Model categories as presentations of homotopy theories

Remark 2.2.1. Model category theory has played an important role in the development of homotopy theory, being one of the main theoretical frameworks researchers used for studying homotopy theories before the widespread adoption of \(\infty \)-category theory. A model category is a 1-category equipped with weak equivalences, fibrations, and cofibrations satisfying various axioms that ensure these classes interact appropriately. From an \(\infty \)-categorical perspective, we may think of a model category \(C\) as providing a ‘model’ for its associated \(\infty \)-categorical localization \(C[W^{-1}]\). In particular, the fibrations and cofibrations in a model category do not affect the ‘underlying homotopy theory’ presented by the model category, and should rather be thought of as tools for gaining control over categorical constructions like pushouts and pullbacks in this localization.