Semi-strictness in three-dimensional category theory and a (non-monoidal) closed structure on the category of Gray-categories.

Adrian Miranda – 27 July 2022

Gray-categories are "semi-strict three-dimensional categories", in the sense that every tricategory is triequivalent to a Gray-category. In this talk I will introduce a closed structure [-, ?] on Gray-Cat, the category of Gray-categories. The objects of [A, B] will be Gray-functors, while the morphisms will be (finite composites of) trinatural transformations satisfying a certain "semi-strictness" property. This semi-strictness property asks for the usual unit and composition laws for a pseudonatural transformation to hold on the nose rather than just up to some invertible 3-cell. As I will show, an arbitrary trinatural transformation between Gray functors out of a cofibrant Gray category is appropriately equivalent to a semi-strict one. There is also a monoidal structure \otimes on Gray-Cat due to Crans, but for general reasons due to Bourke and Gurski these structures cannot be combined to a monoidal closed structure. Instead, I will identify conditions on Gray categories A_1, ...., A_n so that multimaps A_1, ..., A_n ---> B agree for both structures, and I will describe an embedding between the two 2-categories of enriched categories, ie. a 2-fully faithful map \otimes-Cat ----> [-, ?]-Cat. Time permitting I will share some progress towards proving a developing conjecture on a coherence result in dimension four. I will also explain why I think even if the conjecture is true, the techniques from this talk will not be sufficient to define a similar "closed structure of semi-strict maps" on the category of semi-strict four categories.