Homotopy theory of topological spaces represents a rich and interesting interplay between relatively easy-to-define notions such as homotopy of maps, homotopy groups of spaces, fibrations, etc.
In the sixties Quillen realized that topological structures like these could be encoded in a set of axioms which, if satisfied, allow one to talk about 'homotopy theory' in a more abstract setting. Any category that satisfies these axioms is called a '(closed) model category'. Although many instances arise in a geometric context, a perhaps surprising application of model categories is in a more algebraic setting: one of the early success of model categories was the proof that the combinatorial notion of 'simplicial sets' sufficiently 'models' the homotopy category of topological spaces. It turns out that chain complexes of modules also satisfy the axioms (this lead Quillen to the notion of 'homotopical algebra') and hence we can talk about such things as the 'suspension of a chain complex', etc. More recently model categories have been introduced in algebraic geometry in the context of '$A^1$ homotopy' of schemes.
In this talk we will introduce the axioms for a model category and discuss a couple of examples and applications (among those mentioned above).