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Zbl 1050.55004
Lawson, Mark V.; Matthews, Joseph; Porter, Tim
The homotopy theory of inverse semigroups. (English)
[J] Int. J. Algebra Comput. 12, No. 6, 755-790 (2002). ISSN 0218-1967

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 {\it 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, {\bf OG} (every inverse semigroup can be regarded as an ordered groupoid). The category {\bf OG} can be endowed with a cocylinder in such a way that the Kan lifting condition $E(2)$ holds. A notion of ``homotopy equivalence'' is defined in {\bf 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 {\it B. Steinberg} in his ``Fibration Theorem'' [Proc. Edinb. Math. Soc., II. Ser. 44, No. 3, 549--569 (2001; Zbl 0990.20043)].
[J. Remedios (La Laguna)]

MSC 2000:: *55P05 Homotopy extension properties, cofibrations
20M18 Inverse semigroups
55U35 Abstract homotopy theory

Keywords: abstract homotopy theory

Citations: Zbl 0145.43303; Zbl 0990.20043

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