Chapter 2 - Homotopy Theories and Model Categories

Abstract

This chapter explains homotopy theories and model categories. A model category is just an ordinary category with three specified classes of morphisms—fibrations, cofibrations, and weak equivalences—which satisfy a few simple axioms that are deliberately reminiscent of the properties of topological spaces. Surprisingly enough, these axioms give a reasonably general context in which it is possible to set up the basic machinery of the homotopy theory. The machinery can then be used immediately in a large number of different settings, as long as the axioms are checked in each case. Although many of these settings are geometric, some of them are not. This chapter provides a background material, principally a discussion of some categorical constructions (limits and colimits), which come up almost immediately in any attempt to build new objects of some abstract category out of the old ones. The chapter also provides a conceptual interpretation of the homotopy category of a model category. Surprisingly, this interpretation depends only on the class of weak equivalences. This suggests that in a model category, weak equivalences carry the fundamental homotopy theoretic information, while the cofibrations, fibrations, and the axioms they satisfy function mostly as tools to make various constructions.