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Matteo Capucci (he/him) (Oct 27 2022 at 07:54):I'm crossposting here a question from the nForum: are multiple categories and n-fold categories the same thing?
I suspect the answer is yes up to some considerations about strictness.
Multiple categories are Grandis and Parè's brainchild, you can find an introduction here. They have a combinatorial definition in terms of n-cubes. An n-uple category has kinds of -cells, enumerated by multiindices starting from 0. The 1-cells in the zeroth direction have a special role as they are where all strictiness lives: compositors, laxators, and similar coherence morphisms live there. All the other directions are equally weak (at least in the case of a 'weak multiple category'), it seems to me.
On the other hand, n-fold categories are defined inductively by saying an -fold category is an internal (pseudo?)category in the category of -fold categories. -fold categories are sets.
James Deikun (Jan 26 2023 at 00:46):Seems to me they're the same thing (by default) up to 2, then n-fold categories are less specified and include multiple categories but default to something other than multiple categories.
There are a lot of really strange beasts in the n-fold category zoo, like doubly pseudo categories that can compose pinwheel diagrams of squares.