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 $|[\mathbb{A},-]|$ sends a double category $\mathbb{B}$ to the double category whose 0-cells are the double functors $\mathbb{A}\to \mathbb{B}$, whose horizontal and vertical 1-cells are the horizontal and vertical pseudotransformations, respectively, and whose 2-cells are the modifications. Some well-known functors of practical significance are checked to be compatible with this monoidal structure.

35 pages, 5 large and many small diagrams