Australian Category Seminar

Enriched category theory gives rise to a 2-functor $\mathrm{Enr}$ from a suitable 2-category of enrichment bases to the 2-category $\mathbf{2}\text{-}\mathbf{CAT}$ of 2-categories, sending each base $\mathcal{V}$ to the 2-category $\mathrm{Enr}(\mathcal{V}) = \mathbf{V}\text{-}\mathbf{Cat}$ of $\mathcal{V}$-categories. Classically, the monoidal categories are taken as the enrichment bases, but there have been several extensions. In this talk, I will show that the 2-functor $\mathrm{Enr}$ becomes a parametric right 2-adjoint if we take the virtual double categories as the enrichment bases, by decomposing the 2-functor $\mathrm{Enr}_1 \colon \mathbf{VDBL} \to \mathbf{2}\text{-}\mathbf{CAT}/\mathrm{Enr}(1)$ into a few basic right adjoint 2-functors. The powerfulness (or exponentiability) of discrete opfibrations between virtual double categories and the notion of horizontal unit in a virtual double category play essential roles.

(Joint work with Steve Lack)