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3.2. Schreier internal categories 35 We conclude this section by observing that, in a reflexive graph
d0 // X1oos0 //X0,
d1
the fact that both (d0, s0) and (d1, s0) are Schreier split epimorphisms does not imply that K[d0] is a group. As an example we can take, for any monoid M, the Schreier reflexive graph given by:
M
⟨1,0⟩ p1 //
// M × M oo ⟨0,1⟩ // M,
p1
where both the domain and the codomain are given by the morphism p1 such that p1(x,y) = y, according to the simplicial notation (see the Appendix). Then the two split epimorphisms are both Schreier, but M, which is the kernel of p1, is not necessarily a group. Observe that the image by p1 of any element in K[p1] is invertible in M because it is the unit element. In general, given any Schreier split epimorphism (A, B, f, s), the following is a Schreier reflexive graph:
f //
Aoo s //B.
f
3.2 Schreier internal categories
Let us recall that an internal category X1 in any category E is a reflexive graph:
d0 // X1oos0 //X0
d1
such that the following pullback of split epimorphisms, which defines X2 as the
internal object of the composable pairs
oo s0
X2 OO
// X1 OO
(3.2.2)
d0
d2 s1 d1 s0
oos0
X1
// X0
d0
is endowed with a composition map d1 : X2 → X1 satisfying the remaining simplicial identities (see the Appendix):
