The oplax limit of an enriched category

Abstract

We show that 2-categories of the form $\mathscr{B}\mbox{-}\mathbf{Cat}$ are closed under slicing, provided that we allow $\mathscr{B}$ to range over bicategories (rather than, say, monoidal categories). That is, for any $\mathscr{B}$-category $\mathbb{X}$, we define a bicategory $\mathscr{B}/\mathbb{X}$ such that $\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong (\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$. The bicategory $\mathscr{B}/\mathbb{X}$ is characterized as the oplax limit of $\mathbb{X}$, regarded as a lax functor from a chaotic category to $\mathscr{B}$, in the 2-category $\mathbf{BICAT}$ of bicategories, lax functors and icons. We prove this conceptually, through limit-preservation properties of the 2-functor $\mathbf{BICAT}\to 2\mbox{-}\mathbf{CAT}$ which maps each bicategory $\mathscr{B}$ to the 2-category $\mathscr{B}\mbox{-}\mathbf{Cat}$. When $\mathscr{B}$ satisfies a mild local completeness condition, we also show that the isomorphism $\mathscr{B}\mbox{-}\mathbf{Cat}/\mathbb{X}\cong (\mathscr{B}/\mathbb{X})\mbox{-}\mathbf{Cat}$ restricts to a correspondence between fibrations in $\mathscr{B}\mbox{-}\mathbf{Cat}$ over $\mathbb{X}$ on the one hand, and $\mathscr{B}/\mathbb{X}$-categories admitting certain powers on the other.