Introduction

The purpose of this paper is to give general homotopy theoretical methods for

studying diagrams. We have aimed moreover at developing tools that would provide

a convenient framework for studying constructions like: push-outs and pull-backs,

realizations of simplicial and cosimplicial objects, classifying spaces, orbit spaces

and Borel constructions of group actions, fixed points and homotopy fixed points

of groups actions, and singular chains on spaces. All these constructions are very

fundamental in homotopy theory and they all are obtained by taking colimits or

limits of certain diagrams with values in various categories.

One way of organizing homotopy theoretical information is by giving an ap-

propriate model structure on the considered category. Model categories were in-

troduced in the late sixties by D. Quillen in his foundational book [41]. The key

roles are played by three classes of morphisms called weak equivalences, fibrations,

and cofibrations, which are subject to five simple axioms (see Section 2). An im-

portant property of model categories is that one can invert the weak equivalences,

so as to get the homotopy category. Model categories are also very convenient for

constructing derived functors using cofibrant and fibrant replacements (non-abelian

analogs of projective and injective resolutions).

This way of thinking about homotopy theory has become very popular. For

example, recent advances in localization theory (see in particular [3, 4, 5, 9, 13,

33]) show that the category of spaces or spectra can be equipped with various

model category structures, depending on what one wants to focus on. The weak

equivalences for example can be chosen to be the homology equivalences for a certain

homology theory. In this way our attention is placed on these properties which can

be detected by the chosen homology theory.

Although model categories provide a very convenient way of doing homotopy

theory, such structures are difficult to obtain. For example, for a small cate-

gory / and a model category M, quoting [31, page 121] "..., it seems unlikely

that Fun(I, M) has a natural model category structure." Thus to study the homo-

topy theory of diagrams we can not use the machinery of model categories directly.

Instead our approach is to relax some of the conditions imposed on a model cate-

gory so that the new structure is preserved by taking a functor category. At the

same time we are not going to give up too much. We will still be able to form

the localized homotopy category and construct derived functors by taking certain

analogs of the cofibrant and fibrant replacements.

Our methods provide a solution to the problem that motivated us originally:

How to construct the derived functors of the colimit and limit (the homotopy

colimit and limit) in any model category?

These constructions have played important roles for example in the study of

classifying spaces of compact Lie groups. Started in [43] and continued in [30,