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26 Chapter 2. Schreier and homogeneous split epimorphisms in monoids
hand side squares commute, and, since equalizers in Mon are equalizers in Set, this produces a factorization q which satisfies the conditions of a Schreier retraction for the split epimorphism (φ,σ) and makes it a Schreier split epi- morphism.
We get now the main result of this section:
Theorem 2.4.6. The kernel functor reflects commutativity. Given any pair of maps in PtB(Mon) with their two domains in SPtB(Mon):
U
ff
// OOA oo
w v′
l
v
n 88 U′ && xx w′
sf
B
l and n cooperate in the fiber PtB(Mon) if and only if their images by the kernel functor (which is the change of base functor αB∗ : PtB(Mon) → Mon along the initial map αB : 1 → B) cooperate in Mon.
Proof. Bycommutativityoffinitelimits(see,forexample,[25]formoredetails), the kernel functor is left exact (i.e. it preserves finite limits). This implies that it preserves the cooperating pairs (since the cooperator is constructed using only finite limits). Conversely suppose that the following pair cooperates in M on:
K[w] K(l) // K[f] oo K(n) K[w′]
Let us denote by qw : U → K[w] (resp. qw′ : U′ → K[w′]) the Schreier retraction of the kernel kw (resp. kw′ ). We have to define a cooperator φ: U ×B U′ → A in the fibre PtB(Mon). Let us set φ(u,u′) = lqw(u)·n(u′). We check easily that
φ(u, v′w(u)) = lqw(u) · nv′w(u) = lqw(u) · sw(u) = lqw(u) · lvw(u) = = l(qw(u) · vw(u)) = l(u)
and that
φ(vw′(u′), u′) = lqwvw′(u′) · n(u′) = 1 · n(u′) = n(u′).
It remains to show that φ is a monoid homomorphism when the restrictions of
l and n to the kernels cooperate. We have
φ(u·u¯,u′ ·u¯′)=lqw(u·u¯)·n(u·u¯′)=
while
= l(qw(u) · qw(vw(u) · qw(u¯))) · n(u′) · n(u¯′), φ(u, u′) · φ(u¯, u¯′) = lqw(u) · n(u′) · lqw(u¯) · n(u¯′),