### Borisov on Shapes of Cells in *n*-Categories

#### Posted by John Baez

I got an email from the author of this paper:

- Dennis Borisov, Comparing definitions of weak higher categories, I.

asking for comments. Unfortunately I’m completely busy with work on other subjects — and some of you reading this are much better than me at comparing various definitions of $n$-category! So maybe you can say something?

A category is a graph with extra structure, so we may hope that an $n$-category is some sort of generalized graph with extra structure. The details depend, among other things, on what sort of ‘shapes of cells’ we use. Borisov claims that his concept of ‘nested graph’ gives the ‘minimal generalization’ of the concept of graph needed to handle lots of example. If this is correct, it would be good for more people to know about it! If not, it would be good for Borisov to know about it!

His theory of nested graphs is vaguely reminiscent of Berger and Moerdijk’s theory of Geometric Reedy categories. A geometric Reedy category $R$ has ‘shapes of cells’ as objects and ‘face and degeneracy maps’ as morphisms. Every shape of cell has some natural number called its ‘dimension’, any ‘face’ of some shape of cell has lower dimension, while any ‘degeneracy’ of some shape cell has higher dimension. Various axioms are imposed so that the presheaf category $Set^{R^{op}}$ has the structure of a model category; this allows us to do homotopy theory in there. The classic example is $R = \Delta$, where $Set^{R^{op}}$ is the category of simplicial sets.

But this relation between Borisov’s work and the theory of geometric Reedy categories may or may not be deceptive! I don’t know.

## Re: Borisov on Shapes of Cells in n-Categories

Some related links:

For those to whom John’s remarks about Reedy model categories were news: more context, more explanation, more details, more examples and and more references may be found at

$n$Lab:Reedy category,

$n$Lab:Moerdijk-Berger generalized Reedy category,

$n$Lab: Reedy model structure,

$n$Lab: global model structure on functors.

A question about this I had a while ago is still open, by the way:

Question on models for $(\infty,1)$-functor categories.

On the $n$Lab there is also an entry that goes in the direction of the general topic of differing shapes for higher categorties: $n$Lab:geometric shapes for higher structures. But that could use more details.

I notice that Borisov’s general strategy of using “nested graphs” as encoding arbitrary higher shapes seems to be the one we played around with at $n$Lab:hyperstructure a bit.