There is a very well-known result in topology which states that, in the category of topological spaces, every continuous function can be factorized into a homotopy equivalence followed by a fibration, see

*E. H. Spanier* [Algebraic Topology, (McGraw-Hill Series in Higher Mathematics. New York) (1966;

Zbl 0145.43303)]. In this paper, the authors give an analogous factorization for every inverse semigroup homomorphism. To do this, they need to work in the larger category of ordered groupoids and ordered functors,

**OG** (every inverse semigroup can be regarded as an ordered groupoid). The category

**OG** can be endowed with a cocylinder in such a way that the Kan lifting condition

$E\left(2\right)$ holds. A notion of “homotopy equivalence” is defined in

**OG**, and a mapping cocylinder factorization is obtained. Finally, they prove that this factorization (proved using only ideas from homotopy theory) is isomorphic to the one constructed by

*B. Steinberg* in his “Fibration Theorem” [Proc. Edinb. Math. Soc., II. Ser. 44, No. 3, 549–569 (2001;

Zbl 0990.20043)].