Let Top be the category of compactly generated topological spaces and continuous maps. The category, Top, can be given the structure of a simplicially enriched category (or S-category, S being the category of simplicial sets). For A a small category, Vogt (in [22]) constructed a category, Coh (A, Top), of homotopy coherent A-indexed diagrams in Top and homotopy classes of homotopy coherent maps, and proved a theorem identifying this as being equivalent to Ho (TopA), the category obtained from the category of commutative A-indexed diagrams by localizing with respect to the level homotopy equivalences. Thus one of the important consequences of Vogt's result is that it provides concrete coherent models for the formal composites of maps and formal inverses of level homotopy equivalences which are the maps in Ho (TopA). The usefulness of such models and in general of Vogt's results is shown in the series of notes [14–17] by the second author in which those results are applied to give an obstruction theory applicable in prohomotopy theory.
Abstract. This article is an introduction to the categorical theory of homotopy coherence. It is based on the construction of the homotopy coherent analogues of end and coend, extending ideas of Meyer and others. The paper aims to develop homotopy coherent analogues of many of the results of elementary category theory, in particular it handles a homotopy coherent form of the Yoneda lemma and of Kan extensions. This latter area is linked with the theory of generalised derived functors.In homotopy theory, one often needs to "do" universal algebra "up to homotopy" for instance in the theory of homotopy everything H-spaces. Universal algebra nowadays is most easily expressed in categorical terms, so this calls for a form of category theory "up to homotopy" (cf. Heller [34]).In studying the homotopy theory of compact metric spaces, there is the classically known complication that the assignment of the nerve of an open cover to a cover is only a functor up to homotopy. Thus in shape theory, [39], one does not have a very rich underlying homotopy theory. Strong shape (cf. Lisica and Mardešić, [37]) and Steenrod homotopy (cf. Edwards and Hastings [27]) have a richer theory but at the cost of much harder proofs. Shape theory has been interpreted categorically in an elegant way. This provides an overview of most aspects of the theory, as well as the basic proobject formulation that it shares withétale homotopy and the theory of derived categories. Can one perform a similar process with strong shape and hence provide tools for enriching that theory, rigidifyingétale homotopy and enriching derived categories? We note, in particular, Grothendieck's plan for the theory of derived categories, sketched out in 'Pursuing Stacks', [31], which bears an uncanny ressemblance to the view of homotopy theory put forward by Heller in [34].Grothendieck's 'Pursuing Stacks' program, in fact, again raises the spectre of doing categorical construction 'up to homotopy', as his image of a stack is as a sheaf 'up to homotopy' in which the stalks are algebraic models of homotopy types and the whole object has geometric meaning.Problems of homotopy coherence also arise naturally in studying equivariant homotopy theory. If G is a group then the equivariant homotopy of G-complexes can be studied in a useful way by translating to a category of diagrams indexed by the orbit category of G, that is the full subcategory of G-sets determined by If however G is a topological group then this category is more naturally considered to be simplicially enriched and the equivariant homotopy theory of the G-complexes ends up benefitting from results on homotopy coherence to handle the bar-resolutions etc. used to translate from the equivariant setting to the category of diagrams; see [24] and the references it contains. Again the equivariant theory looks like the discrete case but with the categorical arguments done 'up to homotopy'.In this paper we try to lay some of the foundations of such a theory of categories 'up to homotopy' or more exactly 'up to coherent ...
We use the language of homotopy coherent ends and coends, and of homotopy coherent Kan extensions, to give enriched versions of results of Elmendorff. This enables a description of the homotopy type of the space of maps between two G-complexes to be given. (1991). 18G55, 18D20, 55P91. Mathematics Subject Classifications
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