Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

November 30, 2011

The (∞,1)-Category of (∞,n)-Categories

Posted by John Baez

There’s a new paper on the problem of comparing different definitions of (,n)(\infty,n)-category:

My thesis advisor often used the word ‘unicity’ instead of the more common ‘uniqueness’. I’ve never seen anyone else do that until now! But that’s not the most exciting thing about this paper…

According to an email from Chris, they give a few simple axioms that an (,1)(\infty,1)-category should satisfy to be a reasonable candidate for the (,1)(\infty,1)-category of (,n)(\infty,n)-categories. Then they prove that the space of (,1)(\infty,1)-cateories satisfying these axioms is equivalent to B(/2) nB(\mathbb{Z}/2)^n—that is, the product of nn copies of the classifying space of /2\mathbb{Z}/2, namely

B(/2)ℝℙ B(\mathbb{Z}/2) \simeq \mathbb{RP}^\infty

For starters, B(/2) nB(\mathbb{Z}/2)^n is connected, so any two theories obeying their axioms are equivalent. But the fun part is the (/2) n(\mathbb{Z}/2)^n. This comes from functors that send an (,n)(\infty,n)-category to one of its opposites. So, their result can also be seen as sophisticated generalization of the fact that the group of automorphisms of CatCat is /2\mathbb{Z}/2.

(A more sensible thing to look at is the 2-group of autoequivalences of CatCat and natural isomorphisms between these, but I’m pretty sure this is still equivalent to /2\mathbb{Z}/2.)

They also show that a lot of proposed theories of (,n)(\infty,n)-categories satisfy their axioms, and thus are equivalent. These include Barwick’s model of nn-fold complete Segal spaces (use by Lurie in his proof of the cobordism hypothesis), Rezk’s complete Segal Θ n\Theta_n-spaces, Simpson’s nn-fold Segal categories, and a bunch of other models.

Posted at November 30, 2011 3:15 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2465

25 Comments & 0 Trackbacks

Re: The (∞,1)-Category of (∞,n)-Categories

Until recently, I thought that “unicity” was just a mistake, typically made by native French speakers but common enough that it had infected some native English speakers. But then I witnessed someone (maybe Toby?) put up an argument that it was legit; so maybe I was the mistaken one.

Posted by: Tom Leinster on November 30, 2011 4:59 AM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

A more sensible thing to look at is the 2-group of autoequivalences of CatCat and natural isomorphisms between these, but I’m pretty sure this is still equivalent to /2\mathbb{Z}/2.

Yes, this is already true for (0,1)CatPoset(0,1)Cat \simeq Poset, see here.

Just for emphasis: in terms of automorphism nn-groups the above result of course says that even the automorphism ∞-group of (∞,n)Cat is equivalent to 2 n\mathbb{Z}_2^n.

Posted by: Urs Schreiber on November 30, 2011 9:20 AM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

We actually do slightly better than just the automorphism ∞-group. We show the full sub-(∞,1)-category of endofunctors consisting of those endofunctors which happen to be equivalences is equivalent to the discrete group (Z/2)^n.

So any a priori non-invertible natural transformation between these functors is automatically invertible.

Posted by: Chris Schommer-Pries on November 30, 2011 3:51 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

Does anything interesting happen as nn \to \infty? I guess the cardinality converges to 00.

Typo:

  1. Strict n-categories as presehaves of sets
Posted by: David Corfield on November 30, 2011 3:09 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

All along I have, of course, been trying to scan the available (,n)(\infty,n)-options for those best suited for cohesion in practice. I am not sure yet.

Our recent discussion about cohesion in HoTT made me think that the best route to go would be via nn-categories formulated entirely in the internal language of an ambient (,1)(\infty,1)-topos.

This might be interesting, since the HoTT internal formulation should in the end look rather naive and strict, but actually be fully general (weak).

I seem to remember that HoTT-internal 1-categories – which externally should be (,1)(\infty,1)-categories – have already been coded in what should be the evident way, but I forget where.(?) Maybe I am hallucinating this memory, though. Developing this would make for a great “Exercise”.

I’d be tending to have the expectation that there is a straightforward way to HoTT-internall say “globular nn-category” by formulating code that looks like it is talking about the strict definition, with the HoTT machine automatically making everything externally be (,n)(\infty,n). But now that I say this it sounds awfully naive. Has anyone thought about this?

Posted by: Urs Schreiber on November 30, 2011 4:45 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

But now that I say this it sounds awfully naive.

Yes. The strict globular definition involves saying that lots of things are equal. If you write that down naively in type theory, you’ll end up asserting a lot of paths – but no coherences between them. And if you try to make the coherences precise, then you end up reinventing globular operads.

There are some number of people thinking about this question, including me. It’s tricky even to define an internal (,1)(\infty,1)-category because it has to involve infinitely much data, and type theory is essentially finitary; you have to encode infinite things by using inductive or coinductive types. I have confidence that we’ll figure it out sooner or later, but I think it will take more work than an “Exercise”. (It’s definitely not clear to me that the answer will end up looking naive and strict; in fact I currently don’t see any way that it can.)

Posted by: Mike Shulman on November 30, 2011 8:20 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

you’ll end up asserting a lot of paths – but no coherences between them.

Maybe I am thinking of a definition that starts out as described in type-theoretic definition of category

Obj : Type

hom (a b : Obj) : Type

but then continues as

triangles : (a b c : Obj) : Type 

components : forall a b c : Obj, triangles a b c -> hom a b * hom b c

composite : forall a b c : Obj, triangles a b c -> hom a c

and asserts coherence maybe along the lines here by saying that the total space map of

components

is an equivalence.

(I am just playing around in a spare minute, this is not meant to be claiming anything.)

Posted by: Urs Schreiber on November 30, 2011 11:03 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

That sounds to me like a type-theoretic version of Segal spaces (which one would then presumably want to require to be complete). I would make the triangles dependent on their boundary too:

triangle (a b c : Obj) (f : hom a b) (g : hom b c) (h : hom a c) : Type

and so on, thereby building Reedy fibrancy into the definition. I think that’s one of the most promising avenues; the problem is that it seems to require infinitely many types: Obj, hom, triangle, tetrahedron, etc. We can’t write an infinitely long Coq file. (-:

By the way, I didn’t mean to sound dismissive. It’s important to have lots of people thinking about this. I just wanted to say, I think it’s a hard problem.

Posted by: Mike Shulman on December 1, 2011 6:05 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

The paper uses in a crucial way the idea of “correspondences”, which is new to me. I’m wondering what’s going on, even in the “classical” setting of Cat.

Cat is a cartesian closed category, but it is not locally cartesian closed: in general, for a small category C, you don’t expect Cat/C to be cartesian closed. (Right?)

However, in the case of the free-standing arrow [1], the category Cat/[1] is cartesian closed; the category Cat/[1] is equivalent to a category of “correspondences” of categories, hence the name. This fact, and higher generalizations, is the basis for one of the key axioms in the paper.

I don’t have a context to put this in. What sort of conditions on a small category C will ensure that Cat/C is cartesian closed?

Posted by: Charles Rezk on December 3, 2011 7:36 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

It is somewhat rare that Cat/C is cartesian closed. It fails even for C= [2], the free walking composition. Let’s see this explicitly: Recall that [2] is the category with objects 0, 1, and 2 and generated by arrows from 0 to 1, and 1 to 2.

In order for Cat/[2] to be cartesian closed, we need there to be an internal hom, and for that to exist we need for every X in Cat/[2], the fiber product functor

X× [2]():Cat/[2]Cat/[2]X \times_{[2]} (-) \; : \; Cat/[2] \rightarrow Cat/[2]

to commute with colimits. I’ll describe an explicit example where it fails to commute with colimits. Namely we will consider the pushout square which defines [2]. This is a pushout square living over [2] with corners [0], [1], [1], and [2]. These two [1]’s are, respectively, the morphisms from 0 to 1 and 1 to 2 inside [2]. The point [0] maps to the object 1, where the two arrows are glued.

Now we let X = [1], but where the map from X to [2] picks out the composite arrow from 0 to 2. Applying the pullback functor, we see that the above square becomes a square with corners \emptyset, [0], [0], and [1]. It is no longer a pushout square.

So the fiber product functor does not commute with colimits, and hence can’t have a right adjoint. There is no internal hom, so Cat/[2] is not cartesian closed.

One might suspect from this that for Cat/C to be cartesian closed we need C to be a category without any compositions. While I think that is sufficient, the story is more complicated. I suspect that Cat/C is cartesian closed whenever C is a groupoid, as long as you are careful to always take homotopy limits and homotopy colimits, and to mean cartesian closed in the correct weak 2-categorical sense.

Of course this doesn’t quite give what you were asking for: a characterization. I wish I understood better precisely which C have Cat/C cartesian closed, but I don’t have good general picture either.

Posted by: Chris Schommer-Pries on December 4, 2011 4:05 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

Ah! Here is a better question: which functors f:DCf:D\to C have the property that the associated pullback f *:Cat/CCat/Df^*: Cat/C\to Cat/D commutes with colimits. (And of course, I mean everything in the derived sense.)

This is better question, because the class of such ff is closed under composition and (derived) base change. In fact, let MM be the class of fibrations ff in CatCat which have the above property.

I’ll bet that MM is characterized by a lifting property.

Playing around with your example suggests at least a necessary condition to be in MM (other than being a fibration): such fMf\in M should have the right lifting property with respect to d 1:[1][2]d^1:[1]\to [2]. That is, if a morphism in the image of ff factors, then the factorization lifts. This seems to agree with your examples, since if CC is either a groupoid or has no compositions, then any fibration f:DCf:D\to C is in MM.

Posted by: Charles Rezk on December 4, 2011 5:58 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

I mean, then any fibration f:DCf:D\to C has the right lifting property with respect to d 1d^1.

Posted by: Charles Rezk on December 4, 2011 6:03 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

By the way, why is it that even though I always select “Remember personal info (requires cookies)” each time I post, I still have to type in my name, email, etc. again each time?

Posted by: Charles Rezk on December 4, 2011 6:05 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

Charles, perhaps Conduché functors are relevant here?

By the way, why is it that even though I always select “Remember personal info (requires cookies)” each time I post, I still have to type in my name, email, etc. again each time?

This glitch has been around for some time.

Posted by: Todd Trimble on December 4, 2011 8:31 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

Ah, excellent! I was headed toward something like that; now I’m glad I don’t have to work it out.

Though I don’t quite understand the definition as stated on that page; what does “connected by a zigzag of commuting morphisms” mean? I’m guessing this zig-zag is in a suitable category of factorizations of the given map; are morphisms of factorizations required to be “fixed at the ends”? I assume they are, in which case you should be able to say that p:EBp:E\to B is strict Conduche if the fiber of E [2]E [1]× B [1]B [2]E^{[2]}\to E^{[1]}\times_{B^{[1]}} B^{[2]} over each object has connected nerve, and pp is weak Conduche if its fibrant replacement is strict Conduche.

If so, this suggests a derived version; a functor pp of (,1)(\infty,1)-categories is exponentiable if for all k2k\geq2, the fiber of E [k]E [1]× B [1]B [k]E^{[k]}\to E^{[1]}\times_{B^{[1]}} B^{[k]} over each object of the domain has contractible nerve. (Maybe you still only need k=2k=2, I dunno.)

Posted by: Charles Rezk on December 4, 2011 8:58 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

Your surmises in the second paragraph look right to me. You might prefer the formulation Ross Street gives in his note on Powerful Functors over the description given in the nLab. See condition (iii) in the theorem on page 5; I agree “commuting morphisms” in the nLab is a little confusing. I may go in there and make a change.

Posted by: Todd Trimble on December 4, 2011 10:24 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

Here is a belated comment. Chris had written:

One might suspect from this that for Cat/C to be cartesian closed we need C to be a category without any compositions. While I think that is sufficient, the story is more complicated. I suspect that Cat/C is cartesian closed whenever C is a groupoid

It would seem to me that the last assertion is a corollary of the fact that Cat (,1)Cat_{(\infty,1)} is an “absolute distributor” in the sense around p. 25 of Jacob Lurie’s “(,2)(\infty,2)-categories”-article.

By axiom (3) of “distributor” every slice Cat (,n)/XCat_{(\infty,n)}/X over an \infty-groupoid XX is cartesian closed.

Posted by: Urs Schreiber on May 21, 2012 9:14 AM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

My thesis advisor often used the word ‘unicity’ instead of the more common ‘uniqueness’. I’ve never seen anyone else do that until now!

I did it in my doctoral dissertation, but perhaps you did not read that …

Posted by: Toby Bartels on December 4, 2011 11:55 AM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

Now that you mention it, it rings a bell. Maybe it skips a generation?

Posted by: John Baez on December 4, 2011 11:57 AM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

I recall we had a little thread on ‘unicity’ way back when, and I still find myself preferring ‘uniqueness’, but not enough that I feel like revisiting the debate. :-)

Posted by: Todd Trimble on December 4, 2011 10:38 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

I had the pleasure of attending a talk by Chris about this paper at the joint meetings last weekend. I haven’t read the paper very carefully yet, but I thought I would mention a few details that I learned from his talk. I think the whole picture that these guys have come up with is very beautiful—and worth talking some more about, here at a place that takes its name from nn-categories.

The first thing they notice is that from the (,1)(\infty,1)-category (,n)Cat(\infty,n)Cat, we can characterize an important subcategory in purely (,1)(\infty,1)-categorical terms: the 0-truncated objects. It turns out that the 0-truncated (,n)(\infty,n)-categories are precisely those which contain no nonidentity automorphisms (at all levels); they call these gaunt nn-categories. Furthermore, we have that:

  • every gaunt nn-category is equivalent to a strict (n,n)(n,n)-category: first make it skeletal, and then every coherence cell is an automorphism and hence must be an identity.

  • every kk-cell, and moreover every pasting diagram (such as the objects of Θ n\Theta_n), is a gaunt nn-category.

They define a “universal category of pasting diagrams” ϒ n\Upsilon_n which contains Θ n\Theta_n, but is rather larger. Their axioms for the (,1)(\infty,1)-category (,n)Cat(\infty,n)Cat are then, roughly:

  1. (,n)Cat(\infty,n)Cat contains ϒ n\Upsilon_n as a full, strongly-generating subcategory of 0-truncated objects.

  2. Every morphism into a kk-cell is exponentiable.

  3. Certain pushouts of kk-cells exist and are what they ought to be (this gives compositions).

  4. (,n)Cat(\infty,n)Cat is universal with respect to properties (1)–(3): any other (,1)(\infty,1)-category satisfying these is a localization of (,n)Cat(\infty,n)Cat.

The first three axioms alone characterize (,1)(\infty,1)-categories that are localizations of (,n)Cat(\infty,n)Cat, including (for instance) (m,n)Cat(m,n) Cat for all nmn\le m\le \infty.

With the axioms phrased in this way, it is fairly clear that a category satisfying all four axioms must be unique up to (non-unique) equivalence. Moreover, since any automorphism preserves 0-truncated objects, which are strongly generating by the first axiom, any automorphism of (,n)Cat(\infty,n)Cat must be induced by an automorphism of the theory of gaunt nn-categories, and those are strict and therefore easy to reason about. (Some work is required to prove merely from the four axioms that the 0-truncated objects can be identified with gaunt nn-categories.) This is how they show that the space of models is B(/2) nB (\mathbb{Z}/2)^n.

The way they show that any particular purported model of (,n)Cat(\infty,n)Cat satisfies the axioms is to produce one model of the axioms, show that the purported model satisfies the first three axioms and is hence a localization of the known model, then check that the localization is in fact an equivalence.

Moreover, the way they produce one model is quite pleasingly tautological: take presheaves on ϒ n\Upsilon_n and localize them in a universal way so as to force the desired pushout equations (axiom (3)) to hold and preserve axiom (2). Since exponentials in localizations are closely related to pullback-stability properties of the maps being localized at, preserving axiom (2) just means closing up under certain pullbacks. Axiom (4) will then hold by the universality of localization.

Posted by: Mike Shulman on January 10, 2012 9:50 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

I think the whole picture that these guys have come up with is very beautiful—and worth talking some more about, here at a place that takes its name from n-categories.

Wholeheartedly agreed.

It turns out that the 0-truncated (,n)(\infty,n)-categories are precisely those which contain no nonidentity automorphisms

Just to see if I am missing a subtlety, since you say “it turns out”: this is also obvious, no?

But it’s nice how with the kk-truncated objects here one gets kind of the “negative” (in the sense of photography) of the (,k)(\infty,k)-hierarchy. The kk-truncated (,n)(\infty,n)-categories have non-trivial invertible morphisms at most up to degree kk.

The k-connected (,n)(\infty,n)-categories however still have all k\leq k-morphism trivial, it seems.

This is how they show that the space of models is B/2B \mathbb{Z}/2.

You mean B(/2) nB (\mathbb{Z}/2)^n.

Posted by: Urs Schreiber on January 11, 2012 4:15 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

Just to see if I am missing a subtlety, since you say “it turns out”: this is also obvious, no?

Well, since I’m not really comfortably conversant with any model of (,n)(\infty,n)-categories, I’m hesitant to assert that anything about them is “obvious”. But it is very intuitive to me.

You mean B(/2) nB(\mathbb{Z}/2)^n.

Thanks; fixed.

Posted by: Mike Shulman on January 11, 2012 7:09 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

I’m hesitant to assert that anything about them is “obvious”.

What I mean is: the (,1)(\infty,1)-category of (,n)(\infty,n)-categories – for whatever definition – is defined to be such that the “transfors” all have components in the invertible cells. kk-Truncation on an object is detected by the vanishing of such invertible (k+1)(k+1)-transfors into it. That’s the case if there no non-trivial invertible (k+1)(k+1)-cells exist.

Of course this is not a formal proof, but I’d think this is easily made a formal proof for any of the existing models. But I admit that I haven’t checked if one runs into subtleties. That’s why I asked.

Posted by: Urs Schreiber on January 11, 2012 7:36 PM | Permalink | Reply to this

Re: The (∞,1)-Category of (∞,n)-Categories

Well, as I mentioned, it looks like there is a bit of work required to do this starting only from the four axioms. But probably you’re right that in any particular model, it is easy.

Posted by: Mike Shulman on January 11, 2012 7:38 PM | Permalink | Reply to this

Post a New Comment