This course will provide an introduction to model categories. Quillen defined model categories, roughly 30 years ago, as a way to axiomatize homotopy theory. Since then, they have been important in algebraic topology, and have been gaining even more prominence over the past 10 or 15 years. More recently, mathematicians in other fields have started using them; for example, Voevodsky's work on the Milnor conjecture makes use of model categories in an algebro-geometric setting. I have heard rumors of their use in mathematical physics, but I don't know any details.
Model categories are designed for several purposes. I will focus on one of those purposes: if you have a category -- like the category of topological spaces and continuous functions, or the category of chain complexes of modules over a ring and chain maps -- with a particular class of maps that you would like to be isomorphisms -- such as maps of spaces which induce an isomorphism on all the homotopy groups, or maps of chain complexes which induce homology isomorphisms -- then model categories provide a way of inverting that class of maps and ending up with something well-behaved. (It is often possible to invert such classes of maps by doing a simple sort of construction, but to end up with something very badly behaved; the "well-behaved" aspect of model categories is important.) Hence, model categories provide a good way to construct things like derived categories.
Simplicial sets form an important example of a model category, and simplicial sets are important objects to know about regardless. So I plan to spend some time defining simplicial sets and discussing the model category structure on them.