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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 -categories using similar tools.
According to the Grothendieck-Maltsiniotis definition, the notion of an -groupoid depends on the choice of a coherator , which is a category encoding operations and coherences expected in -groupoids, which then arise as presheaves on 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 such that every pair of parallel arrows in has a lifting in . If we index the objects of by and write for the image of in , and for the images of source and target arrows, then a pair of arrows in is parallel if either , or and and . A lifting of such a pair is and arrow s.t. and .
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 , liftings of those pairs of arrows where either and are parallel, or and is an admissible pair? Would you expect presheaves on such a coherator (again, those preserving globular sums) to yield a reasonable notion of -categories?