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September 30, 2008

(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 nn-categories

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 nn-categories can nicely be defined iteratively by successively enriching over strict (n1)(n-1)-categories.

0Cat:=Set 0 Cat := Set 1Cat=SetCat 1 Cat = Set-Cat 2Cat=1CatCat 2 Cat = 1Cat-Cat 3Cat=2CatCat=(1CatCat)Cat 3 Cat = 2Cat-Cat = (1Cat-Cat)-Cat 4Cat=3CatCat=((1CatCat)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 kk for all kk \in \mathbb{N} is called ωCat \omega Cat using the name of the first transfinite ordinal ω\omega.

Now, it so happens that ω\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 (ω+1)Cat:=ωCatCat. (\omega+1) Cat := \omega Cat-Cat \,. This might be interesting for the following reason: objects in ωCatCat\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

(ω+2)Cat:=(ω+1)CatCat=(ωCatCat)Cat (ω+3)Cat:=((ωCatCat)Cat)Cat (ω+4)Cat:=(((ωCatCat)Cat)Cat)Cat \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 (ω+n)Cat(\omega+ n)Cat are \infty-categories all whose comopositions are strict except those along cells of dimension 0k<n0 \leq k \lt n. If we imagine that again we can let nn increase without bounds we should reach (ω+ω)Cat. (\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}\{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 (ω+ω)Cat(\omega+\omega)Cat is. I would like to try to lift the nice “folk” model structure on ωCat\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?

Posted at September 30, 2008 2:03 PM UTC

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Re: (ω+ω)-Categories

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

(ω+ω+1)Cat:=(ω+ω)CatCat? (\omega + \omega + 1)Cat : = (\omega + \omega)Cat - Cat?

Posted by: David Corfield on September 30, 2008 3:17 PM | Permalink | Reply to this

Re: (ω+ω)-Categories

It seems to me that with your idea, the objects of (ω+n)(\omega+n)Cat will actually be \infty-categories whose compositions are strict except those along cells of dimension exactly nn (whatever that means). On ω\omegaCat you have used the weak Crans-Gray tensor product, but as I’ve said elsewhere, the natural monoidal structure inherited by a category of enriched categories is strict relative to the composition in those enriched categories.

Posted by: Mike Shulman on September 30, 2008 4:55 PM | Permalink | Reply to this

Re: (ω+ω)-Categories

Ah right. I suppose one should instead add a weakening at each level. Hm…

Posted by: Urs Schreiber on September 30, 2008 5:08 PM | Permalink | Reply to this

Re: (ω+ω)-Categories

So, in my unbounded naïvety I was imagining that when forming the tensor product of two ωCat\omega Cat-enriched categories we could use the Crans-Gray-tensor product Hom-wise.

So for CC and DD ωCat\omega Cat-enriched categories I was imagining something like defining the Hom-object (CD)((c 1,d 1),(c 2,d )) (C \otimes D)((c_1,d_1),(c_2,d_)) of CDC \otimes D for c 1,d 1Obj(C)c_1,d_1 \in Obj(C) and d 1,d 2Obj(D)d_1,d_2 \in Obj(D) by (CD)((c 1,d 1),(c 2,d )):=C(c 1,c 2)D(d 1,d 2). (C \otimes D)((c_1,d_1),(c_2,d_)) := C(c_1,c_2) \otimes D(d_1,d_2) \,.

What trouble would I run into if I proceeded like this?

Sorry if you explained this before. It seems the message didn’t stick.

Posted by: Urs Schreiber on October 1, 2008 11:44 AM | Permalink | Reply to this

Re: (ω+ω)-Categories

That is how the natural monoidal structure on an enriched category is defined. But now the weakness in the Crans-Gray tensor product has been pushed up to dimension 2, and composites in dimension 1 are now strict. I’m not sure how to explain this any better than I did here.

Posted by: Mike Shulman on October 1, 2008 9:26 PM | Permalink | Reply to this

Re: (ω+ω)-Categories

I’m not sure how to explain this any better

Yeah, sorry, it is weird how I am having a mental block here.

While it is so simple: the categorical dimension of the nn-morphisms in the ωCat\omega Cat-enriched category as seen by the Hom-wise Crans-Gray product is no longer nn but (n1)(n-1). So in particular 1-morphisms are tensored to something of categorical degree (11)+(11)=(11)(1-1) + (1-1) = (1-1) and hence remain 1-morphisms instead of being boosted up to a 2-morphism that would induce the desired weakening.

Okay, next I will try to wiggle my way out of this.

I could try replacing the Hom-objects such as the above C(c 1,c 2)C(c_1,c_2) and D(d 1,d 2)D(d_1,d_2) by what I expect to be their cylinder objects C(c 1,c 2)IC(c_1,c_2) \otimes I and D(d 1,d 2)ID(d_1,d_2) \otimes I before using them in the composition morphism (where II is the interval groupoid I={ab}I = \{a \stackrel{\simeq}{\to} b\}) thereby shifting the categorical dimension up by one while retaining weak equivalence of the Hom-objects.

Hm, and thus I should maybe think about ω\omega-anacategories, where the composition morphism is a morphism out of any acyclic fibration over the tensor product of the objects of morphisms to be composed.

But this attempt will have to wait until tomorrow.

And I should try to figure out if for CC an ω\omega-category indeed CIC \otimes I is its cylinder object with respect to the folk model structure, as one would expect it to be. Is it?

And for CC an ω\omega-groupoid, is hom(I,C)hom(I,C) the path object, as one would expect?

Posted by: Urs Schreiber on October 2, 2008 10:29 AM | Permalink | Reply to this

Re: (ω+ω)-Categories

And I should try to figure out if for CC an ω-category indeed CIC\otimes I is its cylinder object with respect to the folk model structure, as one would expect it to be. Is it?

My intuition would be yes, depending on exactly what you mean by “cylinder object”. If you want C+CCIC+ C \to C\otimes I to be a cofibration, then you might need CC to be a cofibrant ω-category. This is the way it works in other cases, e.g. for topological spaces X+XX×IX+ X \to X\times I is only a cofibration in the usual model structure if XX is itself cofibrant. But some people don’t require that in the definition of model-categorical cylinders.

Posted by: Mike Shulman on October 3, 2008 6:50 PM | Permalink | Reply to this

Re: (ω+ω)-Categories (?)

Hi Urs! Instead of

(w+n+1)cat=(w+n)catcat(w+n+1)cat=(w+n)cat - cat,

why not writing

cat (w+n+1)=cat (w+n))*catcat^{(w+n+1)}=cat^{(w+n))}*cat ?.

I think you would get a better analogy with natural numbers, by aplaying log catlog_cat on both sides, and you would get a better analogy with natural numbers.

Posted by: Daniel de Franša MTd2 on October 1, 2008 2:01 PM | Permalink | Reply to this

Re: (ω+ω)-Categories (?)

Hi Daniel,

I guess I can see where you are coming from: you want to read the iterated enrichment notation in analogy with products. Yes, I suppose there is a certain justification for the notation you propose. It is not standard, however. Right now I don’t feel that I should switch to non-standard notation with even what I said in standard notation being at best shaky.

Posted by: Urs Schreiber on October 1, 2008 5:01 PM | Permalink | Reply to this

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