Over the course of the semester, the following is covered: (1) Review of categories: presheaves, the Yoneda lemma, adjoint functors, inductive and projective limits, additive categories; (2) Abelian categories: injective and projective objects, derived functors and derived categories, triangulated categories, double complexes, spectral sequences. (Optional/time permitting: quasi-abelian categories, dg categories); (3) Model categories; (4) Simplicial methods: model structure on simplicial sets, simplicial groups/modules/rings, Dold-Kan correspondence, derived functors, applications (for example the cotangent complex).