### (ω+ω)-Categories (?)

#### Posted by Urs Schreiber

These days, everybody has his preferred definition of weak $\infty$-categories. I have an idea, too, once mentioned somewhere here on the blog. It’s still very tentative. But when I looked yesterday at Todd Trimble’s operadic definition of weak $\infty$-categories as discussed in

Eugenia Cheng
*Comparing operadic theories of $n$-categories*

arXiv:0809.2070

I had the sensation that the definition I had in mind is not that unrelated, and maybe in fact a nice special case. Or maybe not, I haven’t really tried yet to go through this in detail. But nevertheless I feel like chatting about it.

Strict $n$-categories can nicely be defined iteratively by successively enriching over strict $(n-1)$-categories.

$0 Cat := Set$ $1 Cat = Set-Cat$ $2 Cat = 1Cat-Cat$ $3 Cat = 2Cat-Cat = (1Cat-Cat)-Cat$ $4 Cat = 3Cat-Cat = ((1Cat-Cat)-Cat)-Cat$ and so on.

The limit that this is approaching, where we have strict higher categorical stuctures with cells of degree $k$ for all $k \in \mathbb{N}$ is called $\omega Cat$ using the name of the first transfinite ordinal $\omega$.

Now, it so happens that $\omega$-Cat carries a nice monoidal structure: the Crans-Gray tensor product . This means we can continue enriching, now using this new monoidal structure, to get
$(\omega+1) Cat := \omega Cat-Cat
\,.$
This might be interesting for the following reason: objects in $\omega Cat-Cat$ are $\infty$-categories all of whose composition operations are strict *except* that along 0-cells. The composition along 0-cells is an operation out of the Crans-Gray tensor product and hence inherits the weakening introduced by that. This is precisely the generalization of the phenomenon that makes Gray-categories be 3-categories in which everything is strict except the exchange law for composition of 2-cells along 0-cells.

I think as before the category of enriched categories always inherits a natural monoidal structure itself. So we can keep going and form

$\array{ (\omega+2) Cat := (\omega +1)Cat-Cat = (\omega Cat-Cat)-Cat \\ (\omega+3) Cat := ((\omega Cat-Cat)-Cat)-Cat\\ (\omega+4) Cat := (((\omega Cat-Cat)-Cat)-Cat)-Cat }$ and so on .

Roughly, the objects of $(\omega+ n)Cat$ are $\infty$-categories all whose comopositions are strict except those along cells of dimension $0 \leq k \lt n$. If we imagine that again we can let $n$ increase without bounds we should reach $(\omega+\omega) Cat \,.$ This should have objects which are $\infty$-categories all of whose compositions are weak.

Looking at this, this looks vaguely similar, if more specialized, to the strategy of iterative operadic weakening described by Eugenia on page 5. I am wondering if there might be a family $\{P_i\}$ of operads which reproduces the above idea in this sense.

One would want to know what the weakened notion of limits, colimits etc in the 1-category $(\omega+\omega)Cat$ is. I would like to try to lift the nice “folk” model structure on $\omega Cat$ through the iteration process. Experience with simplicially enriched categories, DG-categories and the like shows that the category of categories enriched over a model category is naturally itself a model category. Is there any precise general statement along these lines?

## Re: (ω+ω)-Categories

So what would happen if you carried on? I.e., what is

$(\omega + \omega + 1)Cat : = (\omega + \omega)Cat - Cat?$