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Stream: theory: category theory

Topic: (∞,n)-categories à la Grothendieck-Maltsiniotis


view this post on Zulip Ivan Kobe (Feb 23 2025 at 12:12):

Hello everyone. In a 2010 paper, Maltsiniotis gave a simplified definition of -groupoids as developed by Grothendieck in Pursuing Stacks, and introduced a notion of -categories inspired by this approach. My question is whether it is possible to obtain a notion of (,n)(∞,n)-categories using similar tools.

According to the Grothendieck-Maltsiniotis definition, the notion of an -groupoid depends on the choice of a coherator CC, which is a category encoding operations and coherences expected in -groupoids, which then arise as presheaves on CC satisfying some left-exactness conditions (i.e., those preserving globular sums). While I don't want to reproduce the full definition of a coherator here, it is in particular a globular extension GC\mathbb{G} \to C such that every pair of parallel arrows in CC has a lifting in CC. If we index the objects of G\mathbb{G} by N\mathbb{N} and write DiD_i for the image of ii in CC, and si,ti:Di1Dis_i, t_i: D_{i-1} \to D_i for the images of source and target arrows, then a pair of arrows f,g:DiXf,g: D_i \to X in CC is parallel if either i=0i=0, or i>0i>0 and fsi=gsifs_i=gs_i and fti=gtift_i=gt_i. A lifting of such a pair is and arrow h:Di+1Xh: D_{i+1} \to X s.t. hsi+1=fhs_{i+1}=f and hti+1=ght_{i+1}=g.

The definition of -categories is almost exactly the same, the only difference being that in the coherator, we now don't require liftings of all pairs of parallel arrows, but only of a strict subclass, called admissible pairs. Being able to lift admissible pairs of arrows is sufficient to derive all grupoidal operations and coherences except inverses. (This notion of -categories was shown by Ara to be equivalent to Batanin's operadic definition.)

What happens if we mix things up a bit? Say, if in the definition of a coherator, we require, for a fixed nn, liftings of those pairs of arrows f,g:DiXf,g: D_i \to X where either ini \geq n and f,gf,g are parallel, or i<ni < n and f,gf,g is an admissible pair? Would you expect presheaves on such a coherator (again, those preserving globular sums) to yield a reasonable notion of (,n)(∞,n)-categories?