Comparing enrichment contexts on Gray-Cat

Adrian Miranda – 10 August 2022

Last week we looked at a 'highlights reel' of the proof that there is a closed structure on Gray-Cat. We saw which bits were easier, which bits were harder, and in what ways. The first thing I'll do this week is give some more detail about how conducting those 'tedious diagram chases' is a more algorithmic and less creative activity than you might think. After this, we'll compare categories enriched over Crans' monoidal structure to categories enriched over the closed structure. We'll see that the fully general notion of a category enriched over the closed structure has some unappealing non-algebraic features, and identify a niceness assumption under which they will be better behaved. We'll also look at degenerate cases including (in decreasing order of degeneracy) symmetric monoidal categories, braided Gray monoids, and the (possibly new?) notion of a monoidal Gray category. We'll see that there is some asymmetry in the notion of a category enriched over the closed structure, even with the niceness assumption. This asymmetry disappears for symmetric monoidal categories, but it has already been noticed by Baez and Neuchl, as well as by Crans, in the context of braided Gray monoids.