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Zbl 0873.18005
Batanin, Mikhail A.
Categorical strong shape theory. (English)
[J] Cah. Topologie Géom. Différ. Catég. 38, No.1, 3-66 (1997). ISSN 0008-0004

Shape theory was introduced by Borsuk in the late 1960's as a way of extending homotopy theoretic methods to non locally nice compact metric spaces. Mardešić and Segal related Borsuk's methods to categories of inverse systems and thereby extended this theory to compact Hausdorff spaces. A categorisation of the theory was initiated by Bacon, Mardešić and others describing a general setting for shape theory as a study of properties of objects from one category by comparison with objects from another which was often a subcategory. In the original example, the objects were from the category of compact metric spaces and homotopy classes of maps and the comparison was with polyhedra and homotopy classes of maps. This led to a greater understanding of Kan extensions of functors where extra structure was involved, e.g. in cohomology theories (Deleanu and Hilton). Finally {\it D. Bourn} and {\it J.-M. Cordier} [ibid. 21, 161-189 (1980; Zbl 0439.55014)] gave a description of the shape category in terms of a Kleisli category for a monad associated to the distributor/profunctor given by the comparison functor.\par Although shape theory was of great use in diverse areas ranging from geometric topology to functional analysis, it hit technical problems due in part to the lack of limits in the homotopy category of polyhedra. The problem was that the methods used led to an inverse system in this homotopy category and it was not clear how to obtain an inverse system in the category of polyhedra, which would be actually commutative yet sufficiently small so as not to obscure the geometric meaning of its construction. Strong shape theory was introduced about 1975 by Edwards and Hastings, Cathey and Segal, and the reviewer, amongst others. This took into account the homotopy coherence problems encountered in shape theory handling them in various ways using homotopical algebra and allowed for a richer homotopy theory in the resulting structure. {\it Yu. T. Lisitsa} and {\it S. Mardešić} [Glas. Mat., III. Ser. 19(39), 335-399 (1984; Zbl 0553.55009)] then proposed a homotopy coherent technology for handling strong shape.\par Related problems of homotopy coherence had become apparent elsewhere especially in homological algebra, in the theory of derived categories and in étale homotopy theory, areas linked in {\it A. Grothendieck}'s ``Pursuing stacks'' manuscript (600 pages) of 1983. By 1990 it was clear that a categorical description of strong shape would help to clarify a very wide range of interrelated areas not just in shape theory and geometric topology. {\it M. A. Batanin} [Cah. Topologie Géom. Différ. Catégoriques 34, No. 4, 279-304 (1993; Zbl 0793.18005)] attacked the problem of giving categorical descriptions of the categories of coherent prohomotopy theory as described by Lisitsa and Mardešić and in the current article gives a complete categorical description of strong shape theory linking it with $A_\infty$-algebras, strong homotopy algebras and related central areas of algebraic topology.
[T.Porter (Bangor)]

MSC 2000:: *18G55 Nonabelian homotopical algebra
55P55 Shape theory
18C20 Algebras associated with monads

Keywords: shape theory; strong shape theory; compact metric spaces; categories of inverse systems; compact Hausdorff spaces; Kan extensions; shape category; Kleisli category; homotopy category of polyhedra; homotopy coherence problems; homotopical algebra; geometric topology; coherent prohomotopy

Citations: Zbl 0439.55014; Zbl 0553.55009; Zbl 0793.18005

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