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3.2. Schreier internal categories 37 category of internal categories.
Actually the notion of Schreier internal category in Mon was already introduced, under the name of homogeneous category, in [22], where it was proved to be equivalent to the notion of crossed module in Mon.
Proposition 3.2.2. On a Schreier reflexive graph there is at most one structure of category. It is sufficient to have the composition map d1 : X2 → X1 with axiom (2), the axioms (1) and (3) come for free.
Proof. When the reflexive graph is a Schreier one, the pullback 3.2.2 is actually a product in the fiber P tX0 (M on). Since the lower horizontal split epimorphism is a Schreier one, the pair (s0,s1) is jointly epimorphic, because PtX0(Mon) is SP tX0 -unital. Accordingly there is a unique possible map satisfying the axioms (2). Actually when we have a map d1 satisfying axiom (2), we can check axiom (1) by composition with the same pair (s0,s1). The pullback 3.2.3 is such that the lower split epimorphism is a Schreier one, and hence the pair (s0,s2) is jointly epimorphic. We get axiom (3) by composition with this pair.
So, given a Schreier reflexive graph, being an internal category is a prop- erty and not an additional structure.
Proposition 3.2.3. Given a Schreier reflexive graph: d0 //
with q: X1 → K[d0] its associated Schreier retraction, it is an internal category if and only if, for any pair (σ, α) ∈ X1 × K[d0], we have:
q(s0d1(σ) · α) · σ = σ · α.
When it is the case, the composition of a composable pair (τ,σ) is given by:
d1(τ,σ)=q(τ)·σ, which implies that K[d0] and K[d1] commute in X1. Proof. Since, for any (τ,σ) ∈ X2, we have (τ,σ) = (q(τ),1)) · (s0d0(τ),σ) the
compositioninthecategoryX1 mustbed1(τ,σ)=q(τ)·σ.Indeed: d1(τ,σ) = d1(q(τ),1)·d1(s0d0(τ),σ) = q(τ)·d1(s0d1(σ),σ) = q(τ)·σ.
On the other hand, in X2, when α is in K[d0], we have that (s0d1(σ), σ) · (α, 1) = (s0d1(σ) · α, σ)
and hence
q(s0d1(σ) · α) · σ = d1(s0d1(σ) · α, σ) = d1(s0d1(σ), σ) · d1(α, 1) =
X1oos0 //X0 d1