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Stream: theory: category theory

Topic: unbiased 2-categories

Daniel Teixeira (Jan 12 2023 at 20:51):

Does anyone have a good reference that review/uses unbiased bicategories, i.e. weak 2-categories where the composition of 1-morphisms is only defined up to 2-morphisms? (contrasted to the usual definition of bicategory, where horizontal composition is associative/unital up to specified isos but a choice of composite is made)

Daniel Teixeira (Jan 12 2023 at 20:52):

for (2,1)-categories I guess the image of the Duskin nerve would suffice, but I'm unaware of a more general construction

Daniel Teixeira (Jan 12 2023 at 20:53):

I know Leinster tackled this a couple decades ago with operads, but I didn't see follow-ups

Daniel Teixeira (Jan 12 2023 at 20:55):

the context is taking an (oo,n)-category and "modding out" morphisms of degree >2; the result is an (oo,2)-category, but we really wanted to move to 2-categories

Nathanael Arkor (Jan 12 2023 at 20:59):

An "unbiased bicategory" is one in which you admit $n$-ary composites for every $n \in \mathbb N$. Having composition of 1-cells only defined up to isomorphism would instead correspond to being a "representable multibicategory/virtual 2-category", where composites are defined by a universal property. Which are you interested in?

Daniel Teixeira (Jan 12 2023 at 21:13):

Thanks, then I'm misusing the word unbiased. What I meant is something in the lines of a 2-category where composition is defined up to a contractible space of choices (not necessarily by an universal property?). Is thisaccomplished by virtual 2-categories?

Nathanael Arkor (Jan 12 2023 at 21:21):

I believe that virtual 2-categories capture the kind of structure you're interested in. A good reference is Cruttwell–Shulman's A unified framework for generalized multicategories: they actually deal with virtual double categories, but you can disregard the vertical arrows there. Then you can identify the composites of 1-cells using the "opcartesian" cells of Definition 5.1 ibid.

Daniel Teixeira (Jan 12 2023 at 21:22):

thank you, I will take a look

Mike Shulman (Jan 13 2023 at 15:45):

In general, a definition of higher category of this style is called "nonalgebraic". There are many ways to realize a nonalgebraic definition of 2-categories, including virtual 2-categories with composites (which are essentially the same as opetopic 2-categories), or the image of the Duskin nerve, or Segal-category objects in Cat. Leinster's paper A Survey of Definitions of n-Category includes some brief discussion of how each definition specializes to $n=2$; six out of his ten definitions are nonalgebraic.

Graham Manuell (Jan 15 2023 at 16:49):

@Daniel Teixeira I think you are referring to something like an [[anabicategory]]. The nlab page has a link to Makkai's paper where they are first discussed.