The Gray Monoidal Product of Double Categories

Abstract

The category of double categories and double functors is equipped with a symmetric closed monoidal structure. For any double category {mathbb {A}}, the corresponding internal hom functor sends a double category {mathbb {B}} to the double category whose 0-cells are the double functors {mathbb {A}} rightarrow {mathbb {B}}, whose horizontal and vertical 1-cells are the horizontal and vertical pseudo transformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.