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Vogt’s theorem on categories of homotopy coherent diagrams. (English) Zbl 0603.55017

Let Top be the category of compactly generated topological spaces and continuous maps and A a small category. R. M. Vogt [cf. Math. Z. 134, 11-52 (1973; Zbl 0276.55006)] constructed a category Coh(A,Top) of homotopy coherent A-diagrams and proved that it is equivalent to the homotopy category \(Ho(Top^{{\mathcal A}})\) of A-diagrams in Top localized at the level homotopy equivalences.
The authors simplify Vogt’s proof using simplicially enriched methods. Based on the first author’s simplicial description of Coh(A,Top) [cf. Cah. Topologie Géom. Différ. 23, 93-112 (1982; Zbl 0493.55009)] they show Vogt’s result to be valid for any locally Kan simplicially enriched category B. Thus giving a reasonably concrete model for the localized category \(Ho(B^{{\mathcal A}})\) of A-diagrams in B. The methods involve a detailed examination of weak Kan conditions on the homotopy coherent nerve of B as introduced by the first author [op. cit.] and a study of certain homotopy coherent Kan extensions.
{Reviewer’s remark: The second named author informs me that J.-M. Cordier has a more powerful result in the case \(B=Top\) giving an isomorphism between \(Ho(Top^ A)\) and a homotopy category of A-diagrams and coherent maps. He gives explicit formulae for the compositions linking his construction with results by Yu. T. Lisitsa and S. Mardešić on strong shape theory [Glas. Mat., III. Ser. 19(39), 335-399 (1984; Zbl 0553.55009)].}
Reviewer: M.Golasiński

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
18D20 Enriched categories (over closed or monoidal categories)
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
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References:

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