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The homotopy theory of inverse semigroups. (English) Zbl 1050.55004
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)].
##### MSC:
 55P05 Homotopy extension properties, cofibrations 20M18 Inverse semigroups 55U35 Abstract homotopy theory; axiomatic homotopy theory
##### Keywords:
abstract homotopy theory