Bifunctor Theorem and strictification tensor product for double categories with lax double functors

Bojana Femić

Theory and applications of categories, Tome 39 (2023), pp. 824-873.

Résumé

We introduce a candidate for the inner hom for the category of double categories and lax double functors, and characterize a lax double functor into it obtaining a lax double quasi-functor. The latter consists of a pair of lax double functors with four 2-cells resembling distributive laws. We extend this characterization to a double category isomorphism. We show that instead of a Gray monoidal product we obtain a product that in a sense strictifies lax double quasi-functors. We explain why laxity of double functors hinders our candidate for the inner hom from making the category of double categories and lax double functors a closed and enriched category over 2-categories (or double categories). We prove a bifunctor theorem by which certain type of lax double quasi-functors give rise to lax double functors on the Cartesian product. We extend this theorem to a double functor between double categories and show how it restricts to a double equivalence. The (un)currying double functors are studied. We prove that a lax double functor from the trivial double category is a monad in the codomain double category, and show that our above double functor recovers the specification in that double category of the composition natural transformation on the monad functor.

Détail

Publié le : 2023-10-03

Classification : 18N10
Keywords: bicategories, double categories, Gray monoidal product