Title: The category Lex as a tangent (2-)category
Abstract: In this talk, I will report on joint work in progress (with Robin Cockett and Ben MacAdam) on how the additive bundle construction (almost) equips the category Lex of lex categories with a tangent category structure. Let Lex be the category of (small) lex categories (i.e. categories with finite limits) and finite-limit-preserving functors. Given a lex category C, an additive bundle in C consists of an object X of C equipped with a commutative monoid object in the slice category C/X over X. The additive bundles in C form a lex category AddBun(C), and the assignment of the lex category AddBun(C) to a lex category C extends to an endofunctor T : Lex —> Lex. This additive bundle endofunctor almost equips the category Lex with the structure of a tangent category, except that a few of the required coherences only hold up to isomorphism rather than equality. I will give an overview of this result and, noting that Lex is in fact a 2-category and that T is a 2-functor, I will also describe our current work in progress towards defining a notion of tangent 2-category and showing that the 2-functor T equips the 2-category Lex with the structure of a tangent 2-category.
