British Mathematical Colloquium, University of
Cambridge, 31 March 2015.
Abstract A fundamental operation of category theory is Isbell conjugacy, which turns a covariant Set-valued functor into a contravariant one and vice versa. The reflexive completion of a category consists of those Set-valued functors on it that are canonically isomorphic to their double conjugate. As the name suggests, and as Isbell showed, this "completion" process is idempotent. Fundamental as reflexive completion is, it has been little studied and exhibits some surprising behaviour. For example, the reflexive completion of a category always has initial and terminal objects, but seemingly need not have any other (co)limits. I will begin by introducing Isbell conjugacy and reflexive completion, and I will then present some new results with Tom Avery, describing when the reflexive completions of two given categories are equivalent. Finally, I will state some open questions about this basic concept.
Joint with Tom Avery (Edinburgh)
Slides In this pdf file.