Higher Dimensional Categories: Model Categories and Weak Factorisation Systems

Loosely speaking, “homotopy theory” is a perspective which treats objects as equivalent if they have the same “shape” which, for a category theorist, occurs when there exists a certain class W of morphisms that one would like to invert, but which are not in fact isomorphisms. Model categories provide a setting in which one can do “abstract homotopy theory” in subjects far removed from the original context of topological spaces. Given a model category, one can form its homotopy category, in which the weak equivalences W become isomorphisms, but it is the additional structure provided by two other distinguished classes of morphisms cofibrations and fibrations that enables one to understand the morphisms that result from formally inverting the weak equivalences, in effect allowing one to “do homotopy theory.” The study of higher dimensional categories, which are a weak notion in their most useful form, can benefit immensely from homotopy theory. Hence, it is worthwhile to first gain a thorough understanding of model categories and their properties, which in turn make use of 2-categorical notions. This is the object of this paper. In Section 2, we begin by introducing a few useful concepts from 2-category theory. Then, in Section 3, we define a model category, which will be one of two central topics in this paper. In Sections 3.1 and 3.2, we develop some of the basic theory of model categories and of chain complexes, which will provide one of the main examples. Sections 3.3 and 3.4 give a thorough discussion of two algebraic examples of model categories: ChR and Cat. Section 3.5 gives Quillen’s well-known small object argument, completing the discussion of model categories. In Section 4, we change perspectives somewhat to discuss weak factorisation systems in general, and in the sections that follow we prove some results connecting factorisations to limits and colimits. Notably, we define a stronger natural weak factorisation system in Section 4.5, which applies to our two example model categories, providing additional algebraic structure. We conclude with a few suggestions for further study and our acknowledgments.