Enrichment preserves fibrations

Soichiro Fujii – 4 October 2023

The notion of (Grothendieck) fibration can be defined internal to any 2-category K, and when K has oplax limits of 1-cells, there is a simple criterion for a 1-cell in K to be a fibration. In this talk, I will show that the 2-functor (-)-Cat : BICAT -> 2-CAT, mapping each bicategory B to the 2-category B-Cat of B-categories, preserves fibrations. Here, BICAT is the 2-category of bicategories, lax functors, and icons, and the fibrations in BICAT are exactly the pseudo functors which are locally (Grothendieck) fibrations. (Joint work with Steve Lack.)