THE HOMOTOPY THEORY OF INVERSE SEMIGROUPS

We show that abstract homotopy theory can be used to define a suitable notion of homotopy equivalence for inverse semigroups. As an application of our theory, we prove a theorem for inverse semigroup homomorphisms which is the exact counterpart of the well-known result in topology which states that every continuous function can be factorized into a homotopy equivalence followed by a fibration. We show that this factorization is isomorphic to the one constructed by Steinberg in his "Fibration Theorem", originally proved using a generalization of Tilson's derived category.