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Stream: theory: category theory

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

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 $(∞,n)$-categories using similar tools.

According to the Grothendieck-Maltsiniotis definition, the notion of an $∞$-groupoid depends on the choice of a coherator $C$, which is a category encoding operations and coherences expected in $∞$-groupoids, which then arise as presheaves on $C$ 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 $\mathbb{G} \to C$ such that every pair of parallel arrows in $C$ has a lifting in $C$. If we index the objects of $\mathbb{G}$ by $\mathbb{N}$ and write $D_i$ for the image of $i$ in $C$, and $s_i, t_i: D_{i-1} \to D_i$ for the images of source and target arrows, then a pair of arrows $f,g: D_i \to X$ in $C$ is parallel if either $i=0$, or $i>0$ and $fs_i=gs_i$ and $ft_i=gt_i$. A lifting of such a pair is and arrow $h: D_{i+1} \to X$ s.t. $hs_{i+1}=f$ and $ht_{i+1}=g$.

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 $n$, liftings of those pairs of arrows $f,g: D_i \to X$ where either $i \geq n$ and $f,g$ are parallel, or $i < n$ and $f,g$ is an admissible pair? Would you expect presheaves on such a coherator (again, those preserving globular sums) to yield a reasonable notion of $(∞,n)$-categories?