Jean-Marc Cordier

A general formulation of homotopy limits

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.

Categorical aspects of equivariant homotopy

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.

Fibrant diagrams, rectifications and a construction of loday

Abstract The notion of a fibrant diagram was introduced by Edwards and Hastings. The limit of a fibrant diagram coincides with its homotopy limit. A process of rectification of homotopy coherent diagrams has been introduced by the authors. The homotopy limit of a coherent diagram is the limit of its rectification. In this paper we show that the rectified diagrams are fibrant and we give applications to homotopy limits and also to a construction of a cat n -group from a homotopy coherent n -cube of spaces.

Pattern recognition and categorical shape theory

Abstract This article sets out to interpret the simpler elements of categorical shape theory in terms of some of the basic theoretical problems of pattern recognition.

Functors between shape categories

Categorical shape theory was introduced in a series of articles by Deleanu, Hilton and Frei ([5], [6], [7] and (81). It provided not only a convenient language for handling purely categorical questions which arose in shape theory, but also fitted very neatly into several categorical areas, notably the study of Kan extensions (cf. Frei-Kleisli [9] and [lo]). In this note we give some methods for inducing functors between shape categories. A preliminary version of these constructions appeared as part of our 1978 Esquisses notes [3], however as there have been several advances since that time (notably arising from the use of distributors in [2]) and as those notes had an extremely limited distribution, it has seemed advisable to write an improved version. We include illustrative examples which show that many of the functors arising naturally in situations involving monads can be interpreted as shape induced functors. This is in line with the examples given by Deleanu, Hilton and Frei which calculate the shape category of a functor having a left adjoint.