A semi-strictly generated closed structure on Gray-Cat

Abstract: We show that the semi-strictly generated internal homs of $\mathbf{Gray}$-categories $[\mathfrak{A}, \mathfrak{B}]\text{ssg}$ defined in \cite{Miranda strictifying operational coherences} underlie a closed structure on the category $\mathbf{Gray}$-$\mathbf{Cat}$ of $\mathbf{Gray}$-categories and $\mathbf{Gray}$-functors. The morphisms of $[\mathfrak{A}, \mathfrak{B}]\text{ssg}$ are composites of those trinatural transformations which satisfy the unit and composition conditions for pseudonatural transformations on the nose rather than up to an invertible $3$-cell. Such trinatural transformations leverage three-dimensional strictification \cite{Miranda strictifying operational coherences} while overcoming the challenges posed by failure of middle four interchange to hold in $\mathbf{Gray}$-categories \cite{Bourke Gurski Cocategorical Obstructions to a Tensor Product of Gray Categories}. As a result we obtain a closed structure that is only partially monoidal with respect to \cite{crans tensor of gray categories}. As a corollary we obtain a slight strengthening of strictification results for braided monoidal bicategories \cite{Gurski Loop Spaces}, which will be improved further in a forthcoming paper \cite{Miranda weak interchange 4-categories}.