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Introduction

Model categories are a framework for making homotopical phenomena precise: what does it mean to “do homotopy theory”?

The motivating problem is the following. In many categories of interest — topological spaces, chain complexes, simplicial sets — one has a class of maps that are not isomorphisms but that one wishes to treat as if they were: weak homotopy equivalences in $\mathbf{Top}\text{,}$ quasi-isomorphisms in $\mathbf{Ch}(R)\text{,}$ and so on. The naive solution is to formally invert these maps, obtaining a localization. But this localization may fail to be a category in any useful sense: the collection of morphisms from one object to another may not even be a set. A model structure is extra data on a category that ensures the localization at the weak equivalences exists as an honest category, with morphism sets one can compute.

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