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Chapter 5
Schreier and homogeneous representability
5.1 The Schreier split extension classifier
This section is devoted to showing that, for any monoid X, the monoid End(X) of endomorphisms of X has a universal property similar to the automorphisms group Aut(G) in the category of groups [5, 6, 3], namely that it allows to clas- sify the Schreier split epimorphisms.
Given a monoid X, we denote by Hol(X) the monoid whose underlying set is X × End(X) and whose monoid operation is the following:
(x1, γ1) · (x2, γ2) = (x1 · γ1(x2), γ1γ2)
where the operation in the second component is the usual composition of ho- momorphisms. It is immediate to prove that this operation is associative and that (1,1X) is the identity for it. It is also easy to see that, in this way, we obtain a split epimorphism:
Hol(X) oo ρX // End(X), (5.1.1) θX
where ρX (γ) = (1, γ) and θX is the projection. The symbol Hol stands for holo- morph, since the construction above is analogous to the one of the holomorph in group theory.
Lemma 5.1.1. The split epimorphism (5.1.1) is a Schreier one. Its kernel is, up to isomorphism, the monoid X itself.
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