this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:

Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CW-complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNN-extensions of groups, and so obtain computations of higher homotopical syzygies in these cases. 1

first author acknowledges the Sao Paulo University hospitality and the FAPESP-Sao Paulo (Brasil) financial support during the time this work was completed. Received by the editors January 1996. Communicated by Y. Felix. 1991 Mathematics Subject Classification : Primary 55N91, 55P15, 55P91; Secondary 55U35, 55Q91, 57S17. Key words and phrases : G-CW-space, weak G-n-equivalence, G-n-type, isotropy ring, local coe#cient system, orbit category, Postnikov decomposition, universal covering. Bull. Belg. Math. Soc. 4 (1997), 265--276 whilst an equivariant theory of n-types from a Quillen model category theory viewpoint has recently been given by A.R. Garzon and J.G. Miranda ([8]). As yet however homological criteria for equivariant weak n-equivalences have not been given and one of the aims of the present paper is to develop an equivariant version of a truncated Whitehead Theorem (cf. [16, 19] for its nontruncated form) with a list of some equivalent conditions for equivariant weak n-eq

Usually, in homotopy theory with cofibrations, cofibrant objects and a suspension functor are necessary to obtain homotopy groups and exact sequences of them. In this sense H.J. Baues defines in his book “Algebraic Homotopy” [1] the concept of “Category with a Natural Cylinder”. His definition requires

The most commonly known triangulated categories arise from chain complexes in an abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called ‘algebraic ’ because they have underlying abelian (or at least additive) categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘non-additive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, compare Definition 3.1; this class includes the algebraic triangulated categories. The purpose of this paper is to explain some systematic differences between these two kinds of triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. These differences are all torsion phenomena, and rationally every topological triangulated category is algebraic (at least under mild size restrictions). Our main tool is a new numerical invariant, the n-order of an object in a triangulated category, for n a natural number (see Definition 1.1). The n-order is a nonnegative integer (or infinity), and an object Y has positive n-order if and only if n · Y = 0; the n-order can be thought of

Many triangulated categories arise from chain complexes in an additive or abelian category by passing to chain homotopy classes or inverting quasi-isomorphisms. Such examples are called ‘algebraic ’ because they have underlying additive categories. Stable homotopy theory produces examples of triangulated categories by quite different means, and in this context the underlying categories are usually very ‘non-additive ’ before passing to homotopy classes of morphisms. We call such triangulated categories topological, and formalize this in Definition 1.4 via homotopy categories of stable cofibration categories. The purpose of this paper is to explain some systematic differences between algebraic and topological triangulated categories. There are certain properties – defined entirely in terms of the triangulated structure – which hold in all algebraic examples, but which can fail in general. The precise statements use the n-order of a triangulated category, for n a natural number (see Definition 2.1). The n-order is a non-negative integer (or infinity), and it measures, roughly speaking, ‘how strongly ’ the relation n · Y/n = 0 holds for the objects Y in a given triangulated category (where Y/n denotes a cone of multiplication by n on Y). Our main results are: • The n-order of every algebraic triangulated category is infinite (Theorem 3.3). • For every prime p, the p-order of every topological triangulated category is at least p − 1 (Theorem 8.2). • For every prime p, the p-order of the p-local stable homotopy category is exactly p − 1

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