In this paper we develop homotopy theoretical methods for
studying diagrams. In particular we explain how to construct homotopy colimits
and limits in an arbitrary model category. The key concept we introduce is
that of a model approximation. A model approximation of a category
$\mathcal{C}$ with a given class of weak equivalences is a model
category $\mathcal{M}$ together with a pair of adjoint functors
$\mathcal{M} \rightleftarrows \mathcal{C}$ which satisfy certain
properties. Our key result says that if $\mathcal{C}$ admits a model
approximation then so does the functor category $Fun(I,
\mathcal{C})$.

From the homotopy theoretical point of view categories with
model approximations have similar properties to those of model categories. They
admit homotopy categories (localizations with respect to weak equivalences).
They also can be used to construct derived functors by taking the analogs of
fibrant and cofibrant replacements.

A category with weak equivalences can have several useful
model approximations. We take advantage of this possibility and in each
situation choose one that suits our needs. In this way we prove all the
fundamental properties of the homotopy colimit and limit: Fubini Theorem (the
homotopy colimit -respectively limit- commutes with itself), Thomason's theorem
about diagrams indexed by Grothendieck constructions, and cofinality
statements. Since the model approximations we present here consist of certain
functors “indexed by spaces”, the key role in all our arguments is
played by the geometric nature of the indexing categories.

Readership

Graduate students and research mathematicians interested in
algebraic topology, category theory, and homological algebra.