The oplax limit of an enriched category
Soichiro Fujii – 1 June 2022
It is known that for any monoidal category V and a monoid T in V, the slice category V/T has a natural monoidal structure, and that there exists a canonical isomorphism of categories between Mon(V/T) and Mon(V)/T. If we regard the monoid T as a lax monoidal functor from the terminal monoidal category to V, then V/T is characterised as the oplax limit of the 1-cell T, in the 2-category of monoidal categories, lax monoidal functors and monoidal natural transformations. In this talk, I will generalise the construction of V/T as follows. For any bicategory W and a category B enriched over W, I will define a bicategory W/B, for which there exists a canonical isomorphism between the 2-categories (W/B)-CAT and (W-CAT)/B. The bicategory W/B can be characterised as the oplax limit of the W-category B, regarded as a lax functor from a suitable indiscrete category to W, in the 2-category of bicategories, lax functors and icons.
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