UC Santa Cruz

An abstract homotopy theory is a situation in which one has a category with a class of ``weak equivalences'' that one would like to invert. One recent description of a homotopy theory is as an ``∞-category,'' which is like a category with extra structure that is made to contain homotopical data. Every ∞-category can be flattened into an ordinary category called its ``homotopy category,'' in a way that inverts the weak equivalences.

Among homotopy theories, there are certain ones that are called stable homotopy theories. Accordingly, there is a notion of stable ∞-categories which formalizes them. The homotopy category of a stable ∞-category has a canonical structure of a triangulated category.

A triangulated category is a category which is equipped with some structure that serves as a ``weak'' or homotopical version of an exact sequence. The theory of stable ∞-categories has some advantages over that of triangulated categories, since ∞-categories retain some homotopical information that is lost in the passage to the homotopy category. This thesis attempts to do two things: (1) to lay out a self-contained exposition of the theory of ∞-categories, and (2) to describe the relationships between stable ∞-categories and the triangulated structure on their homotopy categories.