Page 47 - Textos de Matemática Vol. 45

3.3. Schreier internal groupoids 39
Proof. (b) ⇔ (c) is given by Proposition 3.1.12. Suppose X1 is a Schreier groupoid, then Dec1X1 is a Schreier category and we have Dec1X1 ≃ R[d0] since X1 is a groupoid. Accordingly R[d0] is a Schreier equivalence relation. Conversely suppose that X1 is a Schreier category and R[d0] is a Schreier equivalence relation; then consider the following diagram
X s1 //X OO1 OO2
s1
DD d0s0
d0
θ1 // ##
X1 s0
R[d0 ] BB
d0 s0 d0
//
X0 s0 1
X
s0
where any commutative square of split epimorphisms is a pullback. This says that the image by the change-of-base functor s∗0 of θ1 is an isomorphism. Since the two right hand side split epimorphisms are Schreier ones and the functor s∗0:SPtX1(Mon)→SPtX0(Mon)isconservative(seeProposition2.4.1),θ1 is itself an isomorphism; we have then Dec1X1 ≃ R[d0] and X1 is a groupoid.
Example 3.3.3. Let L and M be monoids, and d: L → M be a morphism between them. Suppose that d is central, i.e. d(L) is contained in the center of M (This means that d(l)·m = m·d(l) for any l ∈ L and any m ∈ M). Then we get the following Schreier internal reflexive graph:
πM //
M × L oo ⟨1,0⟩ // M,
⟨1,d⟩
where the morphism ⟨1, d⟩ is defined by ⟨1, d⟩(m, l) = m · d(l). It is a Schreier internal category if and only if L is a commutative monoid. Indeed, thanks to Proposition 3.2.3, we have that the unique possible composition of arrows is givenbythemorphism1M ×(·L):M×L×L→M×L,associatingwithany triple (m, l1, l2) the pair (m, l1 · l2). It is immediate to see that it is actually a morphism if and only if L is commutative. Moreover, according to the previous proposition, it is an internal groupoid if and only if L (which is the kernel of πM ) is a group.
We can also observe that if the Schreier reflexive graph
πM //
M × L oo ⟨1,0⟩ // M.
⟨1,d⟩
is a Schreier reflexive relation, then d is a monomorphism. This implies that L is commutative, and hence the relation is actually a preorder. According to Proposition 3.1.5, it is an equivalence relation if and only if L is a group.