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,
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