A general formulation of homotopy limits

Abstract

The problem of homotopy coherence has occurred recently in two contexts: explicitly in strong shape theory (Edwards-Hastings [ 1 l), Dydak-Segal [lo] etc.) and implicitly in the simplicial localization of Dwyer-Kan (9). In both cases, the authors take into consideration the notion of higher order homotopies (or “homotopy coherences”). One of the first places in which these higher order homotopy coherences have been used is in the study of homotopy limits (in a restricted sense in Bousfield-Kan (61 and Edwards-Hastings [ 11) and more generally Vogt [ 193 and Porter [ 151). These homotopy limits then appear naturally in strong shape theory. Coming from another direction and following an article of Thomason [ 181 which sheds light on possible relations between lax limits, and homotopy limits, Gray [ 131 has introduced a generalization of homotopy limits (in the sense of Bousfield-Kan the precise definition will be given later). This latter definition has two imperfections; it cuts off the coherence at level 2 and it does not allow the generalization of the replacement schemes necessary for the development of the analogues of the Bousfield-Kan spectral sequences. Our own work in shape theory [S] and coherence [4,7,8] has led us to study a fresh definition which remedies these defects. This general definition is not however entirely new as a particular case of it already appears in Segal’s paper [ 161. Like Gray we feel that the best presentations of homotopy limits are made in terms of indexed limits. However for us, the natural context for these indexed limits is that of profunctors (sometimes also called distributors) [14, l] (it is in thzse terms, in fact, that the replacement scheme seems most natural). The first section recalls some necessary facts about Bousfield-Kan homotopy limits and indexed limits. From a careful inspectio.1 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 16) 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 01 Segal are particular cases of homotopy limits.