Homotopy Theories for Diagrams of Spaces

We show that the category of diagrams of topological spaces (or simplicial sets) admits many interesting model category structures in the sense of Quillen [8]. The strongest one renders any diagram of simplicial complexes and simplicial maps between them both fibrant and cofibrant. Namely, homotopy invertible maps between such are the weak equivalences and they are detectable by the "spaces of fixed points." We use a generalization of the method for defining model category structure of simplicial category given in [5]. 0. The main results. In [5 and 6] D. Kan and W. Dwyer give a classification theory for diagrams of simplicial sets and define a model category structure for such diagrams. Similar structures are discussed also by [7]. In their work [6] ¿/-diagrams are classified up to a certain weak equivalence called here "objectwise weak equivalence" (see below). This same equivalence plays the role of weak equivalence in the sense of Quillen, in their model category structure. The main aim of the present note is to show that their results can be generalized and applied to yield: (i) a whole collection of model category structures on diagrams of spaces with much stronger notions of weak equivalence—rendering such equivalences homotopy invertible under much milder assumptions, (ii) associated with (i), a classification of P-diagram up to stronger notions of weak equivalence, (iii) a Bredon-like result for detecting homotopy invertible maps of diagrams of topological spaces or simplicial sets (compare [2,4]). For example, in TopD, the category of diagrams of topological spaces, one gets the following direct generalization of Bredon's theorem [2, 5.5]: A map /: |A| -» |F| between realization of diagrams of simplicial sets is a weak equivalence if and only if it has a P-homotopy inverse g: |F| -» |A|. A direction of application for the construction of an equivariant instable Adams spectral sequence for equivariant function complexes is discussed briefly in the last section. 0.1. Types of weak equivalences. Let us now briefly recall the concept of weak equivalence in [5], to which we will refer here as objectwise weak equivalence. A map /: X -> F beween two P-diagrams (i.e. functors X. Y: D -* (spaces)) is an objectwise weak equivalence if for each object d e objP the map/(of): X(d) -* Y(d) Received by the editors April 25, 1985 and, in revised form. July 3, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P91, 55P10; Secondary 55N91.