(ω+ω)-Categories (?)
Posted by Urs Schreiber
These days, everybody has his preferred definition of weak -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 -categories as discussed in
Eugenia Cheng
Comparing operadic theories of -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 -categories can nicely be defined iteratively by successively enriching over strict -categories.
and so on.
The limit that this is approaching, where we have strict higher categorical stuctures with cells of degree for all is called using the name of the first transfinite ordinal .
Now, it so happens that -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 This might be interesting for the following reason: objects in are -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
and so on .
Roughly, the objects of are -categories all whose comopositions are strict except those along cells of dimension . If we imagine that again we can let increase without bounds we should reach This should have objects which are -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 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 is. I would like to try to lift the nice “folk” model structure on 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