The aim of the logic seminar for winter quarter will be to deal with approaches to the foundations of category theory as well as applications of category theory to logic and computer science. Following an introduction by Feferman to the foundational issues, Ulrik Buchholtz will begin a review of basic concepts: categories, functors, natural transformations from S. Mac Lane, Categories for the Working Mathematicians, secs. I.1-5. This will be followed in subsequent seminars by a presentation of one standard set-theoretical foundation via small and large categories, from which the review from Mac Lane will move on to universal constructions, limits, colimits, adjoint functors, and the adjoint functor theorem.

We will begin a review of basic concepts: categories, functors, natural transformations from S. Mac Lane, Categories for the Working Mathematicians, secs. I.1-5.

Abstract: I will review strengthened set-theoretical foundations for category theory with "small", "large" distinctions, as well as partial results for unrestricted category theory, without the "small", "large" distinctions and allowing self-membered categories.

We will discuss a general notion of relativization for higher order structures. We will then discuss the way in which Grothendieck Toposes can be relativized. Specifically we will discuss how this yields a notion of a potential isomorphism (and potential maps) between sheaves. We will also discuss the relationship between these potential maps and a generalization of the ordinals.

The design of the programming language Haskell leans on category theory in several different areas. I will describe the way one can interpret (most) of the language design as a category, as well as how endofunctors, monads and fixpoints show up as constructions in the language itself. If time allows, I will finish with describing the work by Conor McBride on introducing formal derivatives of 'nice' endofunctors to produce a new formalism for Zipper datatypes.