The oplax limit of an enriched category

Soichiro Fujii and Stephen Lack

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

Keywords: Enriched categories, bicategories

2020 MSC: 18D20, 18N10

Theory and Applications of Categories, Vol. 40, 2024, No. 14, pp 390-412.

Published 2024-05-17.

TAC Home