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40 Chapter 3. Schreier internal relations, categories and groupoids
Example 3.3.4. An interesting particular case of the Schreier reflexive graph described in the previous example is given when L = M is a commutative monoid and d = 1M ; in this case, the codomain morphism ⟨1, d⟩ is nothing but the multiplication in M:
π1 //
M × M oo ⟨1,0⟩ // M.
·M
The following proposition is related to a known result about internal cat-
egories and groupoids in the category Gp of groups (see [23]). d0 //
Proposition 3.3.5. Let X1 oo s0 // X0 be a Schreier reflexive graph such that d1
K[d0] is a group. The following conditions are equivalent: (a) the kernels of d0 and d1 cooperate in Mon;
(b) the reflexive graph is an internal category;
(c) the reflexive graph is an internal groupoid.
Proof. We already proved (see the end of the proof of Proposition 3.2.3) that, if a Schreier reflexive graph is an internal category, then the kernels of the do- main and of the codomain cooperate. Hence (b) implies (a).
Conversely, suppose that we have a Schreier reflexive graph
oo q k
d0 //
// X1 oo s0 // X0,
K[d0]
with the Schreier split epimorphism associated with the codomain d1 given by
oo t oo s0 K[d1] // X1
// X0,
such that k and l cooperate (we recall that (d1,s0) is a Schreier split epimor-
phism thanks to Proposition 3.1.13). This means that x·y=y·x forallx∈K[d0], y∈K[d1].
We first show that the map defined by
m(a′, a) = q(a′) · a,
d1
l d1