I - Axioms for homotopy theory and examples of cofibration categories

Summary

Axiomatic homotopy theory is the development of the basic constructions of homotopy theory in an abstract setting, so that they may be applied to other categories. And there is, indeed, a strikingly wide variety of categories where these techniques are useful (e.g. topological spaces, differential algebras, differential Lie algebras, chain complexes, modules, sheaves, local algebras….).

The subject is not new and goes back, for example to Kan (1955), Quillen (1967), Heller (1968), and K.S. Brown (1973) each of whom proposes a different set of axioms. In fact, it is not evident what is the most appropriate choice. The best-known approach is that of Quillen who introduces the notion of a (closed) model category, as the starting point for his development of the quite sophisticated ‘homotopical algebra’.

The set of axioms which define a model category is, however, quite restrictive. For instance, they do not apply to topological spaces with the usual definitions of fibrations and cofibrations. There are other examples, as pointed out by K.S. Brown, where they give rise to a ‘somewhat unsatisfactory’ homotopy theory.

We here introduce the notion of a cofibration category. Its defining axioms have been chosen according to two criteria:

1. (1) The axioms should be sufficiently strong to permit the basic constructions of homotopy theory.

2. (2) The axioms should be as weak (and as simple) as possible, so that the constructions of homotopy theory are available in as many contexts as possible.