Edward Morehouse - Fox cartesian structure for Gray monoidal double categories

Описание: Talk at Applied Category Theory 2023

This talk presents some results from the article, Cartesian Gray-Monoidal Double Categories. There, the notion of a locally cubical Gray category is proposed, and it is shown that double categories with a hierarchy of their morphisms, as well as classical (locally globular) Gray categories, are instances of this construction. A one-object locally cubical Gray category is a Gray-monoidal double category, which can be endowed with braided, sylleptic, and symmetric structure, as in the globular case. Equipping a symmetric Gray-monoidal double category with Fox-cartesian structure requires compatible and coherent duplication and deletion. This necessitates the introduction of doubly-lax functors, along with multiple duplicator transformations, and coassociator and cocommutor modifications for these. Intuitively, the reason for this is that in the Gray-monoidal setting we cannot do two things at once, only one at a time, so we must impose and maintain an order on the higher-dimensional cells. An algebraic presentation of the resulting theory is rather complex due to the bureaucracy of linearizing higher-dimensional boundary constraints. Fortunately, it has a relatively simple and compelling representation in the graphical calculus of surface diagrams, which we present.

https://act2023.github.io/papers/pape...
https://act2023.github.io/