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Patrick Nicodemus (Jan 29 2024 at 14:09):What are the known notions of n-category which directly subsume tricategories in an evident way?
I don't mean up to Quillen equivalence, I mean any theorem I want to prove about tricategories, I can prove about these n-categories and I'll get the result for tricategories. I think this requirement implies that I only care about "algebraic" models here unless there's an obvious way to "bias" the geometric model to become algebraic.
Mike Shulman (Jan 29 2024 at 16:44):I think the only one that can do that is a Batanin-Leinster algebra for a suitably chosen globular operad.
Matteo Capucci (he/him) (Jan 30 2024 at 09:37):Noob question, but why isn't this 'trivially' the case? I was expecting all definitions of weak n-categories to specialize to tri- and bicategories!
Patrick Nicodemus (Jan 30 2024 at 13:22):Some models of higher cats are very different shapes than ordinary cats, like quasicategories are just geometrically very different looking than the usual categories.
Like say you have a coherence condition in a bi category that asserts that a pasting diagram commutes (two disks with the same boundary are equal)
and at level 3 you want to weaken this to a 3 cell between the disks (two disks are homotopic by a 3 cell)
At this point it may not be even necessary to demand that you have a unique composite of the pasting diagrams on both sides (i.e. two well defined disks) before you have a morphism between them, and indeed demanding to choose a composite is kind of awkward because the composite of a pasting diagram involves choosing arbitrary composition orders for the morphisms involved. instead you could define a 3 cell directly in terms of the pasting diagrams themselves. So the shape of a 3 cell is no longer a map between 2 cells but a more complicated structure relating pasting diagrams of 2 cells.
Patrick Nicodemus (Jan 30 2024 at 13:23):In quasicategories a 2 cell is a filler for a triangle of 1 cells, a 3 cell is a filler for a tetrahedron whose faces are 2 cells and so on
Patrick Nicodemus (Jan 30 2024 at 13:31):Or you could define composition so that we don't have a single designated composite but it's only defined by a universal property. I think opetopes do this.
Matteo Capucci (he/him) (Jan 30 2024 at 14:02):Uhm I see, makes sense
Mike Shulman (Jan 30 2024 at 17:08):Any correct definition of n-categories has to specialize to tricategories up to "Quillen" equivalence, but Patrick specified that's not the question he's asking now.
Mike Shulman (Jan 30 2024 at 17:08):(I put it in quotes because I'm not sure anyone has actually written down a Quillen model structure for tricategories.)
Kevin Arlin (Jan 30 2024 at 17:42):Indeed quasicategories also don't have single designated composites, and to the extent the composites even have universal properties it's quite implicit.