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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.
@article{TAC_2024_40_a13,
author = {Soichiro Fujii and Stephen Lack},
title = {The oplax limit of an enriched category},
journal = {Theory and applications of categories},
pages = {390--412},
publisher = {mathdoc},
volume = {40},
year = {2024},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2024_40_a13/}
}
Soichiro Fujii; Stephen Lack. The oplax limit of an enriched category. Theory and applications of categories, Bunge Festschrift, Tome 40 (2024), pp. 390-412. http://geodesic.mathdoc.fr/item/TAC_2024_40_a13/