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36 Chapter 3. Schreier internal relations, categories and groupoids
(1) d0d1 = d0d0, d1d1 = d1d2 (incidence axioms)
(2) d1s0 = 1X1 , d1s1 = 1X1 (composition with identities)
This composition must satisfy the associativity axiom; for that consider the following pullback of split epimorphisms:
X3 OO
oo s0
// X2 OO
(3.2.3)
d0
d3 s2 d2 s1
oos0
X2
// X1
d0
The composition map d1 induces a couple of maps (d1,d2): X3 ⇒ X2 such that d0d1 = d0d0, d2d1 = d1d3 and d0d2 = d1d0, d2d2 = d2d3. The associativity is given by the remaining simplicial axiom
(3) d1d1 = d1d2.
So an internal category in E is a 3-truncated simplicial object such that the two squares above are pullbacks. Moreover, given an internal category X1, consider the following diagram:
d0 // Dec1X1 : X2 oo s0 // X1
ε1X1 d2
(3.2.4)
d1 X1: X1oo s0 // X0
d1 d0
//
then the upper reflexive graph is still a category denoted by Dec1X1, while the vertical diagram determines an internal functor denoted by ε1X1 (see the Appendix). According to the pullback defining the object X2, when we work in Mon, the upper category is a Schreier one as soon as so is the lower one.
According to Patchkoria [28], we give the following:
Definition 3.2.1. An internal category in the category of monoids
d0 // X1oos0 //X0
d1
is a Schreier internal category if the split epimorphism (d0,s0) is a Schreier
one. It is a homogeneous internal category if (d0,s0) is homogeneous.
Again, as consequence of Proposition 2.3.3, we have that Schreier (resp. homogeneous) internal categories are closed under finite products inside the
d1