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A general formulation of homotopy limits. (English) Zbl 0575.55006

The first section recalls some necessary facts about Bousfield-Kan homotopy limits and indexed limits [A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations (Lect. Notes Math. 304) (1972; Zbl 0259.55004)]. From a careful inspection of the coherence of a homotopy cone, we introduce in Section 2 a general notion of homotopy limits for a simplicial category. We next show that the replacement scheme [op.cit.] holds in this situation and exhibit an indexing for this notion of limit. We give general conditions of existence and study the cases of the two important simplicial categories Cat and Top. In particular we show that lax limits and a construction of G. B. Segal [Topology 13, 293-312 (1974; Zbl 0284.55016)] are particular cases of homotopy limits. In Section 3 we compare this notion with the notion of J. W. Gray [J. Pure Appl. Algebra 19, 127-158 (1980; Zbl 0462.55008)].

MSC:

55P65 Homotopy functors in algebraic topology
55P55 Shape theory
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
55P60 Localization and completion in homotopy theory
55U10 Simplicial sets and complexes in algebraic topology
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