var textForPages =[" CATEGORICAL METHODS<br/>IN ALGEBRA AND TOPOLOGY Special Volume in Honour of Manuela Sobral<br/>"," Comissa˜o Editorial/Editorial Board<br/>Ana Paula Santana<br/>Ju´lio Severino Neves<br/>Maria Paula Martins Serra de Oliveira<br/>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.<br/>   ","CATEGORICAL METHODS<br/>IN ALGEBRA AND TOPOLOGY<br/>Special Volume in Honour of Manuela Sobral<br/>Maria Manuel Clementino George Janelidze<br/>Jorge Picado<br/>Lurdes Sousa<br/>Walter Tholen (editors)<br/>Textos de Matem´atica/Mathematical Texts<br/>Volume 46<br/>Departamento de Matem´atica da Universidade de Coimbra Portugal 2014<br/>"," DL: 385335/14<br/>ISBN: 978-972-8564-50-6<br/>",""," Manuela Sobral<br/>","FOREWORD<br/>A Workshop on Categorical Methods in Algebra and Topology was held at the University of Coimbra January 24-26, 2014, to honour Professor Manuela Sobral on the occasion of her seventieth birthday. With 57 mathematicians from eleven countries in attendance, the event brought together leading researchers from around the globe whose work is related to Manuela Sobral’s areas of scientific specialization. This volume gives its readers a glimpse at the intense scientific program of the three-day event.<br/>After her first mathematics degree obtained from the University of Coimbra in 1965 and subsequent assistantships at Coimbra and Porto, Manuela Sobral followed her husband Jos´e for a decade-long stay in Mozambique and Angola before returning to Coimbra in 1975. While in southern Africa she began her postgraduate studies by correspondence at the University of South Africa, ob- taining a Master’s degree from UNISA in 1978 and defending her Ph. D. in 1984. Since then she held successive academic appointments at the Univer- sity of Coimbra, as Professora Auxiliar, Associada and, after obtaining her Agrega¸c˜ao in 2002, Catedra´tica. She now carries the title Professora Jubilada.<br/>In her extensive published work Manuela Sobral made significant contribu- tions to Category Theory, especially to the theory of monads and their appli- cations to other fields, such as Grothendieck’s descent theory. With her strong commitment to excellence in teaching she supervised seven Master’s and four Ph. D. theses, as well as several post-doctoral fellowships. Most of her stu- dents went on to pursue successful academic careers. Throughout her three decades as a professor in Coimbra, among many other initiatives, she actively engaged herself in building international connections, playing leading roles in various European and transatlantic projects, under the TEMPUS, ERASMUS and SOCRATES frameworks. Her and her former students’ commitment to col- laboration across boarders has made Coimbra one of the thriving international centres of category theory. For further information on her work we refer the reader to the first article in this volume.<br/>This volume contains research articles covering a broad range of topics in algebra and topology, with most of them employing and developing categorical methods. They make contributions to general, topological, homological, and linear algebra, general and point-free topology, and to category theory itself.<br/>As guest editors we are grateful to the Department of Mathematics of the University of Coimbra and especially to the editors of Textos de Matem´atica, Maria Paula Oliveira, Ana Paula Santana and Ju´lio Neves, for dedicating a spe- cial volume to the proceedings of the Workshop. We acknowledge in particular<br/>","the Centre for Mathematics of the University of Coimbra (CMUC) for support- ing the edition of this volume. We are grateful to all authors who contributed their papers to the volume and thank the referees for their careful reading of the articles and their many constructive suggestions to the authors. Finally we use the occasion of the appearance of this volume to express our gratitude to CMUC and the project on Categorical Methods for Non-Abelian Algebra for assisting with the organization of the Workshop (www.mat.uc.pt/⇠wcmat14).<br/>Maria Manuel Clementino George Janelidze<br/>Jorge Picado<br/>Lurdes Sousa<br/>Walter Tholen<br/>","G. Janelidze<br/>CONTENTS<br/>Mathematical Greetings to Manuela Sobral 1 J. Ad´amek<br/>Colimits of Monads 7 B. Banaschewski<br/>Remarks on strong 0-dimensionality in pointfree topology 35 F. Borceux, M. M. Clementino, and A. Montoli<br/>On the representability of actions for topological algebras 41<br/>D. Bourn<br/>A note on the notion of characteristic subobject in the Mal’tsev and protomodular settings 67<br/>C. Caldeira and J. F. Queiro´<br/>A note on invariant factors 93 M. M. Clementino and W. Tholen<br/>From lax monad extensions to topological theories 99 M. Gran and D. Rodelo<br/>Some remarks on pullbacks in Gumm categories 125 R. Guitart<br/>Mathematical Morphology with Regimes in Algebraic Universes 137 J. Guti´errez Garc´ıa, J. Picado, and A. Pultr<br/>Notes on point-free real functions and sublocales 167 D. Hofmann and P. Nora<br/>Esakia spaces via idempotent split completion 201 N. Martins-Ferreira<br/>From A-spaces to arbitrary spaces via spatial fibrous preorders 221 A. Montoli, D. Rodelo, and T. Van der Linden<br/>On the reflectiveness of special homogeneous surjections of monoids 237 A. P. Santana and I. Yudin<br/>Stratifying ideals and twisted products 245<br/>L. Sousa<br/>On the localness of the embedding of algebras 259<br/>","","MATHEMATICAL GREETINGS TO MANUELA SOBRAL<br/>GEORGE JANELIDZE<br/>Following the talk given at the Workshop on Categorical Methods in Algebra and Topology in Honour of Manuela Sobral on the occasion of her 70th birthday<br/>(Coimbra, 24-26 January 2014)<br/>The Coimbra Workshop was devoted to category theory, but first of all it was devoted to our admiration of Professor Maria Manuela Oliveira de Sousa Antunes Sobral, a mathematician who introduced category theory in her coun- try.<br/>Thanks to beautiful speeches by Eduardo Marques de S´a at the Opening and by Walter Tholen at the Conference Dinner, I could indeed reduce my talk to purely mathematical greetings, presented now in the written form as follows:<br/>It is good to begin with the moment when Manuela decided to do her PhD in mathematics (as we say these days). For family reasons this was happening in Southern Africa, and particularly in South Africa, which was scientifically most advanced in that part of the World. The suggestion for research Manuela got was to study either category theory or homological algebra, which was very nice... I wish I could give such a suggestion to my PhD students... No surprise that Manuela chose category theory, but what happened next is a miracle, which I shall now try to describe in a few sentences:<br/>As we all know, in contrast to set-theoretic mathematics being split into many branches corresponding to many important types of mathematical struc- tures, the purpose of category theory is to unify. Therefore general-categorical constructions and results are supposed to be equally applicable to, say, alge- braic and topological structures, and not to be merely generalized copies of what is done for any single type of structure, even if that type of structure is a very important one. In particular, many years ago some developments in categorical topology were heavily criticized for creating categorical terminol- ogy suitable only for topological and closely related structures. But imagine a PhD student, who only begins to study category theory in an environment where categorical topology clearly dominates. Can we expect such a student to understand, in spite of the fact that categorical topology is so abstract and<br/>1<br/>","","MATHEMATICAL GREETINGS TO MANUELA SOBRAL 3<br/>(I) Grothendieck descent theory: the papers devoted to it are [11]-[14], [16], [17], [19]-[21], [23], [24], [27], [31]. Many of these papers were written jointly either with W. Tholen or with me, and one of them (a survey paper) with both of us.<br/>(II) The paper [15] with J. Ad´amek, R. El Bashir, and J. Velebil characterizes lax epimorphisms in Cat.<br/>(III) The paper [18] with J. MacDonald is a survey on monads, which, to- gether with the survey paper [19] (on descent), are chapters in the book “Categorical Foundations” (Cambridge, 2004).<br/>(IV) The papers [22] and [26] with J. Ad´amek and L. Sousa are devoted to categorical-algebraic logic.<br/>(V) Our paper [25] was inspired by one of our attempts (unsuccessful so far) to characterize effective (co)descent morphisms of distributive lattices via the so-called Priestley spaces.<br/>(VI) The papers [28], [29], [30], [32], all with N. Martins-Ferreira, the last three of them, involving monoids, also with A. Montoli, and the last two also with D. Bourn, are devoted to categorical semidirect products.<br/>In fact the theory of monads plays a significant role in almost all of these papers, as well as in Manuela’s earlier work.<br/>Not saying more about Manuela’s recent work, I cannot, however, resist mentioning one of her first contributions to descent theory:<br/>It is well known that the effective descent morphisms in the category Top of topological spaces were fully characterized by J. Reiterman and W. Tholen in terms of ultrafilter convergence. Before having this remarkable result it was not even known whether every descent morphism in Top is an effective descent morphism, and the Reiterman-Tholen counter-example gave the impression that understanding this problem requires entering the somewhat elusive world of ultrafilters. But Manuela’s counter-example was finite – actually it was a map from a six-element space to a three-element space – showing that life is much easier than expected, and initiating various further developments.<br/>Another miracle of Manuela is her way of building her school of category theory at Coimbra. Unless every Portuguese student is a genius, I cannot imag- ine how one professor could get three students as brilliant as Maria Manuel, Lurdes, and Jorge at the same university. And then Manuela, instead of teach- ing these students alone, which she was perfectly capable of doing, finds three brilliant co-supervisors, respectively: Walter Tholen, Jiˇr´ı Ad´amek, and Bern- hard Banaschewski – and Maria Manuel becomes one of the best students of<br/>","4 G. JANELIDZE<br/>Walter, Lurdes one of the best students of Jiˇr´ı, and Jorge one of the best stu- dents of Bernhard, and the Coimbra School of Category Theory becomes one of the best, stable in time, and most reliable in the World.<br/>Congratulations to Manuela for this wonderful anniversary, and let me wish Manuela, her students, and their students to have many further great achieve- ments, and let me wish all of us to always remain good friends and collabora- tors of Manuela and of all members of the wonderful Category Theory School Manuela created!<br/>References: list of papers of Manuela Sobral<br/>[1] M. Sobral, Algebraic functors and algebraic categories, Proceedings of the Eighth Portuguese-Spanish Conference on Mathematics, Vol. I (Coimbra, 1981), pp. 223–230, Univ. Coimbra, Coimbra, 1981.<br/>[2] M. Sobral, Restricting the comparison functor of an adjunction to projective objects, Quaestiones Mathematicae 6 (1983), no. 4, 303–312.<br/>[3] M. Sobral, On functors inducing the same monad, Proceedings of the ninth conference of Portuguese and Spanish mathematicians, 1 (Salamanca, 1982), 139–141, Acta Salman- ticensia. Ciencias 46, Univ. Salamanca, Salamanca, 1982.<br/>[4] M. Sobral, On adjunctions inducing the same monad, Quaestiones Mathematicae 7 (1984), no. 2, 179–201.<br/>[5] M. Sobral, Projectiveness with regard to a right adjoint functor, Continuous lattices and their applications (Bremen, 1982), 273–278, Lecture Notes in Pure and Applied Mathematics 101, Dekker, New York, 1985.<br/>[6] M. Sobral, Absolutely closed spaces and categories of algebras, Portugalia Matematicae 47 (1990), no. 4, 341–351.<br/>[7] M. Sobral, Reflective subcategories, Proceedings of the XIIth Portuguese-Spanish Con- ference on Mathematics, Vol. II (Portuguese) (Braga, 1987), 158–165, Univ. Minho, Braga, 1987.<br/>[8] M. Sobral, Monads and cocompleteness of categories, Cahiers de Topologie et Geom´etrie Di↵´erentielle Cat´egoriques 32 (1991), no. 2, 139–144.<br/>[9] M. Sobral, Contravariant hom-functors and monadicity, Category theory at work (Bre- men, 1990), 307–319, Res. Exp. Math., 18, Heldermann, Berlin, 1991.<br/>[10] M. Sobral, CABool is monadic over almost all categories, Journal of Pure and Applied Algebra 77 (1992), no. 2, 207–218.<br/>[11] M. Sobral and W. Tholen, E↵ective descent morphisms and e↵ective equivalence rela- tions, Category theory 1991 (Montreal, PQ, 1991), 421–433, CMS Conf. Proc., 13, Amer. Math. Soc., Providence, RI, 1992.<br/>[12] J. Reiterman, M. Sobral and W. Tholen, Composites of e↵ective descent maps, Cahiers de Topologie et Geom´etrie Di↵´erentielle Cat´egoriques 34 (1993), no. 3, 193–207.<br/>[13] M. Sobral, Some aspects of topological descent, Categorical topology (L’Aquila, 1994), Applied Categorical Structures 4 (1996), no. 1, 97–106.<br/>[14] M. Sobral, Another approach to topological descent theory, Applied Categorical Struc- tures 9 (2001), no. 5, 505–516.<br/>[15] J. Ada´mek, R. El Bashir, M. Sobral, and J. Velebil, On functors which are lax epimor- phisms, Theory and Applications of Categories 8 (2001), 509–521.<br/>","MATHEMATICAL GREETINGS TO MANUELA SOBRAL 5<br/>[16] G. Janelidze and M. Sobral, Finite preorders and topological descent. I, Special volume celebrating the 70th birthday of Professor Max Kelly, Journal of Pure and Applied Algebra 175 (2002), no. 1–3, 187–205.<br/>[17] G. Janelidze and M. Sobral, Finite preorders and topological descent. II, E´tale descent. Journal of Pure and Applied Algebra 174 (2002), no. 3, 303–309.<br/>[18] J. MacDonald and M. Sobral, Aspects of monads, in: Categorical foundations, 213–268, Encyclopedia Math. Appl., 97, Cambridge Univ. Press, Cambridge, 2004.<br/>[19] G. Janelidze, M. Sobral, and W. Tholen, Beyond Barr exactness: e↵ective descent mor- phisms, in: Categorical foundations, 359–405, Encyclopedia Math. Appl., 97, Cambridge Univ. Press, Cambridge, 2004.<br/>[20] M. Sobral, Descent for discrete (co)fibrations, Applied Categorical Structures 12 (2004), no. 5–6, 527–535.<br/>[21] X. Guo, M. Sobral, and W. Tholen, Descent equivalence, Cahiers de Topologie et Geom´etrie Di↵´erentielle Cat´egoriques 45 (2004), no. 4, 301–315.<br/>[22] J. Ad´amek, M. Sobral, and L. Sousa, Morita equivalence of many-sorted algebraic the- ories, Journal of Algebra 297 (2006), no. 2, 361–371.<br/>[23] M. Dias and M. Sobral, Descent for Priestley spaces, Applied Categorical Structures 14 (2006), no. 3, 229–241.<br/>[24] G. Janelidze and M. Sobral, Descent for compact 0-dimensional spaces, Theory and Applications of Categories 21 (2008), no. 10, 182–190.<br/>[25] G. Janelidze and M. Sobral, Profinite relational structures, Cahiers de Topologie et Geom´etrie Di↵´erentielle Cat´egoriques 49 (2008), no. 4, 280–288.<br/>[26] J. Ad´amek, M. Sobral, and L. Sousa, A logic of implications in algebra and coalgebra, Algebra Universalis 61 (2009), no. 3–4, 313–337.<br/>[27] G. Janelidze and M. Sobral, Descent for regular epimorphisms in Barr exact Goursat categories, Applied Categorical Structures 19 (2011), no. 1, 271–276.<br/>[28] N. Martins-Ferreira and M. Sobral, On categories with semidirect products, Journal of Pure and Applied Algebra 216 (2012), no. 8–9, 1968–1975.<br/>[29] N. Martins-Ferreira, A. Montoli, and M. Sobral, Semidirect products and crossed mod- ules in monoids with operations, Journal of Pure and Applied Algebra 217 (2013), no. 2, 334–347.<br/>[30] D. Bourn, N. Martins-Ferreira, A. Montoli, and M. Sobral, Schreier split epimorphisms in monoids and in semirings, Textos de Matem´atica, S´erie B, 45. Universidade de Coim- bra, Departamento de Matem´atica, Coimbra, 2013.<br/>[31] G. Janelidze and M. Sobral, What are e↵ective descent morphisms of Priestley spaces?, Topology and its Applications 168 (2014), 135–143.<br/>[32] D. Bourn, N. Martins-Ferreira, A. Montoli, and M. Sobral, Schreier split epimorphisms between monoids, Semigroup Forum 88, no. 3, (2014), 739–752.<br/>[33] N. Martins-Ferreira, A. Montoli and M. Sobral, Semidirect products and Split Short Five Lemma in normal categories, Applied Categorical Structures 22 (2014), no. 5–6, 687–697.<br/>","","COLIMITS OF MONADS<br/>J I Rˇ ´I A D A´ M E K<br/>Dedicated to the seventieth birthday of Manuela Sobral<br/>Abstract. The category of all monads over many-sorted sets (and over other “set-like” categories) is proved to have coequalizers. And a gen- eral diagram has a colimit whenever all the monads involved preserve monomorphisms and have arbitrarily large joint pre-fixpoints. In con- trast, coequalizers fail to exist e.g. for monads over the (presheaf) cate- gory of graphs.<br/>For more general categories we extend the results on coproducts of monads from [3]. We call a monad separated if, when restricted to monomorphisms, its unit has a complement. We prove that every col- lection of separated monads with arbitrarily large joint pre-fixpoints has a coproduct. And a concrete formula for these coproducts is presented.<br/>1. Introduction<br/>Whereas limits in the category Monad(A) of monads over a complete cate- gory A are easy, since the forgetful functor into the category of all endofunctors creates limits, colimits are more interesting. For example, a coproduct of two monads need not exist in Monad (A) – in fact, there are only four (trivial) types of monads over Set having a coproduct with every monad, as proved in [3], see also Theorem 4.4 below. In that paper a formula for coproducts of monads over Set was presented, and we extend it to coproducts of separated monads over general categories A. Separatedness means that a complement of the unit of the monad exists if we restrict ourselves to the category Am of objects and monomorphisms of A. All consistent monads over Set are separated, see [3], where a monad is called consistent if its unit is monic. (The only inconsis- tent monads over Set are the terminal monad, constantly 1, and its submonad given by ; 7! ;.) For other base categories many interesting monads fail to be separated.<br/>Received: 6 September 2014 / Accepted: 11 December 2014.<br/>2010 Mathematics Subject Classification. 18C15, 18C50, 18A30, 08C05. Key words and phrases. Monad, colimit, coproduct.<br/>7<br/> ","","COLIMITS OF MONADS 9<br/>Notation 2.1. (a) Given a category A we write [A, A] for the category of endofunctors on A and natural transformations between them. And Monad (A) denotes the category of monads and monad morphisms. The obvious forgetful functor is denoted by V : Monad (A) ! [A, A].<br/>(b) We use ⇤ to denote the parallel (horizontal) composition of natural trans- formations: given a : F ! F0 and b : G ! G0, where all functors are endo- functor of A, we have a⇤b : FG ! F0G0 given by a⇤b = aG0·Fb = F0b·aG. Recall also the interchange law:<br/>(c ⇤ d) · (a ⇤ b) = (c · a) ⇤ (d · b).<br/>Proposition 2.2. The forgetful functor of Monad (A) creates limits.<br/>Remark. Recall that creation of limits means that for every diagram D in Monad (A) with a limit cone pd : T ! WDd of the underlying diagram in [A, A] there exists a unique structure of a monad on T for which each pd is a monad morphism. Moreover, the resulting cone is a limit in Monad (A). The following two (easy) proofs work, more generally, for the category of monads over an arbitrary monoidal category. We have not found a reference for them, we thus present those proofs.<br/>Proof. For the given diagram<br/>D:D! Monad(A)<br/>denote the objects by<br/>Dd = (Td, μd, ⌘d) (d 2 objD).<br/>Given a limit cone pd : T ! Td, the unit of the monad on T is, necessarily, the<br/>unique natural transformation<br/>⌘T :Id!T withpd·⌘T =⌘d (d2objD).<br/>(Recall that pd’s are required to preserve unit.) And the multiplication μT : T · T ! T is, necessarily, the unique natural transformation for which the squares<br/>μT<br/>Td·Td μd<br/>T · T<br/>pd ⇤pd<br/>✏✏<br/>// T pd<br/>// ✏✏ Td<br/>    commute for all d 2 objD. The verification of the monad axioms is easy. To verify that this is a limit cone, let qd : (S,μS,⌘S) ! (Td,μd,⌘d) be a cone of D. There exists a unique natural transformation q : S ! T with qd = pd · q(d 2<br/>","1 0 J . A D A´ M E K<br/>objD). It is a monad morphism. Indeed, the axiom q · ⌘S = qT follows, since (pd) is a monocone, from<br/>pd ·(q·⌘S)=qd ·⌘S =⌘d =pd ·⌘T. Analogously, the axiom q · μS = μT · q ⇤ q follows from<br/>pd ·(μT ·(q⇤q))=μd ·(pd ⇤pd)·(q⇤q)=pd ·μT ·(q⇤q)<br/>⇤<br/>Corollary 2.3. Limits of monads over a complete category A are computed object-wise (on the level of A).<br/>Proposition 2.4. The forgetful functor of Monad (A) creates absolute coequal- izers.<br/>Remark. Recall that this means that given a parallel pair of monad mor- phisms p, q : S ! T whose coequalizers c in [A, A]:<br/>p<br/>q<br/>is absolute (that is, preserved by every functor with domain [A, A]), there exists a unique monad structure on C making c a monad morphism. Moreover, c is a coequalizer of p and q in Monad (A).<br/>Proof. The unit of C is, necessarily,<br/>⌘C =c·⌘T.<br/>To define the multiplication μC : C · C ! C , use the endofunctor of [A, A] defined by X 7! X · X on objects and by f 7! f ⇤ f on morphisms. Since c ⇤ c is the coequalizer of p ⇤ p and q ⇤ q, we have a unique μC for which c preserves multiplication:<br/> S<br/>))55 T c // C<br/>   S · S<br/>p⇤p ,, c⇤c //<br/>C · C<br/>22 T · T<br/>     q⇤q<br/>μS μT μC<br/>✏✏ p ✏✏ ✏✏<br/>**44 T c // C<br/>(A) is easy. ⇤<br/>Definition 2.5. An object Z is a fixpoint of an endofunctor H if HZ ' Z, and it is a pre-fixpoint of H if HZ is a subobject of Z.<br/>We say that H has arbitrarily large pre-fixpoints provided that for every object X there exists a pre-fixpoint Z of H with Z ' Z + X.<br/> S<br/>The verification that (C,⌘C,μC) is a monad and c is a coequalizer in Monad<br/>  q<br/>","COLIMITS OF MONADS 11<br/>Example 2.6. A monos-preserving endofunctor H of the category SetS of many-sorted sets has arbitrarily large pre-fixpoints i↵ for every cardinal ↵ there exists a pre-fixpoint of H all components of which have at least ↵ elements.<br/>Lemma 2.7. Every accessible endofunctor of a cocomplete category with monic coproduct injections has arbitrarily large pre-fixpoints.<br/>Proof. If H is accessible, then every object B generates a free H-algebra B¯ and B¯ = B + HB¯, see [2]. Given an object A let B be an infinite copower of A. ThentheequalityA+B'BimpliesA+B¯'B¯,andB¯isapre-fixpoint. ⇤<br/>Notation 2.8. (a) For an endofunctor H of A an algebra is a pair (A, a) consisting of an object A and a morphism a : HA ! A. Homomorphisms of algebras are defined by the usual commutative square. The resulting category is denoted by AlgH.<br/>(b) μH denotes the initial algebra (if it exists). By Lambek’s Lemma [10] its algebra structure is invertible, thus, μH is a fixpoint of H.<br/>(c) If H has free algebras, i.e., the forgetful functor AlgH ! A has a left adjoint, then FH denotes the corresponding monad over A. And ⌘ˆ : Id ! FH denotes its unit, whose components are the universal arrows of the free algebras.<br/>Theorem 2.9 (Barr [6] and Kelly [9]). If an endofunctor H has free algebras, then FH is a free monad on H. The converse holds whenever the base category is complete.<br/>Example 2.10. The power-set functor P has no fixpoint, hence, it does not generate a free monad.<br/>Construction 2.11 (see [2]). For every object X of A define the free-algebra chain W : Ord ! A (with objects Wi and morphisms wi,j : Wi ! Wj for all ordinals i  j) uniquely up to natural isomorphism by the following transfinite induction:<br/>The objects are given by<br/>Wo = X<br/>and<br/>Wi+1 =X+HWi,<br/>Wj = colim Wi for limit ordinals j.<br/>i<j<br/>The morphisms are as follows:<br/>w0,1 : X ! X + HX, coproduct injection<br/>wi+1,j+1 = idX + Hwi,j<br/>","","","","COLIMITS OF MONADS 15<br/>This is well-defined because eA ⇤ eA = eRA · ReA is a epimorphism. To verify the unit axioms μR · ⌘R = id, consider the following diagram:<br/>SA eA // RA mA // TA<br/>⌘S ⌘R ⌘T SA RA TA<br/>μSA<br/>✏✏<br/>     // ✏✏ mA⇤mA μRA<br/>axiom. Naturality of ⌘S implies that the upper left-hand square commutes: eRA ·SeA ·⌘SA =eRA ·⌘RSA ·eA =⌘AR ·eA.<br/>Analogously for the upper right-hand square. Consequently, the diagonal pas- sagefromSAtoTAintheabovediagramsatisfies(duetoμTA·⌘TA =id)the equality<br/>mA ·(μRA ·⌘RA)·eA =mA ·eA.<br/>Since nA is strongly monic and eA epic, this implies μRA · ⌘RA = id .<br/>The verification of the other unit axiom μR · R⌘R = id is analogous. The proof of the associativity<br/>μR · RμR = μR · μRR follows from the following diagram:<br/>SSSA eA⇤eA⇤eA // RRRA mA⇤mA⇤mA // TTTA μ S S A μ SA μ R R A R μ RA μ T T A T μ TA<br/>✏✏ eA⇤eA<br/>SSA RRA TTA<br/>// ✏✏ // ✏✏<br/>     // ✏✏<br/>RA mA<br/>μTA TA<br/>SA eA<br/>Its outward square commutes since S and T both satisfy the corresponding<br/>          ✏✏ ✏✏ eA⇤eA ✏✏<br/>// ✏✏ ✏✏ mA⇤mA μRA<br/>// ✏✏<br/>RA mA<br/>// ✏✏ ✏✏ SSA RRA TTA<br/>     μSA<br/>SA eA<br/>μTA TA<br/>// ✏✏<br/>  ","","COLIMITS OF MONADS 17<br/>X'X+A' IXtoget<br/>a`a<br/>Proof. Let `Si = (Si, μi, ⌘i), i 2 I, be such a collection. Then the endo- functor S = Si preserves monomorphisms. And it has arbitrarily large<br/>`i2I<br/>pre-fixpoints: given an object A find X with SiX ⇢ X for all i 2 I and<br/>SiX ⇢<br/>i2I Si generates a free monad FS with the<br/>SX = By Corollary 2.15 the functor S =<br/>universal arrow ⌘ˆ : S ! F S; the coproduct injections are denoted by vi :Si !S (i2I)<br/>The forgetful functor Monad (A) ! [A, A] creates limits, see Proposition 2.2, and we conclude that for the slice category FS / Monad (A) the the correspond- ing forgetful functor<br/>U : FS/ Monad (A) ! FS/[A,A]<br/>also creates limits. Now consider an arbitrarily cocone f = (fi) consisting of monad morphisms fi : Si ! Tf (i 2 I). The functor [fi] : S ! Tf generates a uniquemonadmorphismf¯:FS !Tf withf¯·⌘ˆ=[fi]thatwefactorizeasin Lemma 3.3<br/>S<br/>i2I<br/>i2I<br/>X ' X.<br/>  ⌘S FS<br/>[fi ]<br/>✏✏ ((<br/>// Tf f >><br/>mf<br/>We get a (possibly large) collection of objects (ef , Rf ) of the slice category FS / Monad (A). This collection has a product in FS/[A,A]. Indeed, recall from Remark 3.2 that A is cowellpowered, thus, for every object A all quotients of the object FSA form a complete lattice. Form the meet eA : FSA ! RA of all quotients (ef )A : FS A ! Rf A ranging through all cocones f . For every cocone f above we have a morphism<br/>q fA : R A ! R f A w i t h ( e f ) A = q fA · e A .<br/>The resulting functor R and natural transformations qf : R ! Rf clearly form a product of all ef in FS /[A, A]. Consequently, there exists a product (e, R) of the objects (ef , Rf ) in FS / Monad (A) as f ranges through all cocones: see Proposition 2.2. For the projections qf : R ! Rf define<br/>pf =mf ·qf :R!Tf.<br/> ¯<br/>  ef<br/>R >> f<br/>","","COLIMITS OF MONADS 19<br/>follows from the following diagram<br/>S S μi // S iii<br/>    ui Si \"\"<br/>fi Si RSi ✏✏||pfSi<br/>TfSi<br/>ui<br/>\"\"μR ⌥⌥<br/>RR //R fi<br/>pfR )) T ✏✏ R<br/>pf Tf pf<br/>  Rui<br/>     Tf ui<br/>Tf fi<br/>f<br/> \"\" ✏✏<br/>TfTf μTf<br/>// ✏✏↵↵ Tf<br/> All the inner parts but the upper one (to be proved commutative) commute: recall fi = pf · ui, use the fact that pf is a monad morphism for the lower square, and use the naturality of pf for pfR·Rui = Tfui ·pfSi. Since fi is a monad morphism, the outward square also commutes. This, together with the collective monicity of all pf ’s, proves that the upper square commutes.<br/>For every cocone f = (fi)i2I the monad morphism pf is the desired factor- ization: fi = pf · ui, see (1). This is unique since whenever r : R ! Tf is a monad morphism with fi = r·ui for all i, then f = r·[ui]. From (1) we see that[ui]=e·⌘ˆ,thusr·e·⌘ˆ=f=pf ·e·⌘ˆwhichimpliesr·e=pf ·ebythe universal property of ⌘ˆ; hence r = pf since e is epic. ⇤<br/>Remark 3.8. (a) Kelly described colimits of monads, see [9], Section 27 as follows:<br/>Let D be a diagram in Monad (A) with objects Ti = (Ti,μi,⌘i) for i 2 I.<br/>Form the category CD of all pairs (A,(ai)i2I) where A is an object of A and ai : TiA ! A is an Eilenberg-Moore algebra for Ti (i 2 I) such that for every connecting morphism f : i ! j of the indexing category the<br/>","2 0<br/>J . A D A´ M E K<br/>TiA ai //A OO<br/>triangle<br/>   (b)<br/>commutes. The morphisms of CD are the morphisms of A which are algebra homomorphisms for every Ti. We have the obvious forgetful functor<br/>UD :CD !A.<br/>Kelly proved that if UD has a left adjoint, then the corresponding monad on A is a colimit of D in Monad (A). The converse also holds if A is a complete category.<br/>Given a discrete diagram D of monads Ti (i 2 I) the category CD has as objects multi-algebras<br/>(A,(ai)i2I)whereai :TiA!AliesinATi<br/>and morphisms are those maps in A that are homomorphisms for each of Ti simultaneously. A coproduct of the monads Ti exists in Monad (A) whenever every object of A generates a free multi-algebra.<br/>(Df)A<br/>aj (2) \"\"Tj A<br/>Theorem 3.9. Every diagram with a weakly terminal object has a colimit in Monad (A). In particular, Monad (A) has coequalizers.<br/>Proof. Let D : D ! Monad (A) be a diagram with objects Ti = (Ti,μi,⌘i) for i 2 I, and let Tj be weakly terminal, i.e., for every i 2 I there exists a connecting morphism f : Ti ! Tj in D.<br/>(a) Form the full subcategory C of ATj of all algebras a : TjA ! A for Tj such that for every pair f, g : Ti ! Tj of connecting morphisms of D (i 2 I) we have<br/>a·fA =a·gA (3)<br/>This category is closed in ATj under products, which easily follows from the forgetful functor UTj creating limits. It is also closed under subalgebras. More precisely, let m : (A, a) ! (B, b) be a homomorphism in ATj with m<br/>","COLIMITS OF MONADS 21<br/>monic in A. If (B,b) lies in C, then so does (A,a):<br/>fA<br/>,,22 TjA a // A Tim Tjm m<br/>✏✏ fB ,,✏✏ //✏✏ TiB 22TjB b B<br/>gB<br/>Since the forgetful functor UTj creates limits, the category ATj is com- plete and wellpowered. Let us prove that it is also cowellpowered. Given a factorization of a homomorphism h : (A, a) ! (B, b) in ATj as a strong epimorphism e : C ! B followed by a monomorphism m : C ! B in A, the diagonal fill-in makes e and m homomorphisms:<br/>TjA a //A<br/>Tje e<br/>✏✏✏✏ c // ✏✏✏✏ TjC   C<br/>Tjm m ✏✏ // ✏✏<br/>TjBb B<br/>Thus, if h is a strong epimorphism in ATj then m is an isomorphism (recall that UTj creates limits, thus, reflects isomorphisms), consequently, h is an epimorphism in A. Since A is cowellpowered (see Remark 3.2) we conclude that ATj is cowellpowered.<br/>(b) Every full subcategory of ATj closed under products and subobjects is reflective, see [4], 16.9. Thus, the obvious forgetful functor U : C ! A has a left adjoint.<br/>The theorem now follows from Remark 3.8 and the fact that there exists an isomorphism E of categories such that the triangle<br/>C E //C D<br/>UD U A <br/> TiA<br/>   gA<br/>        ✏✏<br/>      ","2 2<br/>J . A D A´ M E K<br/>commutes. Indeed, E is the “projection to j” E(A, (di)i2I ) = (A, dj ).<br/>From the triangles (2) we deduce that (A,dj) satisfies (3). Thus, E is a well-defined, faithful functor. It is surjective on objects: for every algebra (A,a) in C define, given i 2 I,<br/>ai =a·fA :TiA!Aforanyconnectingmorphismf:Ti !Tj.<br/>Then ai is well-defined due to (3) and, since f is a monad morphism, (A, ai) is an Eilenberg-Moore algebra for Ti. Finally, to prove that E is an isomorphism, we verify that it is full. Let<br/>k : (A,a) ! (B,b)<br/>be a homomorphism in C. Then we need to prove that for every i 2 I this is a homomorphism from (A,ai) to (B,bi), where again bi = b·fB. Use the following diagram<br/>ai<br/>fA // a //✏✏<br/>    TiA TjA A Tik Tjk k<br/>     ✏✏ fB TiB<br/>// ✏✏ b<br/>TjB   B<br/>bi<br/>// ✏✏ OO<br/>    Corollary 3.10. Every diagram of monos-preserving monads with arbitrarily large joint pre-fixpoints has a colimit in Monad (A).<br/>Indeed, apply the usual construction of colimits as a coequalizer of a parallel pair between coproducts; see [11]. Given a diagram D in Monad (A) with monos-preserving objects Si = (Si,μi,⌘i) for i 2 I having arbitrarily large joint pre-fixpoints, then also every collection of monads indexed by I ⇥ J, whereJ isanarbitrarilysetandSi =S(i,j) forall(i,j)2I⇥J,hasarbitrarily large joint pre-fixpoint. (Indeed, for every object A and every cardinal ↵ put ↵0 = ↵+cardJ. By applying Definition 3.4 to A and ↵0 for the former collection indexed by I, we get the required condition for the new collection.) Thus, the two coproducts needed to construct colimD as a coequalizer in Monad (A) exist.<br/>⇤<br/>","","","","2 6 J . A D A´ M E K<br/>Theorem 4.4 (See [3]). A monad over Set has coproducts with all monads i↵ it is trivial.<br/>Moreover, all monads over Set except 1 and 10 are consistent, i.e., the components of the monad unit are monic.<br/>Definition 4.5. A monad (S, μ, ⌘) is called separated if its unit has a comple- ment in the following sense:<br/>(i) S preserves monomorphisms and<br/>(ii) there exists an endofunctor S¯ of Am such that the restriction of S to<br/>Am fulfils<br/>with the unit ⌘ as the left-hand injection.<br/>S = Id + S¯ (1)<br/>Examples 4.6. (1) The exception monad ME is separated: here M¯E is the constant functor of value E.<br/>(2) Every free monad FH which preserves monomorphisms is separated. (In particular, if A has stable monomorphisms, all free monads on monos- preserving functor are separated.) Here F¯ H = H · FH : use Remarks 2.16 and 2.17.<br/>(3) All consistent monads on Set (i.e., all except 1 and 10) are separated. See [3], Proposition IV.5.<br/>(4) Ideal monads of Elgot [7] are separated if they preserve monomorphisms. Recall that an ideal monad S = (S, μ, ⌘) is one for which an endofunctor S¯ of A exists such that (i) S = Id + S¯ in [A, A] with the left-hand injection ⌘ and (ii) μ restricts to a natural transformation μ¯ : S¯S ! S¯.<br/>(5) In particular, the free completely iterative monad S on an endofunctor H given by the greatest fixpoint<br/>SA = ⌫X · (A + HX)<br/>is separated, with S¯ = H · S, whenever it preserves monomorphisms, see<br/>[1].<br/>Notation 4.7. Let Si (i 2 I) be separated monads. For every object A of A<br/>define an endofunctor HA of AIm as follows: a<br/>HA(Xi)i2I = (S¯iYi)i2I where Yi = A + Xj (2)<br/>j2I,j6=i<br/>If HA has an initial algebra, we denote its components by Si⇤A:<br/>μHA = (Si⇤A)i2I<br/>","","","","","","","COLIMITS OF MONADS 33<br/>(J. Ad´amek) Technical University Braunschweig, Germany E-mail address: adamek@iti.cs.tu-bs.de<br/>","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","","ON THE REPRESENTABILITY OF ACTIONS FOR TOPOLOGICAL ALGEBRAS 65<br/>References<br/>[1] M. Barr, Relational algebras, in: Reports of the Midwest Category Seminar, IV, Lecture Notes in Mathematics, vol. 137 (1970), 39–55, Springer.<br/>[2] M. Barr, Exact categories, Lecture Notes in Mathematics, vol. 236, (1970), 1–120, Springer.<br/>[3] F. Borceux, D. Bourn, Mal’cev, protomodular, homological and semi-abelian categories, Mathematics and its applications, vol. 566 (2004), Kluwer.<br/>[4] F. Borceux, D. Bourn, Split extension classifier and centrality, Contemporary Mathe- matics, vol. 431 (2007), 85–104.<br/>[5] F. Borceux, M.M. Clementino, Topological semi-abelian algebras, Adv. Math. 190 (2005), 425–453<br/>[6] F. Borceux, G. Janelidze, G.M. Kelly, On the representability of actions in a semi-abelian category, Theory Appl. Categ. 14 (2005), 244–286.<br/>[7] F. Borceux, G. Janelidze, G.M. Kelly, Internal object actions, Comment. Math. Univ. Carolin. 46 (2005), no.2, 235–255.<br/>[8] J.M. Casas, T. Datuashvili, M. Ladra, Universal strict general actors and actors in categories of interest, Appl. Categ. Structures 18 (2010), 85–114.<br/>[9] M.M. Clementino, D. Hofmann, W. Tholen, The convergence approach to exponentiable maps, Port. Math. (N.S.) 60 (2003), no. 2, 139–160.<br/>[10] M.M. Clementino, A. Montoli, L. Sousa, Semidirect products of (topological) semi- abelian algebras, J. Pure Appl. Algebra 219 (2015), 183–197.<br/>[11] B.J. Day, G.M. Kelly, On topological quotient maps preserved by pullbacks or products, Proc. Cambridge Philos. Soc. 67 (1970), 553–558.<br/>[12] H. Herrlich, E. Lowen-Colebunders, F. Schwarz, Improving Top: PrTop and PsTop, Category theory at work (Bremen, 1990), Res. Exp. Math. 18 (1991), 21–34, Heldermann, Berlin, 1991.<br/>[13] G. Hochschild, Cohomology and representation of associative algebras, Duke Math. J. 14 (1947), 921–948.<br/>[14] J. Isbell, General function spaces, products and continuous lattices, Math. Proc. Cam- bridge Philos. Soc. 100 (1986), no. 2, 193–205.<br/>[15] G. Janelidze, L. M´arki, W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002), no. 2–3, 367–386.<br/>[16] J.L. Loday, Une version non commutative des alg`ebres de Lie: les alg`ebres de Leibniz, Enseign. Math. 39 (1993), no.2, 269–293.<br/>[17] S. Mac Lane, Categories for the working Mathematician, 2nd ed, Springer (1998).<br/>[18] G. Metere, A. Montoli, Semidirect products of internal groupoids, J. Pure Appl. Algebra<br/>214 (2010), 1854–1861.<br/>[19] G. Orzech, Obstruction theory in algebraic categories I, J. Pure Appl. Algebra 2 (1972),<br/>287–314.<br/>[20] C. Pisani, Convergence in exponentiable spaces, Theory Appl. Categ. 5 (1999), no. 6,<br/>148–162.<br/>[21] T. Porter, Extensions, crossed modules and internal categories in categories of groups<br/>with operations, Proc. Edinburgh Math. Soc. 30 (1987), 373–381.<br/>[22] F. Schwarz, Powers and exponential objects in initially structured categories and ap- plications to categories of limit spaces, Proceedings of the Symposium on Categorical Algebra and Topology (Cape Town, 1981), Quaestiones Math. 6 (1983), no. 1–3, 227–<br/>254.<br/>","66 F. BORCEUX, M. M. CLEMENTINO, AND A. MONTOLI<br/>(F. Borceux) IRMP, Universite´ de Louvain, 2 chemin du Cyclotron, 1348 Louvain- la-Neuve, Belgium<br/>E-mail address: francis.borceux@uclouvain.be<br/>(M. M. Clementino and A. Montoli) CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal<br/>E-mail address: mmc@mat.uc.pt; montoli@mat.uc.pt<br/>","A NOTE ON THE NOTION<br/>OF CHARACTERISTIC SUBOBJECT<br/>IN THE MAL’TSEV AND PROTOMODULAR SETTINGS<br/>DOMINIQUE BOURN<br/>D´edi´e, pour son anniversaire, a` une grande dame des math´ematiques<br/>Abstract. We introduce a categorical definition of characteristic sub- objects which is alternative to the one given in [14] for semi-abelian cat- egories and applies to the wider context of Mal’tsev categories. It allows, among other things, to set clearly this notion and to produce examples in the category Rg of non unitary rings and in the category TopGp of topological groups.<br/>Introduction<br/>In a recent work [14] A. Cigoli and A. Montoli investigated the notion of char- acteristic subobject in the context of semi-abelian categories. Mimicking strictly what happens in the categories Gp of groups and R-Lie of Lie R-algebras, their approach was based upon the notion of internal action as described in [4]. In any semi-abelian category the internal actions are in bijection with the split epimorphisms and, actually, it is possible to get rid of the concept of inter- nal action and to get straight to the notion of characteristic subobject just using split epimorphisms and hypercartesian morphisms related to the change of base functors associated with the fibration of points (Definition 2.1). This alternative approach produces alternative proofs and has, among other things, the following benefits:<br/>1) it allows us to extend the notion to the conceptually lighter and wider context of Mal’tsev categories<br/>Received: 29 April 2014 / Accepted: 24 December 2014.<br/>2010 Mathematics Subject Classification. Primary 20D30, 17B05, 13A15, 18A20; Sec-<br/>ondary 18D35,18C99.<br/>Key words and phrases. Mal’tsev, protomodular and action representative category, hy-<br/>percartesian monomorphism, characteristic subobject, commutator and centralizer.<br/>67<br/> ","68 D. BOURN<br/>2) it gives rise to a conceptual proof of the characterization of the characteristic subobjects as those normal subobjects which are stable under composition by normal subobjects on the left hand side (Theorem 2.10)<br/>3) it shows that the notion is stable under the passage to the fibres P tY C of the fibration of points, see Theorem 2.7 for more details<br/>4) it extends in a clear way the notion of characteristic subobjects to the cate- gory Rg of non unitary rings or T opGp of topological groups where the question was up to now far from being cleared (Sections 2.3 and 2.4)<br/>5) it produces some unexpected situations dealing with objects whose any sub- object is a characteristic one (Corollary 3.2)<br/>6) in the exact action representative contexts, such as Gp and R-Lie, it allows to characterize the characteristic subobjects as those normal subobjects whose quotient maps have a (unique) extension to the action groupoids (Corollary 3.5)<br/>7) it allows to characterize in simple terms those pairs of characteristic subob- jects whose commutator is characteristic as well (Theorem 5.3)<br/>8) it brings some new enlightments about the notion of peri-abelian category (Section 5.3).<br/>The article is organized along the following lines. Section 1) gives some recalls about hypercartesian monomorphisms. Section 2) introduces the alternative definition, the first stability properties and the first investigations in Rg and TopGp. Section 3) gives conceptual insights on the classical definition of the characteristic subobjects in groups and in Lie R-algebras dealing with the group AutX and the algebra DerX through the investigation of the characteristic subobjects in the action representative context. Section 4) is dealing with the centralizer of characteristic subobjects while Section 5) is dealing with the commutator of such pairs.<br/>1. Hypercartesian morphisms<br/>Let U : E ! F be any functor. Recall that a map f : X ! Y in E is hypercartesian with respect to U when, given any other map g : X0 ! Y in E with a factorization h : U(X0) ! U(X) in F such that U(g) = U(f)h, there isauniquefactorizationh¯:X0 !XinEsuchthatg=h¯fandU(h¯)=h. Accordingly any hypercartesian map whose image by U is an isomorphism is an isomorphism.<br/>Proposition 1.1. 1) The hypercartesian maps with respect to U are stable under composition; a hypercartesian map above a monomorphism is itself a monomorphism.<br/>","","","","","","","","","","","","","","","","","","","","","","","CHARACTERISTIC SUBOBJECT IN MAL’TSEV AND PROTOMODULAR SETTINGS 91<br/>[10] D. Bourn and M. Gran, Centrality and normality in protomodular categories, Theory and Applications of Categories 9, 2002, 151–165.<br/>[11] D. Bourn and J. R. A. Gray, Aspects of algebraic exponentiation, Bull. Belg. Math. Soc. Simon Stevin 19, 2012, 823–846.<br/>[12] A. Carboni, J. Lambek and M. C. Pedicchio, Diagram chasing in Mal’cev categories, J. Pure Appl. Algebra 69, 1991, 271–284.<br/>[13] A. Carboni, M. C. Pedicchio and N. Pirovano, Internal graphs and internal groupoids in Mal’cev categories, CMS Conference Proceedings 13, 1992, 97–109.<br/>[14] A.S. Cigoli and A. Montoli, Characteristic subobjects in semi-abelian categories, Pr´e- Publica¸c˜oes do Departamento de Matema´tica da Universidade de Coimbra n. 13–50, oct. 2013.<br/>[15] A. S. Cigoli, J. R. A. Gray and T. Van der Linden, On the normality of Higgins commu- tators, J. Pure Appl. Algebra, online http://dx.doi.org/10.1016/j.jpaa.2014.05.025.<br/>[16] J. R. A. Gray, Algebraic exponentiation in general categories, Applied Categorical Struc-<br/>tures 20, 2012, 543–567.<br/>[17] J.R.A. Gray, Algebraic exponentiation for categories of Lie algebras, J. Pure Appl.<br/>Algebra 216, 2012, 1964–1967.<br/>[18] S.A. Huq, Commutator, nilpotency and solvability in categories, Quarterly J. Math.,<br/>19, 1968, 363–369.<br/>[19] G. Janelidze, L. M´arki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra<br/>168, 2002, 367–386.<br/>[20] S. Mantovani, The Ursini commutator as normalized Smith-Pedichio commutator, The-<br/>ory and Applications of Categories 23, 2010, 7–21.<br/>[21] A. Montoli, Action accessibility for categories of interest, Theory and Applications of<br/>Categories 27, 2012, 174–198.<br/>[22] G. Orzech, Obstruction theory in algebraic categories I, and II, J. Pure Appl. Algebra<br/>2, 1972, 287–340.<br/>[23] M. C. Pedicchio, A categorical approach to commutator theory, J. Algebra 177, 1995,<br/>647–657.<br/>[24] J. D. H. Smith, Mal’cev varieties, Lecture Notes in Math. 554, Springer, 1976.<br/>(D. Bourn) Universite´ du Littoral, Laboratoire de Mathe´matiques Pures et Applique´es, CNRS-2956, Bat. H. Poincare´, 50 Rue F. Buisson BP 699; 62228 Calais - France<br/>E-mail address: bourn@lmpa.univ-littoral.fr<br/>","","","","","","A NOTE ON INVARIANT FACTORS 97<br/>This was also proved in [10] in a form valid for EDDs.<br/>But what about the huge family of divisibility relations, mentioned in the<br/>previous section, valid for invariant factors of products of matrices over PIDs (and which actually give the complete answer to the product problem in that setting)? Extending those results to EDDs presents an interesting challenge, necessitating a change in the proofs.<br/>And it would bring an added bonus, since, by a localization argument due to Krull [9], any divisibility relation generally valid in EDDs actually generalizes to GCD domains, rings where every finite collection of elements has a gcd inside the ring. This technique was essentially already used by Kaplansky in [5].<br/>References<br/>[1] R. Bhatia, Linear algebra to quantum cohomology: the story of Alfred Horn’s inequali- ties, Amer. Math. Monthly 108 (2001), 289-318.<br/>[2] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bul- letin AMS 37 (2000), 209-249.<br/>[3] A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962), 225-241.<br/>[4] M. Kapovich, B. Leeb, J. Millson, The generalized triangle inequalities in symmetric<br/>spaces and buildings with applications to algebra, Memoirs AMS 192, no. 896 (2008).<br/>[5] I. Kaplansky, Elementary divisors and modules, Transactions AMS 66 (1949), 464-491.<br/>[6] T. Klein, The multiplication of Schur functions and extensions of p-modules, J. London<br/>Math. Soc. 43 (1968), 280-284.<br/>[7] A. Klyachko, Stable bundles, representation theory and hermitian operators, Selecta<br/>Mathematica 4 (1998), 419-445.<br/>[8] A. Knutson and T. Tao, The honeycomb model of GLn(C) tensor products I: proof of<br/>the saturation conjecture, Journal AMS 12 (1999), 1055-1090.<br/>[9] W. Krull, Allgemeine Bewertunstheorie, J. Reine Angew. Math. 167 (1932), 160-196.<br/>[10] J. F. Queir´o, Invariant factors as approximation numbers, Linear Algebra Appl. 49 (1983), 131-136.<br/>[11] A. P. Santana, J. F. Queiro´ and E. Marques de Sa´, Group representations and matrix spectral problems, Linear and Multilinear Algebra 46 (1999), 1-23.<br/>[12] H. Weyl, Das asymtotische Verteilungsgesetz der Eigenwerte lineare partieller Di↵eren- tialgleichungen, Math. Ann. 71 (1912), 441-479.<br/>(C. Caldeira and J. F. Queir´o) CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal<br/>E-mail address: caldeira@mat.uc.pt; jfqueiro@mat.uc.pt<br/>","","FROM LAX MONAD EXTENSIONS TO TOPOLOGICAL THEORIES<br/>MARIA MANUEL CLEMENTINO AND WALTER THOLEN<br/>To Manuela, teacher and friend<br/>Abstract. We investigate those lax extensions of a Set-monad T = (T,m,e) to the category V-Rel of sets and V-valued relations for a quan- tale V = (V,⌦,k) that are fully determined by maps ⇠ : TV ! V . We pay special attention to those maps ⇠ that make V a T-algebra and, in fact, (V, ⌦, k) a monoid in the category SetT with its cartesian structure. Any such map ⇠ forms the main ingredient to Hofmann’s notion of topological theory.<br/>Introduction<br/>The lax-algebraic setting, originally considered in [5] and [4] as a common syntax for the categories of lax algebras discussed in [2], was generalized by Seal in [9] and in this form adopted in [7] and studied by various authors. A very powerful specialization of the lax-algebraic setting was introduced by Hofmann [6] in the form of his topological theories, which in particular cover Barr’s pre- sentation of topological spaces [1] and the Clementino-Hofmann presentation of approach spaces (see [2, 7]). This paper carefully studies how the Hofmann notion may be characterized within the Seal setting.<br/>Recall that, for an endofunctor T of sets and maps and a (commutative and unital) quantale V = (V, ⌦, k), Seal considers lax functors Tˆ of sets and<br/>Received: 30 September 2014 / Accepted: 27 December 2014.<br/>2010 Mathematics Subject Classification. 18C20, 18C15, 18D20, 18B10.<br/>Key words and phrases. Quantale, monad, lax extension, algebraic lax extension, structure<br/>map, topological theory.<br/>This work was partially supported by the Centro de Matema´tica da Universidade de<br/>Coimbra (CMUC), funded by the European Regional Development Fund through the pro- gram COMPETE and by the Portuguese Government through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia under the project PEst-C/MAT/UI0324/2013. The second author is supported by the National Sciences and Engineering Council of Canada.<br/>99<br/> ","","","","","","","","","","","","","","","","","","","","","","","","FROM LAX MONAD EXTENSIONS TO TOPOLOGICAL THEORIES 123<br/>[8] E. Manes, Taut monads and T0-spaces, Theoret. Comput. Sci. 275 (2002) 79–109.<br/>[9] G. J. Seal, Canonical and op-canonical lax algebras, Theory Appl. Categ. 14 (2005)<br/>221–243.<br/>[10] R. J. Wood, Ordered sets via adjunctions, in: Categorical foundations, Encyclopedia<br/>Math. Appl. 97, pp. 5–47. Cambridge Univ. Press, Cambridge, 2004.<br/>(M. M. Clementino) CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal<br/>E-mail address: mmc@mat.uc.pt<br/>(W. Tholen) Dept of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J1P3<br/>E-mail address: tholen@mathstat.yorku.ca<br/>","","SOME REMARKS ON PULLBACKS IN GUMM CATEGORIES<br/>MARINO GRAN AND DIANA RODELO<br/>Dedicated to Manuela Sobral on the occasion of her seventieth birthday<br/>Abstract. We extend some properties of pullbacks which are known to hold in a Mal’tsev context to the more general context of Gumm categories. The varieties of universal algebras which are Gumm categories are precisely the congruence modular ones. These properties lead to a simple alternative proof of the known property that central extensions and normal extensions coincide for any Galois structure associated with a Birkho↵ subcategory of an exact Goursat category.<br/>1. Introduction<br/>A categorical approach to the property of congruence modularity, well known in universal algebra, was proposed in [5, 6] via a categorical formulation of the so-called Shifting Lemma (recalled in Section 3). The categories satisfying this categorical property are called Gumm categories, since it was the mathe- matician H.P. Gumm who proved that, for a variety of universal algebras, the validity of the Shifting Lemma is equivalent to congruence modularity [13]. As examples of Gumm categories, we also have regular Mal’tsev categories [9, 8] and regular Goursat categories [8], which are defined by the property that any pair of equivalence relations R and S in C (on a same object) 3-permute: RSR = SRS.<br/>In this context [3] D. Bourn established an interesting permutability result (see Theorem 3.2), that we use in the present paper to prove the following<br/>Received: 6 August 2014 / Accepted: 29 October 2014.<br/>2010 Mathematics Subject Classification. 18C05, 08B10, 08C05, 18B10, 18E10.<br/>Key words and phrases. Regular category, Mal’tsev category, Goursat category, Gumm<br/>category, congruence modularity, pullback properties.<br/>The second author was supported by the Centro de Matem´atica da Universidade de Coim-<br/>bra (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 projects PEst-C/MAT/UI0324/2013 and PTDC/MAT/120222/2010 and grant number SFRH/BPD/69661/2010.<br/>125<br/> ","126 M. GRAN AND D. RODELO<br/>property of regular Gumm categories (Proposition 4.1). Given a commutative diagram<br/>Z⇥V U ////X //U OO OO<br/>12<br/>✏✏✏✏ // // ✏✏✏✏ // ✏✏ ZYV<br/>in a regular Gumm category such that the whole rectangle is a pullback and the left square is composed by vertical split epimorphisms and horizontal regular epimorphisms, then both squares and are pullbacks. This property is known to hold in any regular Mal’tsev category, and has been used, for example, in the categorical theory of central extensions [10, 7].<br/>In the present article we also show that this property can be used to give a new proof of a remarkable property of exact Goursat categories, namely the fact that central extensions and normal extensions relative to any (admissible) Birkho↵ subcategory X of C coincide [15]. Let us recall that a full reflective subcategory X of an exact category C is called a Birkho↵ subcategory when X is closed in C under subobjects and regular quotients. In particular, a Birkho↵ subcategory of a variety of universal algebras is just a subvariety. A Birkho↵ subcategory X is admissible, from the point of view of Categorical Galois The- ory, when the reflector I : X ! C preserves pullbacks of regular epimorphisms in X along any morphism in C. The notions of central extension and of normal extension are defined relatively to the choice of the admissible Birkho↵ sub- category X of C, as recalled in Section 5. It is precisely the useful property of pullbacks in regular Gumm categories stated above which allows one to find a simple proof of the coincidence of these two notions in the exact Goursat context (Theorem 5.2 and Corollary 5.3).<br/>In [15] G. Janelidze and G.M. Kelly proved that every Birkho↵ subcategory X of an exact category C with modular lattice of equivalence relations (on any object in C) is always admissible. It was later shown by V. Rossi in [20] that the same admissibility property still holds in the more general context of Gumm categories which are almost exact, a notion introduced by G. Janelidze and M. Sobral in [16]. We conclude the article by relating our observations on Gumm categories with these results concerning the admissibility of Galois structures.<br/>2. Preliminaries<br/>In the present paper the term regular category [1] will be used for a finitely complete category such that any kernel pair has a coequaliser and, moreover, regular epimorphisms are stable under pullbacks. Any morphism f : A ! B in a regular category C has a factorisation f = m·p, with p a regular epimorphism<br/>          1<br/>  1<br/>2<br/>","","128 M. GRAN AND D. RODELO<br/>whenever x, y, t, z are elements in X with (x, y) 2 R ^ T , (x, t) 2 S, (y, z) 2 S and (t,z) 2 R, it then follows that (t,z) 2 T:<br/>xSt TRRT ySz<br/>This notion has been extended to a categorical context in [6]. Indeed, the property expressed by the Shifting Lemma can be equivalently reformulated in any finitely complete category C by asking that a specific class of internal functors are discrete fibrations, as we are now going to recall. For any object X in C and any equivalence relations R, S, T on X with<br/>R^STR<br/>there is a canonical inclusion (i,j): T⇤S ! R⇤S of equivalence relations,<br/>      depicted as<br/>T⇤S // j //R⇤S<br/>     ⇡1 ⇡2 ✏✏ ✏✏//<br/>⇡1 ⇡2 (3.1) //✏✏ ✏✏<br/>T i R,<br/> where T⇤S (respectively, R⇤S) is the largest double equivalence relation on T and S (respectively, on R and S) and ⇡1 and ⇡2 are the projections on T (respectively, on R).<br/>Definition 3.1. [6] A finitely complete category C is called a Gumm category when any inclusion (i,j): T⇤S ! R⇤S as in (3.1) is a discrete fibration. This means that any of the commutative squares in (3.1) is a pullback.<br/>Let us recall that a Mal’tsev category C is a finitely complete category such that every reflexive relation in C is an equivalence relation. A regular cate- gory C is a Mal’tsev category when the composition of (e↵ective) equivalence relations on any object in C is 2-permutable: RS = SR, where R and S are (ef- fective) equivalence relations on a same object (see [9, 8]). The strictly weaker 3-permutability property for (e↵ective) equivalence relations, RSR = SRS, de- fines the notion of regular Goursat categories [8]. Goursat categories, thus in particular Mal’tsev categories, have the property that every lattice of equiva- lence relations (on the same object) is modular (Proposition 3.2 in [8]). This fact implies that any regular Mal’tsev category and, more generally, any regular<br/>","","","","","SOME REMARKS ON PULLBACKS IN GUMM CATEGORIES 133<br/>so that the rectangle formed by the back and right faces is a pullback. We can apply Proposition 4.1 to conclude that both the back and right faces above are pullbacks. By the Barr-Kock Theorem the right face of diagram (5.1) is then a pullback, hence so is the front face of (5.1). ⇤<br/>As a consequence of Proposition 5.1 we give a new proof of Theorem 4.8 of [15] stating that every central and split extension is a trivial extension for the Galois structure associated with any Birkho↵ subcategory of an exact Goursat category. Let us briefly recall the main definitions, and we refer to [15] for more details.<br/>When C is an exact category and X a full replete subcategory of C<br/>I<br/>one calls X a Birkho↵ subcategory of C when X is stable in C under subobjects and regular quotients. Equivalently, all X-components ⌘X of the unit ⌘: 1C ) I of the adjunction are regular epimorphisms (the right adjoint is assumed to be a full inclusion and will not be mentioned explicitly), and the naturality square<br/>//<br/>Coo ? ?_X, (5.2)<br/>  X ⌘X ////IX<br/>f If<br/>✏✏✏✏ // // ✏✏✏✏ Y⌘Y IY<br/>(5.3)<br/>    is a pushout for any regular epimorphism f : X   // // Y .<br/>A regular epimorphism f : X   // // Y is called a trivial extension when the<br/>naturality square (5.3) is a pullback. It is called a central extension when it is “locally” trivial: there exists a regular epimorphism y : Z   // // Y such that the pullback of f along y is a trivial extension.<br/>Theorem 5.2. (Theorem 4.8 of [15]) Let C be an exact Goursat category, and X a Birkho↵ subcategory of C. Then every central and split extension is necessarily a trivial extension.<br/>Proof. Let f : X   // // Y be both a central extension and a split epimorphism.<br/>By definition, there exists a regular epimorphism y : Z   // // Y such that the pullback of f along y is a trivial extension. So, in the following commutative<br/>","","","136 M. GRAN AND D. RODELO<br/>[18] G. Janelidze, W. Tholen, Facets of descent I, Appl. Cat. Structures 2 (1994), no. 3, 245–281.<br/>[19] P.T. Johnstone and M.C. Pedicchio, Remarks on continuous Mal’cev algebras, Rend. Istit. Mat. Univ. Trieste 25 (1993), no. 1-2, 277–297.<br/>[20] V. Rossi, Admissible Galois Structures and Coverings in Regular Mal’cev Categories, Appl. Categ. Structures 14 (2006), no. 4, 291–311.<br/>[21] W. Rump, Almost abelian categories, Cahiers Topologie G´eom. Di↵´erentielle Cat´egor. 42, (2001), no. 3, 163–225.<br/>[22] J.D.H. Smith, Mal’cev Varieties, Lect. Notes Math. 554, Springer Verlag (1976).<br/>(M. Gran) Institut de Recherche en Mathe´matique et Physique, Universite´ Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Bel- gium<br/>E-mail address: marino.gran@uclouvain.be<br/>(D. Rodelo) CMUC, Universidade de Coimbra, 3001–501 Coimbra, Portugal, and Departamento de Matema´tica, Faculdade de Cieˆncias e Tecnologia, Universi- dade do Algarve, Campus de Gambelas, 8005–139 Faro, Portugal<br/>E-mail address: drodelo@ualg.pt<br/>","MATHEMATICAL MORPHOLOGY WITH REGIMES IN ALGEBRAIC UNIVERSES<br/>RENE´ GUITART<br/>Friendly dedicated to Manuela Sobral, on the occasion of her 70th birthday<br/>Abstract. Usual structures can be described through the elementary notion of a regime of assimilations in the category of sets. This allows a translation principle between modality in discursive analysis and mor- phology (dilatations and erosions) in visual analysis. Especially it is a natural context to analyze images (as subsets of regimes).<br/>This notion of regime and the associated calculus of augmentation and diminution are also definable in any algebraic universe, and this allows us to reach the main purpose of this paper, which is to show that in fact Mathematical Morphology and the associated construction of hyperspaces, initially developed in a set theoretical context, can be developed in any algebraic universe.<br/>In any algebraic universe mathematical morphology exists as essen- tially the calculus of inverse relations, and consequently we automatically obtain morphologies in any category of fuzzy sets or any topos.<br/>Introduction<br/>In this paper we present an extension of morphological analysis in two di- rections.<br/>The first direction of development is about the global setting, according to which we proceed to analyze and transform images and data, which usually is the calculus of relations in the category Set of sets and maps. We show that the structural morphological operations, and basically dilatation and erosion, make sense in a very general setting, the context of an algebraic universe. This extension automatically with no new e↵orts allows to obtain toposified or fuzzified versions of morphological analysis.<br/>Received: 12 August 2014 / Accepted: 9 January 2015.<br/>2010 Mathematics Subject Classification. 18B, 18C, 03G30, 03E72, 06A15, 54, 54E15,<br/>54E35, 52A.<br/>Key words and phrases. Mathematical morphology, convexity, uniform structure, hyper-<br/>space, regime, algebraic universe, topos, fuzzy set.<br/>137<br/> ","","","","MATHEMATICAL MORPHOLOGY WITH REGIMES IN ALGEBRAIC UNIVERSES 141<br/>As a consequence, the monad P = (9, a, S) is “strong” i.e. with a strengthening (in the sense of Kock),<br/>s t AE : E A ! P ( E ) P ( A ) , or equivalently an internal image calculus,<br/>imA,E :PA⇥EA !PE. 1.2. The canonical universe of sets.<br/>Proposition 1.1. The category Set of sets and maps is equipped with a canon- ical structure of universe as follows: F ⇥ E is the cartesian product of the two sets E and F, PE = P(E) is the powerset of E, with elements the subsets of E, the coupling cE : E2 ! P(E) is given by<br/>cE ((y, x)) = {y, x}, the two maps and ⇡ are given by<br/>E(A)={B;9x(x2A^x2B)}, ⇡E(A)={B;8x(x2B(x2A)}, and Cf is defined by inverse image,<br/>Cf(B) = {x 2 E;f(x) 2 B}.<br/>Proposition 1.2. In the algebraic universe of sets from Proposition 1.1, any structure (in the Bourbaki’s sense), algebraic or topological, can be defined equa- tionally. Especially we can study hyperspaces and continuous relations.<br/>Remark 1.3. The proposition allows, in principle, to write any set theoretical construction or specification by an equation or a set of equations in the context of the set canonical universe, in such a way that, these equations can serve of specification of “the same thing” in an arbitrary algebraic universe. Concretely, this allows us to introduce operations and notions in arbitrary algebraic uni- verses by using the set theoretical notations. In the continuation of this paper at several moments we do that, mainly because for the standard reader it is clearly more intuitive. Nevertheless, if we can emphasize the part played by and ⇡, which is the core of the structure in the universe, we do not forget to do so. It will be the case notably for topological data and the construction of hyperspaces.<br/>1.3. The case of an elementary topos.<br/>Proposition 1.4. An elementary topos E is equipped with a canonical structure of algebraic universe, given, in the internal language, by the same formula as in the case of Set in Proposition 1.1.<br/>","","","","","","","","","","","","","","","","","","","","","","","","","","NOTES ON POINT-FREE REAL FUNCTIONS AND SUBLOCALES<br/>JAVIER GUTIE´RREZ GARC´IA, JORGE PICADO, AND ALESˇ PULTR<br/>Dedicated to Manuela Sobral<br/>Abstract. Usingthetechniqueofsublocaleswepresentasurveyofsome known facts (with a few new ones added) on point-free real functions. The subjects treated are, e.g., images and preimages, semicontinuity, algebraic structure (point-free real arithmetics), zero and cozero parts, z- embeddings, z-open and z-closed maps, disconnectivity, small sublocales and supports.<br/>Introduction<br/>The frame of reals, L(R) (“point-free real numbers”), was originally intro- duced by Joyal in an unpublished manuscript [24] and thoroughly studied by Banaschewski in [3] (see also Johnstone [23]). As one might expect, it was not defined as the lattice ⌦(R) of open sets in the standard real line R but as a pri- marily algebraic entity, the free frame generated by pairs of rational numbers (which one can intuitively view as rational intervals) factorized by natural re- lations (see 2.3 below). Under the Axiom of Choice, L(R) is indeed isomorphic with ⌦(R), but the point is to have the frame of point-free reals as a frame in its own right and to be able to avoid choice whenever possible (it should be noted that one can prove in a choice-free way for instance that L(R) is the completion of the frame of rationals or that it is continuous, that is, locally compact, see [3]).<br/>Once one has the frame of real numbers, one can also represent contin- uous real functions on a general frame L, namely as frame homomorphisms<br/>Received: 30 July 2014 / Accepted: 23 December 2014.<br/>2010 Mathematics Subject Classification. Primary 06D22; Secondary 13J25, 26A15, 54C30,<br/>54D15.<br/>Key words and phrases. Frame, locale, sublocale lattice, localic map, frame of reals, real<br/>function, upper semicontinuous, lower semicontinuous, ring of continuous functions, com- pletely separated sublocales, z-embedding, z-open map, z-closed map, perfectly normal frame, small sublocale, support.<br/>167<br/> ","168 J. GUTIE´RREZ GARC´IA, J. PICADO, AND A. PULTR<br/>h: L(R) ! L. This was originally done by Banaschewski ([3] – see [27] for fur- ther references). However, the classical theory of real functions, not necessarily continuous, calls for a point-free counterpart as well. An appropriate defini- tion was presented in [15] and developed in subsequent papers (e.g. [18, 6]). A classical (general) real function on a space (X,⌦(X)) is a continuous real function on the discrete space (X,P(X)). The lattice P(X) of all subsets of X has a natural counterpart in S(L)op where S(L) is the co-frame of all sublo- cales of L. Hence, a (general) real function on L can be represented as a frame homomorphism L(R) ! S(L)op.<br/>The present paper is inspired by [26]. Using extensively the technique of sublocales, we present a survey of some facts on point-free real functions. Most of the results are not new; the originality is essentially in the presentation. Our main goal is to show how zero sets may be considered in the localic setting (as zero sublocales) and then how several important notions and results about real functions may be rewritten and directly proved using this tool.<br/>After some necessary preliminaries we introduce the point-free real func- tions and prove a few facts, in particular some results concerning images and preimages of sublocales are discussed. Then, semicontinuous functions and their relation with the continuous ones are mentioned. In the following section, point- free algebraic operations on L(R) are studied, with special attention paid to the addition, multiplication, maximum and minimum. Next we turn to cozero and zero sublocales. The concept of cozero element is a well-known standard topic and its sublocale counterpart is straightforward, but there are no reasonable zero elements while in the context of sublocales we obtain a sensible notion. This approach allows to formulate the basics of the theory in a way very much parallel to the classical book of Gillman and Jerison [12]. We illustrate this in a miscellany of topics.<br/>Regarding general background, we refer to Picado and Pultr [27] for frames and locales and to Banaschewski [3] and Ball and Walters-Wayland [1] for specific information on continuous functions on frames.<br/>1. Preliminaries I. Free constructions<br/>We will work with point-free real numbers as they are usually described in literature, that is, by generators subject to relations. Since the free generators come from a set that is in fact a meet-semilattice (while its elements are used in the free construction simply as elements of a set) we think that it may be useful for the reader to compare the free frames over sets with free frames over semilattices.<br/>","","","NOTES ON POINT-FREE REAL FUNCTIONS AND SUBLOCALES 171<br/>of a frame homomorphism h = f ⇤ : M ! L. This can be done since frame ho- momorphisms preserve suprema; but of course not every mapping preserving infima is a localic one. Here is a characterization (see [27] or [26]).<br/>Letf:L!M havealeftadjointf⇤:M!L.Thenitisalocalicmapi↵ (1) f[Lr{1}]✓Mr{1},and<br/>(2) f(f⇤(a)!b)=a!f(b)<br/>(! is the Heyting operation in the frames L resp. M).<br/>2.2. The frame of sublocales. A sublocale of a frame L is a subset S ✓ L such that V<br/>(1) M✓Simplies M2S,and<br/>(2) ifa2Lands2Sthena!s2S.<br/>This concept expresses the intuition of a natural subobject of L understood as a generalized space; in the category of locales and localic maps the inclusion j : S ✓ L is a localic extremal monomorphism, hence indeed a sub-locale (in the frame perspective, it is the image of a nucleus). The set of all sublocales ordered by inclusion, denoted by<br/>S (L), is a co-frame, with the lattice operations<br/>V Si = T Si and W Si ={VA|A✓ S Si}. i2J i2J i2J i2J<br/>We have the closed resp. open sublocales<br/>c(a)=\"a resp. o(a)={x|a!x=x}={a!x|x2L}<br/>modelling closed resp. open subspaces. They are complements of each other, and the o(a) are in a natural one-one correspondence with the elements of L, preserving joins and finite meets.<br/>We will need, rather, the opposite of S(L), the frame of sublocales, denoted<br/>S(L),<br/>W Si = T Si and V Si ={VA|A✓ S Si}.<br/>withST i↵S◆T and<br/>i2J i2J i2J i2J<br/>Note that S(L) is isomorphic with the frame of congruences on L and we have a natural frame embedding<br/>cL : L ! S(L) (a 7! c(a)). (2.2.1)<br/>This means that there is a one-one correspondence between the elements of L and the closed sublocales of L, agreeing in arbitrary joins and finite meets in<br/>","","","","","","","","","","","","","","","","","","","","","","","","","","","","NOTES ON POINT-FREE REAL FUNCTIONS AND SUBLOCALES 199<br/>Acknowledgements. Support by the Ministry of Economy and Competitive- ness of Spain (through grant MTM2012-37894-C02-02), the University of the Basque Country UPV/EHU (through grants GIU12/39 and UFI 11/52), the Centre for Mathematics of the University of Coimbra (funded by the Euro- pean program COMPETE and by the Portuguese Government through the FCT project PEst-C/MAT/UI0324/2013) and the project P202/12/G061 of the Grant Agency of the Czech Republic is gratefully acknowledged.<br/>References<br/>[1] R. N. Ball and J. Walters-Wayland, C- and C⇤-quotients in Pointfree Topology, Dissert. Math, Vol. 412, 2002.<br/>[2] B. Banaschewski, Compactifications of frames, Math. Nachr. 149 (1990) 105–116.<br/>[3] B. Banaschewski, The Real Numbers in Pointfree Topology, Textos de Matema´tica, Vol.<br/>12, University of Coimbra, 1997.<br/>[4] B. Banaschewski, T. Dube, C. Gilmour and J. Walters-Wayland, Oz in pointfree topol-<br/>ogy, Quaest. Math. 32 (2010) 215–227.<br/>[5] B. Banaschewski and C. Gilmour, Pseudocompactness and the cozero part of a frame,<br/>Comment. Math. Univ. Carolinae 37 (1996) 577–587.<br/>[6] B. Banaschewski, J. Guti´errez Garc´ıa and J. Picado, Extended real functions in pointfree<br/>topology, J. Pure Appl. Algebra 216 (2012) 905–922.<br/>[7] B. Banaschewski, S. S. Hong and A. Pultr, On the completion of nearness frames, Quaest.<br/>Math. 21 (1998) 19–37.<br/>[8] B. Banaschewski and A. Pultr, A constructive view of complete regularity, Kyungpook<br/>Math. J. 43 (2003) 257–262.<br/>[9] R. L. Blair and E. van Douwen, Nearly realcompact spaces, Topology Appl. 47 (1992)<br/>209–221.<br/>[10] T. Dube, On the ideal of functions with compact support in pointfree function rings,<br/>Acta Math. Hungar. 129 (2010) 205–226.<br/>[11] T. Dube and J. Walters-Wayland, Coz-onto frame maps and some applications, Appl.<br/>Categ. Struct. 15 (2007) 119–133.<br/>[12] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, 1960.<br/>[13] J. Guti´errez Garc´ıa and T. Kubiak, General insertion and extension theorems for localic<br/>real functions, J. Pure Appl. Algebra 215 (2011) 1198–1204.<br/>[14] J. Guti´errez Garc´ıa, T. Kubiak and J. Picado, Pointfree forms of Dowker’s and Michael’s<br/>insertion theorems, J. Pure Appl. Algebra 213 (2009) 98–108.<br/>[15] J. Guti´errez Garc´ıa, T. Kubiak and J. Picado, Localic real functions: a general setting,<br/>J. Pure Appl. Algebra 213 (2009) 1064–1074.<br/>[16] J. Guti´errez Garc´ıa, T. Kubiak and J. Picado, Perfectness in locales, in preparation.<br/>[17] J. Guti´errez Garc´ıa and J. Picado, On the algebraic representation of semicontinuity, J.<br/>Pure Appl. Algebra 210 (2007) 299–306.<br/>[18] J. Guti´errez Garc´ıa and J. Picado, Rings of real functions in pointfree topology, Topology<br/>Appl. 158 (2011) 2264–2278.<br/>[19] J. Guti´errez Garc´ıa and J. Picado, Insertion and extension results for point-free complete<br/>regularity, Bull. Belg. Math. Soc. Simon Stevin 20 (2013) 675–687.<br/>[20] R. W. Heath and E. A. Michael, A property of the Sorgenfrey line, Compositio Math. 23<br/>(1971) 185–188.<br/>","200 J. GUTIE´RREZ GARC´IA, J. PICADO, AND A. PULTR<br/>[21] J. R. Isbell, Atomless parts of spaces, Math. Scand. 31 (1972) 5–32.<br/>[22] J.R. Isbell, I. Kˇr´ıˇz, A. Pultr and J. Rosicky´, Remarks on localic groups, in: Catego- rical Algebra and its Applications (Proc. Int. Conf. Louvain-La-Neuve 1987, ed. by F.<br/>Borceux), Lecture Notes in Math. 1348, pp. 154-172.<br/>[23] P. T. Johnstone, Stone Spaces, Cambridge University Press, Cambridge, 1982.<br/>[24] A. Joyal, Nouveaux fondaments de l’analyse, Lectures Montr´eal 1973 and 1974 (unpub-<br/>lished).<br/>[25] J. Picado and A. Pultr, Sublocale sets and sublocale lattices, Arch. Math. (Brno) 42<br/>(2006) 409–418.<br/>[26] J. Picado and A. Pultr, Locales treated mostly in a covariant way, Textos de Matem´atica,<br/>Vol. 41, University of Coimbra, 2008.<br/>[27] J. Picado and A. Pultr, Frames and Locales: Topology without points, Frontiers in Math-<br/>ematics, vol. 28, Springer Basel, 2012.<br/>[28] M. Weir, Hewitt-Nachbin Spaces, Mathematics Studies, vol. 17, North-Holland, 1975.<br/>(J. Guti´errez Garc´ıa) Department of Mathematics, University of the Basque Country UPV/EHU, Apdo. 644, 48080 Bilbao, Spain<br/>E-mail address: javier.gutierrezgarcia@ehu.es<br/>(J. Picado) CMUC, Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal<br/>E-mail address: picado@mat.uc.pt<br/>(A. Pultr) Department of Applied Mathematics and ITI, MFF, Charles Univer- sity, Malostranske´ na´m. 24, 11800 Praha 1, Czech Republic<br/>E-mail address: pultr@kam.ms.mff.cuni.cz<br/>","ESAKIA SPACES VIA IDEMPOTENT SPLIT COMPLETION<br/>DIRK HOFMANN AND PEDRO NORA<br/>Dedicated to Manuela Sobral<br/>Abstract. Under Stone/Priestley duality for distributive lattices, Esakia spaces correspond to Heyting algebras which leads to the well- known dual equivalence between the category of Esakia spaces and mor- phisms on one side and the category of Heyting algebras and Heyting morphisms on the other. Based on the technique of idempotent split com- pletion, we give a simple proof of a more general result involving certain relations rather than functions as morphisms. We also extend the notion of Esakia space to all stably compact spaces and show that these spaces define the idempotent split completion of compact Hausdor↵ spaces. Fi- nally, we exhibit connections with split algebras for related monads.<br/>Introduction<br/>These notes evolve around the observation that Esakia duality for Heyting al- gebras arises more naturally when considering the larger category SpecDist with objects spectral spaces and with morphisms spectral distributors. In fact, as we observed already in [25], in this category Esakia spaces define the idempotent split completion of Stone spaces. Furthermore, it is well-known that SpecDist is dually equivalent to the category DLat?,_ of distributive lattices and maps preserving finite suprema and that, under this equivalence, Stone spaces corre- spond to Boolean algebras. This tells us that the category of Esakia spaces and spectral distributors is dually equivalent to the idempotent split completion of the category Boole?,_ of Boolean algebras and maps preserving finite suprema.<br/>Received: 1 August 2014 / Accepted: 8 January 2015.<br/>2010 Mathematics Subject Classification. 03G05, 03G10, 18A40, 18C15, 18C20, 54H10. Key words and phrases. Boolean algebra, distributive lattice, Heyting algebra, dual equiv-<br/>alence, Stone space, spectral space, Esakia space, Vietoris functor, idempotent split comple- tion, split algebra.<br/>Partial financial assistance by Portuguese funds through CIDMA (Center for Research and Development in Mathematics and Applications), and the Portuguese Foundation for Science and Technology (“FCT – Funda¸c˜ao para a Ciˆencia e a Tecnologia”), within the project PEst-OE/MAT/UI4106/2014, and by the project NASONI under the contract FCOMP-01- 0124-FEDER-028923 is gratefully acknowledged.<br/>201<br/> ","202 D. HOFMANN AND P. NORA<br/>However, the main ingredients to identify this category as the full subcate- gory of DLat?,_ defined by all co-Heyting algebras were already provided by McKinsey and Tarski in 1946.<br/>In order to present this argumentation, we carefully recall in Section 1 vari- ous aspects of spectral spaces and Stone spaces which are the spaces occurring on the topological side of the famous duality theorems of Stone for distributive lattices and Boolean algebras. Special emphasis is given to the larger class of stably compact spaces and their relationship with ordered compact Hausdor↵ spaces. We also briefly present the extension of Stone’s result to categories of continuous relations, an idea attributed to Halmos. These continuous rela- tions and, more generally, spectral distributors, are best understood using the Vietoris monad which is the topic of Section 2. In particular, we identify ad- junctions in the Kleisli category of the lower Vietors monad on the category of stably compact spaces and spectral maps. Based on this description, we present Esakia spaces as the idempotent split completion of Stone spaces, and in Section 3 we use this fact to deduce Esakia dualities using the technique of idempotent split completion. Moreover, we extend the notion of Esakia space to all stably compact spaces and show in Section 4 that the category of (general- ised) Esakia spaces and spectral distributors is the idempotent split completion of the category of compact Hausdor↵ spaces and continuous relations. Finally, the idempotent split completion of Kleisli categories is ultimately linked to the notion of split algebra for a monad, which is the topic of Section 5.<br/>1. Stone and Halmos dualities<br/>The aim of this section is to collect some well-known facts about duality theory for Boolean algebras and distributive lattices and about the topological spaces which occur as their duals. As much as possible we try to indicate original sources.<br/>Naturally, we begin with the classical Stone dualities stating (in modern lan- guage) that the category Stone of Stone spaces (= zero-dimensional compact Hausdor↵ topological spaces) and continuous maps is dually equivalent to the category Boole of Boolean algebras and homomorphisms (see [42])<br/>Stoneop ' Boole;<br/>and that the category Spec of spectral spaces and spectral maps is dually equivalent to the category DLat of distributive lattices1 and homomorphisms<br/>(see [43])<br/>Specop ' DLat.<br/> 1We note that for us a lattice is an ordered set with finite suprema and finite infima, hence every lattice has a largest element > and a smallest element ?.<br/>","","","","","","","","","","","","214 D. HOFMANN AND P. NORA<br/>4. Generalised Esakia spaces as idempotent split completion<br/>With the results of the last section in mind, we would like to conclude that GEsaDist is the idempotent split completion of the category CompHausRel. This follows indeed from Theorem 3.2, as soon as we know that the cate- gory StCompDist is idempotent split complete. Similarly to the case of spectral spaces, it is easier to argue in the dual category. We write StContDLatW,⌧ to denote the category having as objects continuous distributive lattices where the way-below relation is stable under finite infima, and as morphisms those maps which preserve suprema and the way-below relation. Note that every continuous distributive lattice is a frame. The following result can be found in [28].<br/>Theorem 4.1. The category StCompDist is dually equivalent to the category StContDLatW,⌧ .<br/>Proposition 4.2. The category StContDLatW,⌧ is idempotent split complete.<br/>Proof. Let e: L ! L be an idempotent morphism in StContDLatW,⌧. Then e splits in the category of sup-lattices and sup-preserving maps, that is, there is a complete lattice M and sup-preserving maps r: L ! M and s: M ! L so that e = sr and rs = 1M . Then M is certainly a distributive lattice, and, since the embedding s : M ! L preserves suprema, for all x, y 2 M one has<br/>s(x)⌧s(y) ) x⌧y.<br/>Consequently, since e: L ! L preserves the way-below relation, so does r: L ! M. We show now that s: M ! L preserves the way-below relation. To this end, let x ⌧ y in M. Since L is a con_tinuous lattice,<br/>s(y) = {b 2 L | b ⌧ s(y)},<br/>and note that {b 2 L | b ⌧ s(y)} is directed. Hence, y = rs(y) is the directed supremum of {r(b) 2 L | b ⌧ s(y)}. Therefore there exist some b ⌧ s(y) with x  r(b) and, since e preserves the way-below relation, we obtain<br/>s(x)  sr(b) = e(b) ⌧ e(s(y)) = s(y).<br/>This shows that s preserves the way-below relation, and from that it follows that M is a continuous lattice. Finally, we prove that the way-below relation in M is stable under finite infima. Note that r(>) = > since r is surjective, therefore,since>⌧>inL,weobtain>⌧>inM.Letnowx⌧x0 and y⌧y0 inM.Then<br/>x^y=r(s(x)^s(y))⌧r(s(x0)^s(y0))=x0 ^y0. ⇤<br/>","","","","218 D. HOFMANN AND P. NORA<br/>References<br/>[1] Barr, M. (1970), Relational algebras, in Reports of the Midwest Category Seminar, IV, pages 39–55, Lecture Notes in Mathematics, Vol. 137. Springer, Berlin.<br/>[2] Bezhanishvili, G., editor (2014), Leo Esakia on duality in modal and intuitionistic logics, Dordrecht: Springer, xiii + 334 pages.<br/>[3] Bezhanishvili, G., Bezhanishvili, N. and Harding, J. (2012), Modal compact Hausdor↵ spaces, Journal of Logic and Computation, to appear.<br/>[4] Bezhanishvili, G. and Jansana, R. (2013), Esakia style duality for implicative semilat- tices, Appl. Categ. Struct. 21 (2), 181–208.<br/>[5] Blackburn, P., de Rijke, M. and Venema, Y. (2001), Modal logic, Cambridge: Cambridge University Press, xxii + 554 pages.<br/>[6] Borceux, F. and Dejean, D. (1986), Cauchy completion in category theory, Cahiers Topologie G´eom. Di↵´erentielle Cat´eg. 27 (2), 133–146.<br/>[7] Bru¨mmer, G. C. L., Giuli, E. and Herrlich, H. (1992), Epireflections which are comple- tions, Cahiers Topologie G´eom. Di↵´erentielle Cat´eg. 33 (1), 71–93.<br/>[8] Cignoli, R., Lafalce, S. and Petrovich, A. (1991), Remarks on Priestley duality for dis- tributive lattices, Order 8 (3), 299–315.<br/>[9] Clementino, M. M. and Hofmann, D. (2004), On extensions of lax monads, Theory Appl. Categ. 13, No. 3, 41–60.<br/>[10] Cornish, W. H. (1975), On H. Priestley’s dual of the category of bounded distributive lattices, Mat. Vesn., N. Ser. 12, 329–332.<br/>[11] Davey, B. A. and Galati, J. C. (2003), A coalgebraic view of Heyting duality, Studia Logica 75 (3), 259–270.<br/>[12] Day, A. (1975), Filter monads, continuous lattices and closure systems, Canad. J. Math. 27, 50–59.<br/>[13] Doˇsen, K. (1990), Normal modal logics in which the Heyting propositional calculus can be embedded, in Mathematical logic. Proceedings of the summer school and conference dedicated to the ninetieth anniversary of Arend Heyting (1898-1980), held in Chaika, Bulgaria, September 13-23, 1988, pages 281–291, New York: Plenum Press.<br/>[14] Engelking, R. (1989), General topology, volume 6 of Sigma Series in Pure Mathematics, Heldermann Verlag, Berlin, second edition, viii+529 pages, translated from the Polish by the author.<br/>[15] Esakia, L. (1974), Topological Kripke models, Dokl. Akad. Nauk SSSR 214, 298–301.<br/>[16] Escard´o, M. H. (1997), Injective spaces via the filter monad, in Proceedings of the 12th Summer Conference on General Topology and its Applications (North Bay, ON, 1997),<br/>volume 22, pages 97–100.<br/>[17] Escard´o, M. H. (1998), Properly injective spaces and function spaces, Topology Appl.<br/>89 (1-2), 75–120.<br/>[18] Fawcett, B. and Wood, R. J. (1990), Constructive complete distributivity. I, Math. Proc.<br/>Cambridge Philos. Soc. 107 (1), 81–89.<br/>[19] Fleisher, I. (2000), Priestley’s duality from Stone’s., Adv. Appl. Math. 25 (3), 233–238.<br/>[20] Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. W. and Scott, D. S.<br/>(1980), A compendium of continuous lattices, Springer-Verlag, Berlin, xx+371 pages.<br/>[21] Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. W. and Scott, D. S. (2003), Continuous lattices and domains, volume 93 of Encyclopedia of Mathematics<br/>and its Applications, Cambridge University Press, Cambridge, xxxvi+591 pages.<br/>","ESAKIA SPACES VIA IDEMPOTENT SPLIT COMPLETION 219<br/>[22] Halmos, P. R. (1956), Algebraic logic. I. Monadic Boolean algebras, Compositio Math. 12, 217–249.<br/>[23] Hochster, M. (1969), Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142, 43–60.<br/>[24] Hofmann, D. (2013), A four for the price of one duality principle for distributive spaces, Order 30 (2), 643–655.<br/>[25] Hofmann, D. (2014), The enriched Vietoris monad on representable spaces, J. Pure Appl. Algebra 218 (12), 2274–2318.<br/>[26] Johnstone, P. T. (1986), Stone spaces, volume 3 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, xxii+370 pages, reprint of the 1982 edition.<br/>[27] Jung, A. (2004), Stably compact spaces and the probabilistic powerspace construction, in J. Desharnais and P. Panangaden, editors, Domain-theoretic Methods in Probabilistic Processes, volume 87, 15pp.<br/>[28] Jung, A., Kegelmann, M. and Moshier, M. A. (2001), Stably compact spaces and closed relations, in Papers from the 17th conference on the mathematical foundations of pro- gramming semantics, (Aarhus University, Aarhus, Denmark, May 23–26, 2001), Elsevier, Electron. Notes Theor. Comput. Sci. volume 45, pages 209–231.<br/>[29] Kock, A. (1995), Monads for which structures are adjoint to units, J. Pure Appl. Algebra 104 (1), 41–59.<br/>[30] McKinsey, J. and Tarski, A. (1946), On closed elements in closure algebras, Ann. Math. (2) 47, 122–162.<br/>[31] Nachbin, L. (1950), Topologia e Ordem, Univ. of Chicago Press, in Portuguese, English translation: Topology and Order, Van Nostrand, Princeton (1965).<br/>[32] Priestley, H. A. (1970), Representation of distributive lattices by means of ordered stone spaces, Bull. London Math. Soc. 2, 186–190.<br/>[33] Priestley, H. A. (1972), Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. (3) 24, 507–530.<br/>[34] Pultr, A. and Sichler, J. (1988), Frames in Priestley’s duality, Cah. Topol. G´eom. Di↵´er. Cat´eg. 29 (3), 193–202.<br/>[35] Rosebrugh, R. and Wood, R. J. (1994), Constructive complete distributivity IV, Appl. Categ. Structures 2 (2), 119–144.<br/>[36] Rosebrugh, R. and Wood, R. J. (2004), Split structures, Theory Appl. Categ. 13, No. 12, 172–183.<br/>[37] Sambin, G. and Vaccaro, V. (1988), Topology and duality in modal logic, Ann. Pure Appl. Logic 37 (3), 249–296.<br/>[38] Schalk, A. (1993), Algebras for Generalized Power Constructions, Ph.D. thesis, Tech- nische Hochschule Darmstadt.<br/>[39] Scott, D. (1972), Continuous lattices, in Toposes, algebraic geometry and logic (Conf., Dalhousie Univ., Halifax, N. S., 1971), Springer, Lect. Notes Math. volume 274, pages 97–136.<br/>[40] Simmons, H. (1980), Reticulated rings, J. Algebra 66, 169–192.<br/>[41] Simmons, H. (1982), A couple of triples, Topology Appl. 13 (2), 201–223.<br/>[42] Stone, M. H. (1936), The theory of representations for Boolean algebras, Trans. Amer.<br/>Math. Soc. 40 (1), 37–111.<br/>[43] Stone, M. H. (1938), Topological representations of distributive lattices and Brouwerian<br/>logics, Cˇasopis pro pˇestov´an´ı matematiky a fysiky 67 (1), 1–25.<br/>[44] Vietoris, L. (1922), Bereiche zweiter Ordnung, Monatsh. Math. Phys. 32 (1), 258–280.<br/>","220 D. HOFMANN AND P. NORA<br/>[45] Wyler, O. (1981), Algebraic theories of continuous lattices, in Continuous lattices (Conf., Bremen, 1979), Springer, Lect. Notes Math. volume 871, pages 390–413.<br/>[46] Z¨oberlein, V. (1976), Doctrines on 2-categories, Math. Z. 148 (3), 267–279.<br/>(D. Hofmann and P. Nora) Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal<br/>E-mail address: dirk@ua.pt; a28224@ua.pt<br/>","FROM A-SPACES TO ARBITRARY SPACES VIA SPATIAL FIBROUS PREORDERS<br/>NELSON MARTINS-FERREIRA<br/>Dedicated to Manuela Sobral<br/>Abstract. The well-known equivalence between preorders and Alexan- drov spaces is extended to an equivalence between arbitrary topological spaces and spatial fibrous preorders, a new notion to be introduced.<br/>1. Introduction<br/>The categorical equivalence between preorders and A-spaces (i.e. Alexandrov spaces) is essentially a classical result established in [1]. A preorder is simply a reflexive and transitive relation while an A-space is a topological space in which any intersection of open sets is open. The latter property trivially holds for fi- nite topological spaces and the equivalence between finite topological spaces and finite preorders was used in [6, 7] to solve some open problems in topolog- ical descent theory. In [4], p. 61, Ern´e writes “Hence the question arises: How can we enlarge the category of A-spaces on the one hand and the category of quasiordered sets on the other hand, so that we still keep an equivalence be- tween the topological and the order-theoretical structures, but many interesting ‘classical’ topologies are included in the extended definition?” and proposes the notions of B-space and C-space.<br/>With a di↵erent motivation, and not being restricted to the order-theore- tical structures, we propose a new structure, which we call fibrous preorder, and which generalizes the one of a preorder. With the appropriate morphisms, called fibrous morphisms, and a suitable equivalence between them, we observe<br/>Received: 28 July 2014 / Accepted: 10 January 2015.<br/>2010 Mathematics Subject Classification. 18B35, 54H99, 16B50.<br/>Key words and phrases. Preorder, quasiorder, topological space, fibrous preorder, fibrous<br/>morphism, spatial fibrous preorder, equivalence of categories.<br/>Research supported by IPLeiria(ESTG/CDRSP) and FCT grant SFRH/BPD/43216/<br/>2008. Also by the FCT projects: PTDC/EME-CRO/120585/2010, PTDC/MAT/120222/ 2010 and PEst-OE/EME/UI4044/2013.<br/>221<br/> ","222 N. MARTINS-FERREIRA<br/>that the category Top, of topological spaces and continuous maps, is sitting between the category of preorders and the category of fibrous preorders:<br/>Preord   // Top   // FibPreord.<br/>The main result of this work is the description of the subcategory of fibrous preorders which is equivalent to the category of topological spaces. Inspired by what are called spatial frames in point-free-topology (see e.g. [10]), the fibrous preorders arising in this way are called spatial fibrous preorders.<br/>A fibrous preorder is a generalization of a preorder. It was obtained while looking for a simple description of topological spaces in terms of internal cat- egorical structures, that is a structure which can be defined in an arbitrary category with finite limits — as for instance the notion of internal category or internal groupoid, internal preorder or internal equivalence relation (see e.g. [2]).<br/>This work is organized as follows: in section 2 we describe the category of fibrous preorders, by defining its objects and morphisms and an equivalence relation on each hom-set of fibrous morphisms; the equivalence of morphisms induces an equivalence between the objects (Proposition 2.5); at the end we recover the classical Alexandrov theorem stating that every A-space is equiv- alent to a preorder. In section 3 we introduce the notion of spatial fibrous preorder and prove that (up to equivalence) it defines a subcategory of the category of fibrous preorders which is isomorphic to the category of topological spaces. In section 4 we provide some examples illustrating how spatial fibrous preorders can be used to work with topological spaces described by systems of open neighbourhoods.<br/>2. Fibrous preorders and fibrous morphisms<br/>The following definition is a generalization of the notion of preorder, i.e. a reflexive and transitive relation. The word fibrous is a derivation of the word fibre and it is motivated by the presence of a morphism p: A ! B (see below), suggesting that A may be considered as a fibre over the base B. Moreover when p is an isomorphism the classical notion of preorder is recovered.<br/>Definition 2.1. A fibrous preorder is a sequence R @ //A p //B<br/>in which A and B are sets, p and @ are maps, R ✓ A⇥B is a binary relation (and as usual we simply write (a, b) 2 R as aRb) such that the following conditions hold:<br/>(F1) p@(a, b) = b;<br/>  ","FROM A-SPACES TO ARBITRARY SPACES VIA SPATIAL FIBROUS PREORDERS 223<br/>(F2) aRp(a);<br/>(F3) @(a, b)Ry ) aRy;<br/>for every a 2 A and b, y 2 B with aRb.<br/>Clearly this is a generalization of a preorder. The axiom (F2) generalizes the reflexivity axiom and (F3) the transitivity axiom. Indeed, when p: A ! B is the identity map 1B : B ! B, then @ has to be the second projection, (F2) reads as aRa and (F3) as aRb and bRy implies aRy.<br/>Concerning the definition of morphisms, one possibility would be to say that they are maps with a certain property, which in the case of topological spaces would correspond to the property of a map being continuous. However, by requiring that a morphism is a map with an extra structure, as considered here, we have the following three advantages: (a) we can still look at it as a property, by asking whether or not the extra structure exists; (b) in the case of topological spaces, and more specifically in metric spaces, one is able to distinguish between di↵erent notions of continuity, such as uniform continuity (see for instance the example of natural spaces in Section 4); (c) it is a more natural notion to define in an abstract category. Furthermore, in the case of preorders the extra structure is uniquely determined, up to a unique isomorphism of pullbacks, provided it exists.<br/>For convenience of writing we will sometimes refer to the structure of a fi- brous preorder A = ( R @ // A p // B ) simply as a five-tuple (R,A,B,@,p).<br/>Definition 2.2. Let A = (R,A,B,@,p) and A0 = (R0,A0,B0,@0,p0)) be two fibrous preorders. A fibrous morphism between A and A0 is a pair (f,') with f:B!B0 amapfromBtoB0 and':A0f !Aamapfrom<br/>  to A such that<br/>and<br/>A0f ={(a0,b)2A0⇥B|p0(a0)=f(b)} p'(a0, b) = b<br/>'(a0,b)Ry ) a0R0f(y)<br/>(2.1) (2.2)<br/>foralla0 2A0 andb,y2Bwithp0(a0)=f(b).<br/>In other words a fibrous morphism from A to A0 is a map f from B to B0 together with a span<br/>A0oo⇡1 A0 ' //A f<br/>  ","","","","FROM A-SPACES TO ARBITRARY SPACES VIA SPATIAL FIBROUS PREORDERS 227<br/>Proof. Straightforward verification. ⇤<br/>In the next section we describe the fibrous preorders arising from a topo- logical space, called spatial fibrous preorders and prove that the category of topological spaces is isomorphic to the category of spatial fibrous preorders. Before that we illustrate how the classical result of [1] can be obtained via this new setting.<br/>Proposition 2.8. Let (B,⌧) be a topological space and consider the fibrous preorder, say G(B,⌧), described in Proposition 2.7. It is an Alexandrov space if and only if G(B,⌧) is isomorphic to a preorder.<br/>Proof. If (B,⌧) is an Alexandrov space then there exists a map u: B ! A<br/>assigning to each point x 2 B the element (✓x, x) 2 A with ✓x the intersection of all open neighbourhoods of x, moreover this map satisfies the requirements of Proposition 2.6, and hence it is isomorphic to a preorder.<br/>For the converse, assume that G(B,⌧) is isomorphic to a preorder. Then u: B ! A provides, for each b 2 B, a smallest neighbourhood. Hence, (B,⌧) is<br/>an Alexandrov space.<br/>⇤<br/>3. The main result<br/>The so called spatial frames are the frames that are isomorphic to the topol- ogy of some space (see e.g. [10]). Here our main result is the description of the full subcategory of FibPreord0 which is equivalent to the category of topological spaces.<br/>Definition 3.1. A fibrous preorder ( R @ // A p // B ) is said to be spatial when there exists s : B ! A and m : A ⇥B A ! A with A ⇥B A = {(a, a0 ) 2 A ⇥ A | p(a) = p(a0)} such that<br/>(F4) ps(y) = y;<br/>(F5) pm(a, a0) = p(a) = p(a0); (F6) m(a, a0)Ry ) aRy&a0Ry;<br/>foreverya,a0 2Aandy2Bwithp(a)=p(a0).<br/>Observe that the identification of fibrous morphisms, as in Definition 2.3, implies that the extra structure, (s, m), in the definition of a spatial fibrous pre- order, (R, A, B, @, p, s, m), is uniquely determined by its fibrous preorder part, that is (R, A, B, @, p). This means that the category of spatial fibrous preorders and (equivalent) fibrous morphisms is a full subcategory of FibPreord0, which we denote by SpFibPreord0.<br/>  ","","","","","","FROM A-SPACES TO ARBITRARY SPACES VIA SPATIAL FIBROUS PREORDERS 233<br/>The p-adic topology. The p-adic topology on the set of integers is obtained as<br/>N(n,x)={z2Z|z=x+kpn, k2Z}<br/>If instead of a map N : N ⇥ B ! P(B) we consider a family of binary rela-<br/>with B = Z.<br/>tions Rn over B, then we have examples of the following type, with I = N.<br/>Indexed families of preorders. A more general example is obtained as fol- lows. Let I be a unitary magma, B a set, (Ri)i2I a family of binary relations Ri ✓B⇥B,and(@i:Ri !I)afamilyofmapssuchthat:<br/>(1) xRix<br/>(2) xRijy ) xRiy&xRjy (3) xRib&bR@i(x,b)y ) xRiy<br/>foralli,j2I andx,y,b2B.<br/>A morphism, say from (@i : Ri ! I)i2I to (@i00 : Ri00 ! I0)i02I0 , consists of a<br/>map f : B ! B0 together with a family of maps (fj : B ! I0)j2I0 such that xRfj(x)y ) f(x)Rj0 f(y).<br/>In this case we construct a fibrous preorder as follows: A = I ⇥B, p(i, x) = x, s(x) = (1, x), m(i, j, x) = (ij, x),<br/>(i, x)Ry , xRiy<br/>and @(i, x, y) = (@i(x, y), y) if xRiy.<br/>In the paper [5], in preparation, the topological spaces arriving from a struc-<br/>ture such as the one above, which is called an abstract system of neighbourhoods, are considered and some of its categorical properties are studied.<br/>5. Conclusion<br/>In this note we have introduced the notions of (spatial) fibrous preorder and fibrous morphism, showing that the category of topological spaces is the quotient category of the category of spatial fibrous preorders, obtained by iden- tifying two fibrous morphisms whenever they have the same underlying map. The examples show that this notion provides a convenient setting to work with the intuitive notion of base of open neighbourhoods. However, as explained in the introduction, the main motivation that leads to the definition of fibrous preorder is the purpose of finding a purely categorical notion of topological space. In a future work we plan to specify the internal version of a fibrous pre- order, by replacing the relation R ✓ A ⇥ B with a jointly monomorphic pair of morphisms and by giving the appropriate translation of axioms (F1)-(F3)<br/>","234 N. MARTINS-FERREIRA<br/>and (F4)-(F6). In particular, the additional structure of spatial fibrous pre- order is nothing but a comonoid structure in the monoidal category of fibrous preorders and fibrous morphisms, with an appropriate tensor product. This means that a fibrous preorder (R,A,B,@,p) is spatial if the canonical span<br/>B ⇥ B oo   B   // 1 can be lifted to fibrous preorders. Further studies will then take place in FibPreord(C) and SpFibPreord(C) for an arbitrary ca- tegory C with finite limits. For instance if C is the category of finite sets then Preord(C) ' SpFibPreord(C) ' FibPreord(C), as it follows easily from Proposition 2.6.<br/>Another possible application for the notion of (spatial) fibrous preorder is in the characterization of descent and e↵ective descent morphisms in the category of topological spaces and other kinds of categories of spaces. As the results of Manuela Sobral and her collaborators [3, 6, 7, 8, 11, 9, 12, 13, 14] show, there are some advantages in considering the notions of topological spaces as generalized preorders.<br/>Acknowledgements<br/>The author is indebted to George Janelidze and Manuela Sobral for their support during the period of maturing of the ideas presented in this work. Thanks are also due to the anonymous referee for many helpful contributions; the example of pre-topological spaces, in particular, is due to him or her.<br/>References<br/>[1] P. Alexandrov, Diskrete Ra¨ume, Mat. Sbornik (N.S.) 2 (1937) 501–518.<br/>[2] F. Borceux and D. Bourn, Mal’cev, Protomodular, Homological and Semi-Abelian Cat-<br/>egories, Math. Appl. 566, Kluwer, 2004.<br/>[3] M. Dias and M. Sobral, Descent for Priestley Spaces, Applied Categorical Structures 14<br/>(2006) 229–241.<br/>[4] M. Ern´e, The ABC of order and topology, in: Category Theory at Work (ed. by H.<br/>Herrlich and H.-E. Porst), Heldermann Verlag, Berlin, 1991, 57–83.<br/>[5] G. Gutierres and N. Martins-Ferreira, A relational approach to the theory of topological<br/>groups and other algebraic structures, in preparation.<br/>[6] G. Janelidze and M. Sobral, Finite preorders and Topological descent I, Journal of Pure<br/>and Applied Algebra 175 (2002) 187–205.<br/>[7] G. Janelidze and M. Sobral, Finite preorders and Topological descent II: ´etale descent,<br/>Journal of Pure and Applied Algebra 174 (2002) 303–309.<br/>[8] G. Janelidze and M. Sobral, Profinite relational structures, Cahiers de Topologie et<br/>G´eom´etrie Di↵´erentielle Cat´egoriques 49 (2008) 280–288.<br/>[9] G. Janelidze and M. Sobral, Descent for compact 0-dimensional spaces, Theory and<br/>Applications of Categories 21 (2008) 182–190.<br/>[10] J. Picado and A. Pultr, Frames and Locales: Topology without points, Frontiers in Math-<br/>ematics, Springer Basel, 2012.<br/>","FROM A-SPACES TO ARBITRARY SPACES VIA SPATIAL FIBROUS PREORDERS 235<br/>[11] J. Reiterman, M. Sobral and W. Tholen, Composites of e↵ective descent maps, Cahiers de Topologie et G´eom´etrie Di↵´erentielle Cat´egoriques 34 (1993) 193–207.<br/>[12] M. Sobral, Descent for Discrete (Co)fibrations, Applied Categorical Structures 12 (2004) 527–535.<br/>[13] M. Sobral, Another approach to topological descent theory, Applied Categorical Struc- tures 9 (2001) 505–516.<br/>[14] M. Sobral, Some aspects of topological descent, Applied Categorical Structures 4 (1996) 97–106.<br/>[15] M. Sobral and W. Tholen, E↵ective descent morphisms and e↵ective equivalence rela- tions, Canadian Mathematical Society Conference Proceedings (AMS, Providence 1992) 421–431.<br/>(N. Martins-Ferreira) Escola Superior de Tecnologia e Gesta˜o, Centro para o Desenvolvimento Ra´pido e Sustentado do Produto, Instituto Polite´cnico de Leiria, 2411 - 901 Leiria, Portugal<br/>E-mail address: martins.ferreira@ipleiria.pt<br/>","","","","","","","","","244 A. MONTOLI, D. RODELO, AND T. VAN DER LINDEN<br/>(A. Montoli) CMUC, Universidade de Coimbra, 3001–501 Coimbra, Portugal E-mail address: montoli@mat.uc.pt<br/>(D. Rodelo) CMUC, Universidade de Coimbra, 3001–501 Coimbra, Portugal, and Departamento de Matema´tica, Faculdade de Cieˆncias e Tecnologia, Universi- dade do Algarve, Campus de Gambelas, 8005–139 Faro, Portugal<br/>E-mail address: drodelo@ualg.pt<br/>(T. Van der Linden) Institut de Recherche en Mathe´matique et Physique, Univer- site´ catholique de Louvain, chemin du cyclotron 2 bte L7.01.02, B–1348 Louvain- la-Neuve, Belgium<br/>E-mail address: tim.vanderlinden@uclouvain.be<br/>","","","","","","","","","","","","","","258 A. P. SANTANA AND I. YUDIN<br/>[4] E. Cline, B. Parshall, and L. Scott. Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math., 391:85–99, 1988.<br/>[5] E. Cline, B. Parshall, and L. Scott. Stratifying endomorphism algebras. Mem. Amer. Math. Soc., 124(591):viii+119, 1996.<br/>[6] S. Donkin, A. P. Santana, and I. Yudin. Homological properties of quantised Borel-Schur algebras and resolutions of quantised Weyl modules. J. Algebra, 402:120–157, 2014.<br/>[7] W. Geigle and H. Lenzing. Perpendicular categories with applications to representations<br/>and sheaves. J. Algebra, 144(2):273–343, 1991.<br/>[8] G. Hochschild. Relative homological algebra. Trans. Amer. Math. Soc., 82:246–269,<br/>1956.<br/>[9] S. Mac Lane. Homology. Springer-Verlag, Berlin-New York, first edition, 1967. Die<br/>Grundlehren der mathematischen Wissenschaften, Band 114.<br/>[10] A. P. Santana and I. Yudin. Characteristic-free resolutions of Weyl and Specht modules.<br/>Adv. Math., 229(4):2578–2601, 2012.<br/>[11] J. Sz´ep. On the structure of groups which can be represented as the product of two<br/>subgroups. Acta Sci. Math. Szeged, 12 (Leopoldo Fejer et Frederico Riesz LXX annos<br/>natis dedicatus, Pars A):57–61, 1950.<br/>[12] G. Zappa. Sulla costruzione dei gruppi prodotto di due dati sottogruppi permutabili tra<br/>loro. In Atti Secondo Congresso Un. Mat. Ital., Bologna, 1940, pages 119–125. Edizioni Cremonense, Rome, 1942.<br/>(A. P. Santana and I. Yudin) CMUC, Department of Mathematics, University of Coimbra, Coimbra, Portugal<br/>E-mail address: aps@mat.uc.pt; yudin@mat.uc.pt<br/>","ON THE LOCALNESS OF THE EMBEDDING OF ALGEBRAS<br/>LURDES SOUSA<br/>Dedicated to Manuela Sobral<br/>Abstract. Let N : A ! B be a faithful functor between categories. Given an object B of B, we may ask whether there is an embedding B ! NA with A 2 A. In some cases the answer is well known. For instance, an abelian semigroup may be embedded in an abelian group if and only if it is cancellative. And every Lie algebra over a field K is embeddable in an associative K-algebra with identity. Many other examples are known. This paper concentrates on the localness of the embeddability. That is, it studies conditions under which the following property holds: B 2 B is embeddable in NA for some object A of A, whenever every finitely generated subobject of B is so.<br/>1. Introduction<br/>The following problem has been investigated for various algebraic categories: Let N : A ! B be a faithful functor; given an object B of B, determine if there is a monomorphism B ,! NA with A 2 A. The following two results on this subject are well known:<br/>(a) An abelian semigroup may be embedded in an abelian group if and only if it is cancellative.<br/>(b) Poincar´e-Birkho↵-Witt Theorem: Every Lie algebra over a field K is embeddable in an associative K-algebra with identity.<br/>There are many other examples on the embeddability of algebras in the literature. J. MacDonald studied the subject from a categorical point of view<br/>Received: 30 September 2014 / Accepted: 8 January 2015.<br/>2010 Mathematics Subject Classification. 18B15, 03C05, 18C05.<br/>Key words and phrases. Embedding theorems, categories of algebras, finitely generated<br/>subobjects.<br/>This work was partially supported by the Centro de Matema´tica da Universidade de<br/>Coimbra (CMUC), funded by the European Regional Development Fund through the pro- gram COMPETE and by the Portuguese Government through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia under the projects PEst-C/MAT/UI0324/2013 and MCANA PTDC/MAT/120222/2010.<br/>259<br/> ","","","","","","ON THE LOCALNESS OF THE EMBEDDING OF ALGEBRAS 265<br/>Proof. Let C be an object of C with local A-behaviour. Consider a quasi- identity of Q,<br/>(ui(x) = vi(x), i = 1,...,k) ) (u(x) = v(x)), (2.6)<br/>where x = (x1, . . . , xn), and ui(x), vi(x), u(x) and v(x) are terms on the vari- ables x1,...,xn. Given C 2 C, let c1,...,cn 2 C, and put c = (c1,...,cn). We write<br/>uC (c)<br/>for denoting the element of C obtained from u(x) by replacing every xi by ci, and every operation symbol ✓ 2 ⌃ by the operation ✓C .<br/>Suppose that<br/>uCi (c) = viC(c), i = 1,...,k. We want to prove that then uC (c) = vC (c). Put<br/>X={c1,...,cn}[{uCi (c),i=1,...,k}[{uC(c),vC(c)}.<br/>By hypothesis we have a commutative diagram as in (2.5). Without loss of<br/>generality, we may assume that d is an inclusion map. Then,<br/>h(uDi (c)) = uCi (h(c1), ..., h(cn)) = uCi (c1, ..., cn) = uCi (c),<br/>and, analogously, h(viD(c)) = viC(c). Consequently, h(uDi (c)) = h(viD(c)). Con- sider the subset<br/>Z=X[{uDi (c),i=1,...,k}[{viD(c),i=1,...,k}[{uD(c),vD(c)}<br/>of UD, and let f : D ! A be as in (ii) of Definition 2.6. By hypothesis,<br/>ker(Uf·s)=ker(Uh·s),andsotheequalityh(uDi (c))=h(viD(c))implies f(uDi (c))=f(viD(c)).<br/>And then, since f is a homomorphism,<br/>uAi (f(c1), ..., f(cn)) = viA(f(c1), ..., f(cn)), i = 1,...,k.<br/>Hence, since A satisfies the given quasi-identity (2.6), uA(f(c1), ..., f(cn)) = vA(f(c1), ..., f(cn)).<br/>This is the same as f(uD(c)) = f(vD(c)). But, again by the fact that ker(Uf · s) = ker(U h · s), this implies that<br/>h(uD(c)) = h(vD(c)).<br/>That is, taking into account the commutativity of (2.5), where h is a homo-<br/>morphism and the other two maps are inclusion maps, uC(c) = vC(c).<br/>⇤<br/>","266 L. SOUSA<br/>3. Main result<br/>Before stating and proving the main result, we need some properties on nontrivial right adjoints over Set. (By nontrivial we mean that there is some C 2 C such that UC has at least two elements.) In particular, we will see that, for faithful nontrivial adjunctions (F,U,⌘,\") : Set ! C, if f : X ! Y is a monomorphism of Set, then the square<br/>X ⌘X //UFX (3.1) f UFf<br/>✏✏ ⌘Y // ✏✏<br/>Y UFY<br/>is a pullback. In the terminology of [5], this means that every monomorphism f:X!Y issplitovertheidentitymorphismidY :Y !Y.<br/>Part (a) of the following lemma is showed in Manes [10] (Proposition 5.2 and Proposition 5.42).<br/>Lemma 3.1. Let (F,U,⌘,\") : Set ! C be a nontrivial adjunction with U faithful. Then:<br/>(a) The unit ⌘ is pointwise injective and F preserves monomorphisms.<br/>(b) Every monomorphism f : X ! Y of Set is split over the identity mor-<br/>phism idY .<br/>Proof. (a) Given X 2 Set, and two di↵erent elements x,y 2 X, let C be an object of C such that UC has at least two elements, a and b. Define h : X ! UC byh(x)=aandh(z)=bforallz6=x.Nowleth# bethemorphisminCsuch that Uh# · ⌘X = h. Since h(x) 6= h(y), then ⌘X(x) 6= ⌘X(y).<br/>Let now m : X ! Y be an injective map. If X 6= ;, then m is a split monomorphism, thus the same is true for Fm. If X = UFX = ;, UFm is a monomorphism since it has empty domain, and then Fm is a monomorphism since U being faithful reflects isomorphisms. If X = ; and UFX 6= ;, consider the diagram<br/>; ⌘; //UF; << OO<br/>         m tUFm<br/>✏✏ //✏✏<br/>Ut# Y ⌘Y UFY<br/> where t is any map from Y to UF;. Then we have U(t#Fm)⌘; = Ut#·⌘Y ·m = tm = ⌘;. Thus t#Fm = idF;, so Fm is a monomorphism.<br/>","","","ON THE LOCALNESS OF THE EMBEDDING OF ALGEBRAS 269<br/>Thus, EY is contained in X; let i : EY ! X be the inclusion map. And let a : X ! Y be the morphism such that sa = ⌘X. Then we have that UFj·UFi·nY ·a = UFdY ·nY ·a = UFj·s·a = UFj·⌘X. Since j is a monomorphism, so is UFj, and, thus, UFi·nY ·a = ⌘X. By (b) of Lemma 3.1, the square part of the commutative diagram<br/>X<br/>  t nYa<br/>!!E ))// UFE<br/>idX Y ⌘EY ⇠⇠ ✏✏<br/>Y<br/>i UFi<br/>   // ✏✏<br/>X ⌘X UFX<br/> isapullback.Hence,thereisamapt:X!EY withit=idX.Sinceiisan<br/>inclusion, we conclude that X = EY . Theorem 3.3. Let<br/>N<br/>C<br/>sumptions of Section 2. Moreover, assume that:<br/>(H0) U is nontrivial, preserves directed colimits, and has an intersection preserving left adjoint F;<br/>(H1) U creates U-separated epimorphisms;<br/>(H2) U0 locally detects B-morphisms;<br/>(H3) A contains all objects in C with local A-behaviour.<br/>Then, in B, an object B is a subobject of some object NA with A 2 A, whenever every finitely generated subobject of B is so.<br/>Proof. Let B 2 B be such that every finitely generated subobject of B is a subobject of some object NA with A 2 A.<br/>1. Involving an inverse limit of nonempty finite sets. Let mX :X,!UFU0B, X2F,<br/>A ✏ o<br/>be a commutative diagram of categories and functors satisfying General As-<br/>// ??B<br/>M !!<br/>⇤<br/>    L<br/>U0<br/>U // Set<br/> ","","","","","","ON THE LOCALNESS OF THE EMBEDDING OF ALGEBRAS 275<br/>[8] J. MacDonald, Conditions for a universal mapping of algebras to be a monomorphism, Bull. Amer. Math. Soc. 80 (1974), 888-892.<br/>[9] J. MacDonald, Embeddings of Algebras, Galois theory, Hopf algebras, and semiabelian categories, Fields Institute Communications 43, Amer. Math. Soc., 2004, 359-373.<br/>[10] E. Manes, Algebraic Theories, Springer-Verlag, 1976.<br/>[11] B. H. Neumann, An embedding theorem for algebraic systems, Proc. London Math. Soc.<br/>(3) 4 (1954), 138-153.<br/>(L. Sousa) CMUC, University of Coimbra, Portugal, and<br/>ESTV, Polytechnic Institute of Viseu, 3504-510 Viseu, Portugal E-mail address: sousa@estv.ipv.pt<br/>","","TEXTOS DE MATEMA´TICA<br/>1 J.A. GREEN. Classical Invariants, 1993.<br/>2 E.M. SA´. Interlacing Problems for Invariant Factors, 1998.<br/>3 J.A. GREEN. Classical Groups, 1995.<br/>4 J.A. GREEN. Hall Algebras and Quantum Groups, 1994.<br/>5 W.TUTSCHKE.InhomogeneousEquationsinComplexAnalysis,1995.<br/>6 W. BERGWEILER. An Introduction to Complex Dynamics, 1995.<br/>7 J. CNOPS, H. MALONEK. An Introduction to Cli↵ord Analysis, 1995.<br/>8 P. MELLON. An Introduction to Several Complex Variables, 1995.<br/>9 J.A. GREEN. Shu✏e Algebras, Lie Algebras and Quantum Groups,<br/>1995.<br/>10 F. CRAVEIRO DE CARVALHO (ed.). Dia de Topologia e Geometria,<br/>Coimbra, May 21, 1996.<br/>11 F.A. OLIVEIRA, P. OLIVEIRA, M.F. PATR´ICIO, J.A. FERREIRA.<br/>(eds.). First Meeting on Numerical Methods for Partial Di↵erential<br/>Equations, Coimbra, 25-27 September 1995, 1997.<br/>12 B. BANASCHEWSKI. The Real Numbers in Pointfree Topology, 1997.<br/>13 A.P. MOURO, C. VREUGDENHIL. Numerical Methods for Advec-<br/>tion-Dominated Problems, 1998.<br/>14 J. DE GRAAF. Evolution Equations, 1998.<br/>15 J. DE GRAAF. Evolution Equations in Harmonic Function Spaces,<br/>1998.<br/>16 R. HEERSINK. Initial Value Problems in Scales of Banach Spaces,<br/>1998.<br/>17 E. WEGERT. Complex Methods for Boundary Value Problems I, 1999.<br/>","18 L. VON WOLFERSDORF. Complex Methods for Boundary Value Problems II, 1999.<br/>19 A.P. SANTANA, A.L. DUARTE, J.F. QUEIRO´ (eds.). Matrices and Group Representations, Coimbra, 6-8 May 1998, 1999.<br/>20 M.J.H. ANTHONISSEN, J. DE GRAAF. Hopper’s Shape Evolution Equation, 1999.<br/>21 M. SOBRAL, M.M. CLEMENTINO, J. PICADO, L. SOUSA (eds.). School on Category Theory and Applications, Coimbra, July 13-17, 1999.<br/>22 A. DURA´N. Symmetries of Di↵erential Equations and Numerical Ap- plications, 1999.<br/>23 I. VAISMAN. Lectures on Symplectic and Poisson Geometry (Notes edited by F. Petalidou), 2000.<br/>24 J. VAILLANT, J. CARVALHO E SILVA (eds.). Op´erateurs Di↵´eren- tiels et Physique Math´ematique, 2000.<br/>25 P. MELLON. An Introduction to Bounded Symmetric Domains, 2000.<br/>26 W. SPRO¨ßIG, K. GU¨RLEBECK. Cli↵ord Analysis and Boundary<br/>Value Problems of Partial Di↵erential Equations, 2000.<br/>27 O. MARTIO. Modern Tools in the Theory of Quasiconformal Maps,<br/>2000.<br/>28 N.J. WILDBERGER. Algebraic Structures Associated to Group Ac-<br/>tions and Sums of Hermitian Matrices, 2001.<br/>29 E.H. TWIZELL (ed.). Aspects of Computational Bio-Mathematics.<br/>(Notes compiled by P. M. Rosa), 2001.<br/>30 A.M. ALMEIDA, R. RODRIGUES. Classificac¸˜ao e Complexidade de<br/>Problemas, 2001.<br/>31 G.THORBERGSSON.GeometryofSubmanifoldsinEuclideanSpaces,<br/>2001.<br/>","32 H.ALBUQUERQUE,R.CASEIRO,F.J.CRAVEIRODECARVALHO, J.M. NUNES DA COSTA (eds.). The J. A. Pereira da Silva Birthday Schrift, 2002.<br/>33 R. W. CARTER. Representations of the Monster, 2002.<br/>34 A.J.G. BENTO, A.M. CAETANO, S.D. MOURA, J.S. NEVES (eds.).<br/>The J. A. Sampaio Martins Anniversary Volume, 2004.<br/>35 F. BORCEAUX. Algumas Teorias de Galois dos Corpos e dos An´eis,<br/>2004.<br/>36 G. JANK. Matrix Riccati Di↵erential Equations, 2005.<br/>37 R.W. CARTER. Cluster Algebras, 2006.<br/>38 R. KAHLE, I. OITAVEM (Eds.). Days in Logic ’06:Two Tutorials,<br/>2006.<br/>39 O. AZENHAS, A.L. DUARTE, J.F. QUEIRO´ , A.P. SANTANA (eds.).<br/>Mathematical Papers in Honour of Eduardo Marques de S´a, 2006.<br/>40 R. KAHLE. The applicative realm, 2007.<br/>41 J. PICADO, A. PULTR. Locales Treated Mostly in a Covariant Way,<br/>2008.<br/>42 J. BURESˇ, R. LA´VICˇKA, V. SOUCˇEK . Elements of Quaternionic<br/>Analysis and Radon Transform, 2009.<br/>43 J. CARDOSO, K. HU¨PER, P. SARAIVA (eds.). Mathematical papers<br/>in honour of F´atima Silva Leite, 2011.<br/>44 C. FONSECA, S. FURTADO, A. KOVACˇ EC, CHI-KWONG LI (eds.).<br/>The Nat´alia Bebiano Anniversary Volume, 2013.<br/>45 DOMINIQUE BOURN, NELSON MARTINS-FERREIRA, ANDREA<br/>MONTOLI AND MANUELA SOBRAL (eds.). Schreier split epimor- phisms in monoids and in semirings, 2013.<br/>","","","                                "];