then f is a monomorphism, hence an isomorphism by our hypothesis.
9.2 Regular epimorphisms
A map f : X → Y in E is a regular epimorphism when it is the coequalizer of its kernel equivalence relation; when it is the case we denote this map by f : X Y . In any variety of universal algebra, as the categories M on of monoids, Gp of groups, SRng of semirings or Rng of rings, the regular epimorphisms are precisely the surjective homomorphisms. The regular epimorphisms are not stable under pullbacks in general; but they are in any variety of universal algebra. In this context the proposition of the previous section has a partial converse (see Theorem 3 in [7] and Proposition 2.7 in [21]):
","9.3. Fibrations 103
Proposition 9.2.1. Suppose the regular epimorphisms are stable under pullbacks in E. Consider the following commutative diagram where both the whole rect- angle and the left hand side square are pullbacks:
• // // • // • // // //
•••
Then so is the right hand side one, provided that the lower left hand side hori- zontal arrow is a regular epimorphism.
We have also the following partial converse to Corollary 9.1.2, which is known as the Barr-Kock Theorem, see [1]:
Theorem 9.2.2. Suppose the regular epimorphisms are stable under pullbacks in E. Consider the following commutative diagram, where any of the left hand side squares is a pullback:
p1 R(x)
p0
// f R[f]oo s0 // X // // Y
xy
// Y ′.
regular epimorphism. Moreover, when x is a monomorphism, so is y. 9.3 Fibrations
We recall here from [2] the notions of cartesian morphisms and fibrations.
Definition 9.3.1. Let F : D → E be a functor. Given an object E ∈ E, the fiber of F at E is the subcategory FE of D whose objects are the objects D ∈ D such that F(D) = E and whose arrows are the arrows f: D → D′ in D such that F(f) = 1E.
Definition 9.3.2. Let F: D → E be a functor and α: J → E a morphism in E. A morphism f : Y → X in D is cartesian over α if F (f ) = α and, for any g: Z → X in D such that F(g) factors as F(g) = αβ for some β in E, there existsauniquemorphismh:Z→Y suchthatF(h)=βandg=fh.
Definition 9.3.3. A functor F : D → E is a fibration when, for every morphism α: J → E and every object X in the fiber FE, there exists in D a cartesian morphism over α.
′oop′0
R [ f ] s ′0 // X′
//
Then the right hand side one is also a pullback, provided that the map f is a
p′ 1
f′
","104 Chapter 9. Appendix
The main example we are interested in is the so-called fibration of points. Let E be a finitely complete category. The category PtE is the category whose objects (called points) are the split epimorphisms in E (with a fixed splitting), and whose arrows are the commutative squares between them. It is not difficult to see that the functor
¶E : P tE → E,
associating with any split epimorphism its codomain, is a fibration. Indeed,
givenamorphismα:J→EinEandasplitepimorphism Xoo s //E,the f
morphism (α′,α), where α′ is given by the following pullback square P α′ //X
OO OO
f′s′ fs //
JαE
is cartesian over α.
Given any morphism p: E → B in E, we can define a functor, called the
change-of-base functor
p∗ : P tB E → P tE E
by pulling back along p any split epimorphism with codomain B. Thanks to the commutativity of limits, we get that any change-of-base functor is left exact, which means that it preserves finite limits.
9.4 Simplicial objects
We need a uniform notation for internal relations, categories and groupoids in E. For that we chose the simplicial one. A simplicial object in a category E is a graded set of objects Xn,n ∈ N, together with a family of (face) maps di:Xn+1 →Xn, 0≤i≤n+1,andof(degeneracy)mapssi:Xn →Xn+1, 0≤ i ≤ n:
d0d0 d0
oos0 oo s0 oos0 d0 //
Xn Xn−1 .... .... X2 d1 // X1 oo s0 X0
.... Xn+1
dn+1 dn d2
ooLLooKK ooLL sn sn−1 s1
// d1
","9.4. Simplicial objects
subject to the following identities: didj+1 =djdi, i≤j
sj+1si =sisj, i≤j
105
p0 oo s0
R[f ]n−1
p0
oos0 p0 //
p1 // R[f] oo s0 X // Y oo KK // f
pn
disj =sj−1di, idisj =1, i=j,j+1 disj =sjdi−1, i>j+1
Any map f : X → Y in E determines a simplicial object by means of its iterated kernel equivalence relations:
.... R[f ]n
oo JJ sn−1
....
....
R[f ]2
Given any (n + 1)-uple (x0, x1, ..., xn), we denote by (x0, x1, ..., xˆi, ..., xn) the n-uple where xi is erased. In the category Set, the set R[f]n is the set of (n+1)- uples (x0, x1, ..., xn) of elements of X such that f(x0) = f(x1) = ... = f(xn). The face map pi is defined by pi(x0,x1,...,xn) = (x0,x1,...,xˆn−i,...,xn) and thedegeneracymapsi bysi(x0,x1,...,xn)=(x0,x1,...,xn−i,xn−i,...,xn).
A morphism of simplicial objects is a graded set of maps fn : Xn → Yn, n ∈ N which makes the following diagram commutative:
d0d0 d0
oos0 oo s0 oos0 d0
s1 p2
p1
// //
d1
.... Xn+1 dn+1
Xn Xn−1 .... .... X2 d1 // X1 oo s0
X0
ooLLooKK ooLL
sn
sn−1 dn
s1 d2
fn+1
.... Yn+1
sn sn−1 s1 dn+1 dn d2
fn
d0d0 d0
s0
fn−1
f2 f1
f0
oos0 YLLn
oo
oooo oo
oo s0 d0
// Y
0 Yn−KK1 .... .... Y2 d1 // LL1 oo s
Y
// 0
d1
This way, we get the category SimplE of simplicial objects in E. Any
","106 Chapter 9. commutative square in E like the following right hand side one:
Appendix
Rn[f].... Rn (x)
n ′ R[f]....
....R[f]oo
p0 //X f //Y s0 //
p1
R(x) xy
p′0 // ....R[f] s′0 //X′
p′ 1
//Y′.
f′
′ oo
determines a simplicial map (x, R(x), ..., Rn(x)) between the respective associ- ated simplicial objects.
A n-simplicial (or n-truncated simplicial) object, for n ≥ 1, has a simi- lar definition except that everything is only defined up to n. Hence we get a category denoted by SimplnE.
Given a split epimorphism (f,s): X Y, the associated simplicial ob- ject is endowed with a further family of degeneracy maps sn : R[f ]n−1 → R[f ]n satisfying the same simplicial identities and making it an augmented split sim- plicial object:
oos0
oos0
p0 //
R[f] oo s0 X oo f
.... R[f]n
TT oo
sn−1
.... R[f]2
SS oo
RR
p1
// s
R[f]n−1 ....
sn s2s1
// Y
starting with s1 : X → R[f ] defined by s1 (x) = (sf (x), x).
9.5 Internal preorders, categories and monoids
A 1-truncated simplicial object is just a reflexive graph:
d0 // X1oos0 //X0
d1
The object X0 is called the “object of objects”, while the object X1 is called the “object of arrows”; the map d0 is called the “domain” map, while the map d1 is called the “codomain” map. It is a reflexive relation as soon as the pair (d0,d1) is jointly monomorphic. If E is a variety of universal algebra, an inter- nal reflexive relation is just a reflexive relation which is compatible with all the operations.
s1
","9.5. Internal preorders, categories and monoids 107
An internal category X1 is a 3-truncated simplicial object: d0 d0
X3
oo s0
oo LLoo LL //
oo s0 X2 d1
d0 //
// X1 oo s0 X0
X2
// X1 X3
oo s0
// X2
(9.5.1)
s2 s1 d1
d3 d2
where the two following commutative squares are pullbacks of split epimor-
phisms:
oo s0
OO OO OO OO
d0
d2 s1 d1 s0
d0
d3 s2 d2 s1
oo s0 X1
d0
// X0
X2
oo s0
// X1
d0
The pullback on the left hand side defines X2 as the “objects of compos- able pair of arrows” and the map d1 : X2 → X1 as the “composition” map, while the pullback on the right hand side defines X3 as the “object of com- posable triples of arrows”. An internal functor is a morphism between such 3-truncated simplicial objects. An internal category is a preorder as soon as the morphisms d0 and d1 (from X1 to X0) are jointly monomorphic. It is then sufficient to have a 2-truncated simplicial object, the 3-truncation part coming for free. If E is a variety of universal algebra, then an internal preorder is just a preorder which is compatible with all the operations.
An internal monoid is an internal category such that X0 is the terminal object 1 which implies that d0 = d1 is the terminal map τX :
d0
oos0 oos0 τX //
X × X d1 // X oo 1 oo KK oo MM s0
s2 s1
d3 d2
d0
X × X × X
The map d0 : X × X → X becomes the first product projection and the map d2 the second one; the map d1 is then the internal binary operation, while the map s0 defines the unit element. The equations d1s0 = 1X = d1s1 give the unit axioms.
The monoid is commutative when, moreover, we have d1tw = d1, where the map tw : X × X → X × X is the twisting isomorphism which exchanges the projections.
","108 Chapter 9. Appendix
Example 9.5.1. If E is the category Top of topological spaces and continuous maps, an internal monoid in E is what is usually called a topological monoid, i.e. a topological space X endowed with a monoid operation ·: X × X → X which is continuous.
Example 9.5.2. If E is the category M on of monoids, an internal monoid in E is nothing but a commutative monoid. Indeed, the monoid operation ·: X×X → X of a monoid X is a homomorphism of monoids if and only if X is commutative. This is the classical Eckmann-Hilton argument [18].
When we are working in a (pointed) unital category C, the cooperator m: A×A → A of a commutative object satisfies by definition the unit axioms. The commutativity mtw = m of the binary operation can be checked using the fact that the pair (⟨1A , 0⟩, ⟨0, 1A ⟩) is jointly strongly epimorphic, while the associativity can be checked using the fact that the pair (⟨1A,0⟩, ⟨0,1A×A⟩) is jointly strongly epimorphic. This is why a commutative object in a unital category is endowed with a canonical structure of an internal commutative monoid. The example of the internal monoids in M on, that are the commutative monoids, is just an instance of this fact.
9.6 Internal groupoids, equivalence relations, groups
An internal category X1 in E is a groupoid when, in addition, the following square determined by the composition map d1 is a pullback in E, see [7]:
X d1 // X (9.6.2) 21
d0 X1
d0 X0,
// d0
Actually it appears to be only a 3-truncated simplicial object:
X3
d0 d0 oos0 oo s0
d0 //
X2 d1 // X1 oo s0 X0
ooLLoo
s2 s1
LL // d1
d3 d2
","9.6. Internal groupoids, equivalence relations, groups 109
such that the following part is an iterated kernel relation:
oo X3
oo d2 d3
d0 d0
s0 oo s0 d0 //
d1// //oos
// X2
LL oo
d1 X1 0 X0 LL //
s1 d1 d2
which implies that any commutative square is a pullback. An internal groupoid is an equivalence relation as soon as the morphisms d0 and d1 (from X1 to X0) are jointly monomorphic. It is then sufficient to have a 2-truncated simplicial object, the 3-truncation part coming for free:
R[d0 ]
p0
oo s0 d0 //
p1 // X1 oo s0
oo LL //
s1 d1 d2
X0
If E is a variety of universal algebra, then an internal equivalence is just a con- gruence, i.e. an equivalence relation which is compatible with all the operations.
According to the previous observation an internal monoid X is an internal group as soon as the following square is a pullback:
X×X d1 //X d0 τX
// XτX 1
In set theoretical terms, this means that the map from X × X to itself sending (m, y) to (m, m · y) is a bijection. This implies that any element has a right inverse, and thus an inverse.
Given a left exact functor U : E → E′, it preserves internal categories, groupoids, monoids and groups, preorders and equivalence relations. Suppose moreover it reflects pullbacks, i.e. suppose that a commutative square in E is a pullback as soon as its image by U is a pullback; it is clear that, in those cir- cumstances, the functor U reflects the internal categories, groupoids, monoids and groups, preorders and equivalence relations.
","110 Chapter 9. Appendix 9.7 The shift functor
Any category X1 produces a category Dec1X1, given by the upper row along with a downward vertical functor, denoted by ε1X1 (see [7]):
d0
X4
X3
oo X3
d0
s0 d0
//
d1 // X2 oo s0 X1
LL oo LL
s1 d1
//
d4 d3 d2 d1
d3 d0
d2 d0
oo X2
X
// X
s0 d0
d1 // 1oos 0
0 s1 d1
LL oo LL d2
//
d3
where X4 is given by the following pullback, which defines X4 as the “objects
of quadruple composables pairs”.
X d0 // X (9.7.3)
43
X1 is a groupoid if and only if so is Dec1X1.
d4 X3
d3 X2,
// d0
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","","Index
S-protomodular category, 98 C′ -unital category, 8
accessible Schreier split extension, 71
Baer sums in monoids, 89 Baer sums in semirings, 95
cartesian morphism, 103 centralizer of reflexive relations, 54 change-of-base functor, 23 commutative object, 7
connector, 44
conservative functor, 23 cooperating morphisms, 6 cooperator, 6
direction, 84
direction functor, 86
double centralizing relation, 47
faithful Schreier split extension, 71 fiber, 103
fibration, 103
fibration of homogeneous points, 23 fibration of points, 8
fibration of Schreier points, 23
homogeneous internal category, 36 homogeneous internal groupoid, 38 homogeneous reflexive graph, 29 homogeneous split epimorphism, 11 homogeneous split extension classi-
internal category, 35 internal group, 109 internal groupoid, 38 internal monoid, 107 internal reflexive graph, 29
jointly strongly epimorphic morphisms, 6
kernel equivalence relation, 5 kernel functor, 26
Mal’tsev category, 9 protomodular category, 97 reflexive relation, 29
Schreier equivalence relation, 29 Schreier internal category, 36 Schreier internal groupoid, 38 Schreier reflexive graph, 29 Schreier split epimorphism, 11 Schreier split extension classifier, 49 Schreier split short five lemma, 22 semidirect product of monoids, 52 semidirect product of semirings, 77 semiring, 57
shift functor, 110
simplicial notation, 104
simplicial ob ject, 104
special homogeneous surjection, 79 special Schreier extension, 81 special Schreier short five lemma,
fier, 52
81
115
","116 Index special Schreier surjection, 79
strongly split epimorphism, 7 trivial semiring, 58
unital category, 5
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