Page 35 - Textos de Matemática Vol. 45

2.4. The fibrations of Schreier and homogeneous points 27
so we have to show that:
lqw(vw(u) · qw(u¯)) · n(u′) = n(u′) · lqw(u¯)
First notice that w(u) = w′(u′) since (u,u′) in U ×B U′, so we have to show that:
lqw(vw′(u′) · qw(u¯)) · n(u′) = n(u′) · lqw(u¯)
We have:
lqw(vw′(u′) · qw(u¯)) · n(u′) = lqw(vw′(u′) · qw(u¯)) · nqw′ (u′) · nv′w′(u′) = (∗).
According to the assumption lqw(vw′(u′)·qw(u¯)) and nqw′ (u′) commute. Whence (∗) = nqw′ (u′) · lqw(vw′(u′) · qw(u¯)) · nv′w′(u′) =
= nqw′ (u′) · lqw(vw′(u′) · qw(u¯)) · lvw′(u′) =
= nqw′ (u′) · l(qw(vw′(u′) · qw(u¯)) · vw′(u′)) = nqw′ (u′) · l(vw′(u′) · qw(u¯)) = = nqw′ (u′) · nv′w′(u′) · lqw(u¯) = n(qw′ (u′) · v′w′(u′)) · lqw(u¯) = n(u′) · lqw(u¯).

We noticed that the fiber SPtB(Mon) is unital. Accordingly, there is, inside this fiber, a natural notion of internal commutative object.
Corollary 2.4.7. A Schreier split epimorphism Aoo s ////B,
f
seen as an object in the fiber P tB (M on), is commutative if and only if the kernel K[f] of f is a commutative monoid. Accordingly any Schreier split epimorphism has at most one (commutative) monoid structure in this fiber.
Proof. As we already observed in the proof of the previous theorem, the ker- nel functor preserves finite limits. Consequently, it preserves the commutative objects. So that, when the Schreier split epimorphism is commutative, so is its kernel. The converse comes from the reflection of commutativity for Schreier split epimorphisms. The last point comes from the fact that, in a unital cate- gory, an object is commutative if and only if it has an internal monoid structure (see for example [4], Theorem 1.4.5).
Proposition 2.4.8. Suppose that the Schreier split epimorphism (A,B,f,s) is commutative in the fiber SPtB(Mon). Then the unique possible multiplication is:
m(a,a′)=q(a)·a′, for (a,a′)∈R[f].