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January 22, 2009

Limits in the 2-Category of 2-Hilbert Spaces

Posted by John Baez

guest post by Jamie Vicary

I’m interested in understanding the limits that exist in 2Hilb, the 2-category of finite-dimensional 2-Hilbert spaces. (The category 2Vect of finite-dimensional 2-vector spaces over Vect {}_{\mathbb{C}} would be just as good.)

One sort of limit that people study in 2-categories is the ‘iso-inserter’, which is like a weakened version of an equaliser.

Definition. In a 2-category, given parallel 1-cells f,g:ABf,g:A \to B, their iso-inserter is a 0-cell SS equipped with a 1-cell s:SAs: S \to A and an invertible 2-cell σ:fsgs\sigma : f s \Rightarrow g s, satisfying these conditions:

  1. For any 1-cell z:ZAz: Z \to A and invertible 2-cell ζ:fzgz\zeta: f z \Rightarrow g z, there is a unique 1-cell c:ZSc:Z \to S such that sc=zs c=z and σc=ζ\sigma c = \zeta.
  2. Given 1-cells c,d:ZSc,d:Z \to S and a 2-cell ν:scsd\nu:s c \Rightarrow s d such that (σd)(fν)=(gν)(σc)(\sigma d) (f \nu) = (g \nu) (\sigma c), there is a unique 2-cell μ:cd\mu:c \Rightarrow d with sμ=νs \mu = \nu.

Unfortunately, it seems to me that 2Hilb doesn’t have all iso-inserters; just try calculating the iso-inserter of id Hilb\mathrm{id} _{\mathrm{Hilb}} with itself! Just from the first condition, you can sort of tell that the 0-cell SS should be a 2-Hilbert space having objects which are Hilbert spaces equipped with an automorphism, and 1-cells which are linear maps that ‘get along’ with the choices of automorphism. Unfortunately, this requirement of ‘getting along’ is quite stringent, and SS winds up being infinite-dimensional. Since we defined 2Hilb to only have finite-dimensional objects, this object doesn’t exist.

This doesn’t mean we should give up, because iso-inserters are a very constrained type of limit, and they won’t be the right type to consider in some 2-categories. The most powerful class of limits are called ‘bilimits’, but you have to be quite brave to wheel them onto the battlefield; because of the way they’re defined, they can be quite difficult to get to grips with in a particular 2-category, and as far as I’m aware, no direct definition (along the lines of the one given above for an iso-inserter) has been published for any nontrivial sort of bilimit. For this reason people often make do with simpler classes of limits, such as the ‘pie-limits’, of which the iso-inserter is an example. These are easier to define explicitly, but are quite ‘brittle’: the comparison 1-cells are unique up to isomorphism, rather than up to equivalence.

For some 2-categories this ‘brittleness’ is too much to take, and it’s possible for a 2-category to have bilimits but not pie-limits. Hopefully, this is what’s going on with 2Hilb. Fortunately, for these categories, there is a powerful coherence theorem we can use: every 2-category with bilimits is biequivalent to a 2-category with pie-limits. So, here’s my question: is there a 2-category, biequivalent to 2Hilb, that has all (finite) pie-limits?

Perhaps such a 2-category doesn’t exist. This would be a real shame, because 2Hilb is a categorification of Hilb, which has excellent completeness properties, and which itself is a categorification of \mathbb{C}, which also has excellent completeness properties. The formal connection between these different types of ‘completeness’ is strengthened by Bruce Bartlett’s recent observation, described in Proposition 3.5 of his thesis, that 2Hilb is the 2-category of Cauchy-complete H*-categories, which is a nice categorification of the statement that Hilb is the category of Cauchy-complete inner-product spaces over \mathbb{C}.

All the facts about 2-categorical limits that I’ve mentioned here are discussed in these two papers:

  • G. M. Kelly, Elementary observations on 2-categorical limits, Bull. Austral. Math. Soc. 39 (1989), 301–317.
  • J. Power, 2-categories, BRICS Notes Series, NS-98-7.
Posted at January 22, 2009 7:31 PM UTC

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Re: Limits in the 2-Category of 2-Hilbert Spaces

Have you thought about whether 2Hilb\mathbf{2Hilb} is Cauchy-complete in some appropriate 2-categorical sense?

Indeed, what is the appropriate 2-categorical sense?

I ask this because, given that 2Hilb\mathbf{2Hilb} doesn’t seem to have all the limits you want, you might start by asking whether it satisfies even very weak completeness conditions; and Cauchy-completeness can be regarded as a very weak (co)completeness condition.

Posted by: Tom Leinster on January 26, 2009 2:08 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Thanks for the comment, Tom!

If I remember right, it’s true that an ordinary category is Cauchy-complete iff it admits all absolute weighted limits. One problem is that what this means depends on how you view the category as being enriched — for example, any category enriched over Vect is also enriched over Set, but these two points of view lead to different notions of Cauchy-completeness. I suppose this problem would remain when the notion of Cauchy-completeness is categorified, and it would serve to make everything quite complicated! 2Vect should probably be properly regarded as enriched over itself, but I prefer to get away with thinking of it as enriched over Cat — I suppose it’s possible that this is spoiling the limits.

Cauchy completeness often has to do with splitting idempotents, so I suppose this is a sort of limit I could check for. The question then is what flavour of limit to choose — I suppose I just need to bite the bullet and choose the most general possible one!

Posted by: Jamie Vicary on January 26, 2009 10:15 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Jamie: the way you’ve phrased your question, few people will know enough to understand it. We can phrase it in a way that more algebraists will appreciate.

Most algebraists don’t know what a 2-Hilbert space is. But since the 2-category of finite-dimensional 2-Hilbert spaces is equivalent to that of finite-dimensional 2-vector spaces, and since it’s easy to explain the notion of 2-vector space, we can rephrase it like this.

Fix a field kk, say the complex numbers. Let VectVect be the category of finite-dimensional vector spaces over this field.

Define a category to be a (finite-dimensional) 2-vector space if it is equivalent to Vect nVect^n for some finite nn. Note that Vect nVect^n is an abelian category of a pathetically nice sort.

Let 2Vect2Vect be the 2-category with:

  • 2-vector spaces as objects
  • exact functors as morphisms
  • natural transformations as 2-morphisms

We want to know what bilimits this 2-category has, or how to expand it a bit to have more.

There are many other ways to describe 2Vect2Vect (or more precisely, equivalent 2-categories), some of which suggest possible answers. For example, we can describe 2Vect2Vect as the 2-category with:

  • categories of representations of finite-dimensional semisimple algebras as objects
  • exact functors as morphisms
  • natural transformations as 2-morphisms

It’s also the 2-category with:

  • categories of representations of finite-dimensional semisimple algebras as objects
  • left exact functors as morphisms
  • natural transformations as 2-morphisms
These are 2-categories that certain algebraists should know and love. There are big fat books on representations of associative algebras.

Moreover, this suggests embedding 2Vect2Vect in a larger 2-category that might have more bilimits, such as:

  • categories of representations of finite-dimensional algebras as objects
  • exact functors as morphisms
  • natural transformations as 2-morphisms

or perhaps the one with:

  • categories of representations of finite-dimensional algebras as objects
  • left exact functors as morphisms
  • natural transformations as 2-morphisms

The last one here is closely related to the bicategory with:

  • finite-dimensional algebras as objects
  • bimodules as morphisms
  • bimodule homomorphisms as 2-morphisms

It’s also worth pondering the 2-category with:

  • abelian kk-linear categories as objects
  • left exact functors as morphisms
  • natural transformations as 2-morphisms

To make progress, I think you need to consider a lot of these options. I also think that Elgueta’s work would be interesting to consider here. His 2-category of ‘generalized 2-vector spaces’ might have more bilimits than that of of 2-vector spaces.

Posted by: John Baez on January 26, 2009 7:32 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

The last one here is closely related to the bicategory with:

- finite-dimensional algebras as objects - bimodules as morphisms - bimodule homomorphisms as 2-morphisms

Once we are here, maybe it helps (maybe not, though) to go one step further even:

for some fixed symmetric monoidal closed category VV (such as VectVect) the 2-category with:

- objects are VV-enriched categories VV, thought of as placeholders for the category of VV-functors [C,V][C,V] which is the category of modules over CC;

- morphisms CDC \to D are profunctors, i.e. VV-functors C op×DVC^{op} \times D \to V, composition is composition of profunctors;

- 2-morphisms are VV-natural transformations.

For V=VectV = Vect this generalizes John’s last couple of examples from algebras to algebroids.

While it may look like complete overkill, I find this way of looking at the question helpful for one reason:

The very general 2-category CC of 2-vector spaces which is modeled on Vect (in that End C(1)=VectEnd_C(1) = Vect) is

2Vect=VectMod, 2 Vect = Vect-Mod \,,

the 2-category whose objects are categories equipped with an action of the monoidal category VectVect.

All the 2-categories John listed are subcategories of this big one, which is hard to get under control.

Now, the enriched perspective shows what is special about the sub-2-category whose objects are algebroid modules: these are the 2-vector spaces with basis. The algebroid CC is the “basis” for the 2-vector space CModC-Mod, since CMod[C,V]C-Mod \simeq [C,V] just as for a vector space WW with basis there is a set SS such that W=[S,k]W = [S,k] (kk the ground field).

All 2-vector spaces with basis, that seems to be a reasonable 2-category to consider.

(And this reminds me that somebody should write an nnLab entry on profunctors…)

Posted by: Urs Schreiber on January 26, 2009 9:40 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

A quick google search for profunctors and bilimits produces Profunctors, open maps and bisimulation which at least states (section 5.1) that the bicategory Prof(V)Prof(V) of VV-enriched categories, profunctors and natural transformations has all pseudo/bi-products and -coproducts.

(They look just at V=SetV = Set, though.)

But somebody here will know: which kind of weak limits does Prof(V)Prof(V) have, more generally?

Posted by: Urs Schreiber on January 27, 2009 11:39 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Side note: if you want to find Australian stuff you probably need to search for “module” in addition to “profunctor” (which unfortunately is a much less unique word), and some people use “distributor” as well. I personally dislike “profunctor” because it is nothing like a pro-object in a functor category.

According to Cauchy characterization of enriched categories, VModV-Mod has lax colimits, and products which are also coproducts (analogously to in an additive category). I don’t know about other limits; I would suspect that it has few.

I think that VModV-Mod is often better regarded as either a double category, an equipment (I, II) or an enhanced 2-category, especially when you want to consider what sorts of limits it has. These approaches all keep track of the functors as well as the profunctors, which actually play an important role when considering limits.

Posted by: Mike Shulman on January 27, 2009 2:33 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Thanks Mike, good points. I am fond of your notion of framed bicategories and it’s good to know that VModV-Mod in the general enriched/profunctorial way is an example, as one would hope.

VModV−Mod has lax colimits, and products which are also coproducts (analogously to in an additive category). I don’t know about other limits; I would suspect that it has few.

Hm. This will probably just show my ignorance, but anyway: given how bimodules are so close to spans it feels like morphisms should be very relaxed about which direction they go. Why does that not reflect itself in a correspondence between colimits and limits?

Posted by: Urs Schreiber on January 27, 2009 6:21 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Well, I guess if VV is symmetric then you are right: VModV Mod is equivalent to VMod coopV Mod^{coop} via the operation taking AA to A opA^{op}, so it should have oplax limits that are the same as its lax colimits. The framing makes it asymmetric, though: the coprojections of the lax colimit are maps (i.e. functors), but the projections of the oplax limit are “co-maps.”

Posted by: Mike Shulman on January 28, 2009 3:19 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

VModV Mod is equivalent to VMod coopV Mod^{coop} # via the operation taking AA to A opA^{op}, so it should have oplax limits that are the same as its lax colimits.

All right, good. So now we can go back to Jamie’s question and see if this is of any help:

Jamie is looking at the sub-2-category

KV2Vect=Bimod {k n} nVectMod KV2Vect = Bimod_{\{k^n\}_n} \subset Vect-Mod

of VectModVect-Mod (= 2-vector spaces with basis = Vect-enriched categories, profunctors, transformations) on objects of the form Bk n\mathbf{B}k^n namely VectVect-enirched categories with a single object whose endomorphism algebra is that of n×nn \times n diagonal matrices).

Now, we know that VectModVect-Mod has all lax limits. So, to see if a particular lax limit (are lax limits of any use to you, Jamie?) exists in KV2VectKV2Vect one strategy might be to check if it exists after inclusion into VectModVect-Mod and happens to lie again in KV2VectKV2Vect (that may not be sufficient for it to exist in KV2Vect, I suppose, but may be a start).

Maybe with a concrete example of a diagram in KV2VectKV2Vect for which Jamie needs the limit, if any, we could say more.

Posted by: Urs Schreiber on January 28, 2009 8:03 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

By the way:

…if VV is symmetric then…

If VV is not symmetric, don’t we have to be careful with saying C opC^{op} for CC VV-enriched? Kelly (p. 12) considers C opC^{op} only for the case that VV is symmetric monoidal.

What’s generally the assumption when we talk about the bicategory VModV-Mod of distributors?

By the way, I created nnLab: distributor and started adding a bit material. Help is appreciated.

Posted by: Urs Schreiber on January 28, 2009 10:23 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

You can certainly develop a lot of the basic theory replacing “symmetric monoidal” with “braided monoidal”, including for instance consideration of C opC^{op}.

A typical standing assumption in developing a satisfactory theory of enriched (bi)modules/profunctors/distributors is that the hom-base VV is complete cocomplete symmetric monoidal closed. I think some people use the word ‘cosmos’ to refer to such VV, although I’m not sure that completeness is usually assumed as part of the definition (you don’t need it to define the bicategory VV-Mod).

I can probably add something to the article on distributors pretty soon.

Posted by: Todd Trimble on January 28, 2009 2:49 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Well, the standard assumption if you’re an Australian would be that VV is a bicategory with colimits in the hom-categories preserved on both sides by composition. The amusing thing is that while you need at least a braiding to define C opC^{op}, you don’t need to define C opC^{op} to talk about distributors, if you think of them as bimodules.

Posted by: Mike Shulman on January 28, 2009 3:57 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Incidentally, Mike, is the condition on VV you just named the defining condition for ‘cosmos’ as the term is usually used? I seem to remember seeing such a definition, although I think the etymology of this term traces further back, and was originally an acronym: cocomplete symmetric monoidal, plus an -os ending in analogy with ‘topos’.

Posted by: Todd Trimble on January 28, 2009 4:59 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

you don’t need to define C opC^{op} to talk about distributors

Not as long as we don’t try to compose them, I suppose.

Posted by: Urs Schreiber on January 28, 2009 7:17 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

the standard assumption if you’re an Australian would be that VV is a bicategory with colimits in the hom-categories preserved on both sides by composition. The amusing thing is that while you need at least a braiding to define C opC^{op}

Just to be sure: you implicitly switched from bicategories back to monoidal categories in the second sentence, right? There is no reasonable way to say C opC^{op} when CC is a generic bicategory-enriched category, is there?

Posted by: Urs Schreiber on January 28, 2009 7:21 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Correct, there is no reasonable definition of C opC^{op} when CC is enriched over a bicategory VV. (In fact, I would question whether that notation is justified when VV is just braided monoidal, since then there are “left opposites” and “right opposites” that aren’t in general the same.) However, you can talk about and compose distributors with no trouble, without the need for any braiding.

For simplicity of exposition, let VV be a cocomplete, non-symmetric, non-braided, monoidal category. (The generalization to bicategories is easy.) Let CC, DD, and EE be VV-enriched categories. A distributor from CC to DD is defined to consist of, for each pair of objects cCc\in C and dDd\in D, an object H(d,c)VH(d,c)\in V, together with action maps H(d,c)D(d,d)H(d,c)H(d,c) \otimes D(d',d) \longrightarrow H(d',c) and C(c,c)H(d,c)H(d,c)C(c,c') \otimes H(d,c) \longrightarrow H(d,c') satisfying suitable associativity and unitality conditions. If HH is such a distributor and KK is a distributor from DD to EE, then their composite H DKH\otimes_D K is defined by the coequalizer d 1,d 2DH(d 2,c)D(d 1,d 2)K(e,d 1) dDH(d,c)K(e,d)(H DK)(e,c)\coprod_{d_1,d_2\in D} H(d_2,c) \otimes D(d_1,d_2) \otimes K(e,d_1) \;\rightrightarrows\; \coprod_{d\in D} H(d,c) \otimes K(e,d) \longrightarrow (H\otimes_D K)(e,c) This composition is always unital, and is associative as long as \otimes preserves colimits in VV on both sides.

I don’t know what “the usual” definition of “cosmos” is. I’ve read that Benabou originally used it to mean “complete and cocomplete closed symmetric monoidal category.” In some of Street’s papers (“Cauchy characterization of enriched categories” I think) he uses it to mean a bicategory which is equivalent to VModV Mod for a suitable bicategory VV (rather than for VV itself); the thought is that VV is more like the “site” of a topos than the topos itself. I never heard about it being an acronym, but it sounds plausible.

Posted by: Mike Shulman on January 28, 2009 8:30 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Thanks, Mark and Urs, for your very useful comments!

The sort of limits I’m really interested in are ‘geometrical’ ones. For example, given two monics a:AMa: A \hookrightarrow M and b:BMb: B \hookrightarrow M, I’d like to calculate their ‘essential intersection’. Usually the intersection of two subobjects is given by the pullback, but that won’t do here as the images of aa and bb might contain elements which aren’t equal, but only isomorphic.

This doesn’t sound like a lax colimit to me, but hopefully it’s nevertheless possible in Vect-Mod, and perhaps also in 2Vect.

Posted by: Jamie Vicary on January 29, 2009 10:08 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

This doesn’t sound like a lax colimit

It does sound like a lax limit, though!

as the images of aa and bb might contain elements which aren’t equal, but only isomorphic.

Yes, that’s what the laxness is supposed to take care of:

The lax limit/pullback of

B b A a M \array{ && B \\ && \downarrow^b \\ A &\stackrel{a}{\to}& M }

is not a commuting diagram, but is a 2-cell

laxlim B b A a M \array{ laxlim &\to& B \\ \downarrow &\Downarrow& \downarrow^b \\ A &\stackrel{a}{\to}& M }

which is the universal such 2-cell in some more-or-less obvious way. So, indeed, an object of laxlimlaxlim need not hit the same objects as we chase it around the two boundaries of this square: the resulting objects only need to be connected by a morphism.

In a pseudo-limit that morphism would be forced to be an isomorphism. But I don’t know about pseudo-limit in VectModVect-Mod at the moment. Above we (only) seem to have concluded that VectModVect-Mod has all lax limits. So the lax pullback of

B b A a M \array{ && B \\ && \downarrow^b \\ A &\stackrel{a}{\to}& M }

should exist in VectModVect-Mod. One should try to compute it there and see if it lands in your sub-bicategory of KV-2-vector spaces if aa and bb are taken from this sub-bicategory.

So I’d say next we need to get our hands on the concrete comutation of lax limits in VectModVect-Mod

Posted by: Urs Schreiber on January 29, 2009 12:04 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Urs wrote

In a pseudo-limit that morphism would be forced to be an isomorphism.

Right, exactly, pseudo-limits are what we need! That’s what I was getting at when I said that what I wanted didn’t seem like a lax colimit (obviously I should have said “lax limit”, sorry.)

I think that the iso-inserter that I described is an example of a pseudo-limit. (At least, it’s unique up to isomorphism and features an invertible 2-cell.) Pseudo-pullbacks are familiar from Jeff Morton’s and others’ work on composing spans of groupoids, but I think that we’ll still have the same problem computing them in 2Hilb as we do with the iso-inserters: an object of the pullback is a pair of objects equipped with an isomorphism, and since isomorphic objects in 2-Hilbert spaces have an infinite number of isomorphisms between them, the pseudo-pullback will be infinite-dimensional.

As I wrote here, I think this problem is caused by trying to compute Cat-weighted limits, rather than 2Hilb-weighted ones. I’m working on this! Maybe someone else has more facility with this sort of thing and can work it out faster.

I’m a bit confused about the page on 2-categorical limits on the nnLab. It implies that we should never be interested in bilimits in a strict 2-category. But isn’t it quite possible to have a strict 2-category with bilimits, but nevertheless lacking strict, pseudo or lax limits? After all, even in a strict 2-category, equivalent objects need not be isomorphic.

Posted by: Jamie Vicary on January 29, 2009 1:24 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

I’m a bit confused about the page on 2-categorical limits on the nLab. It implies that we should never be interested in bilimits in a strict 2-category.

Sorry about that; I wrote that unclearly. You are certainly right that there are strict 2-categories having only bicategorical limits. I tried to improve the page (taking into account your objection elsewhere too), but please help if it is still unclear.

I also edited the page to make clear the important distinction between “lax pullbacks” and “comma objects.” What Urs described above is not a lax pullback, but a comma object, which is not an example of a lax limit. Likewise, the iso-inserter you described is not a pseudo-limit, although it is equivalent to the pseudo-equalizer if they both exist.

Posted by: Mike Shulman on January 29, 2009 4:25 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Thanks Mike!

the iso-inserter you described is not a pseudo-limit, although it is equivalent to the pseudo-equalizer if they both exist.

Great! But I’m guessing you’re using ‘equivalent’ in its non-technical sense, right? Surely it should be isomorphic to the pseudo-equaliser, unless I’m very confused.

Posted by: Jamie Vicary on January 29, 2009 4:37 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Nope, only equivalent, not isomorphic. Let f,g:ABf,g:A\;\rightrightarrows\; B be functors in Cat. The objects of the iso-inserter are pairs (a,ϕ)(a,\phi) where aAa\in A and ϕ\phi is an isomorphism f(a)g(a)f(a)\cong g(a). The objects of the pseudo-equalizer are quadruples (a,b,γ,δ)(a,b,\gamma,\delta) where aAa\in A, bBb\in B, γ:f(a)b\gamma:f(a)\cong b, and δ:g(a)b\delta:g(a)\cong b.

Posted by: Mike Shulman on January 29, 2009 8:22 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Ah, I see! Thanks for making that clear.

Posted by: Jamie Vicary on January 30, 2009 9:49 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

What Urs described above is not a lax pullback, but a comma object

Ah, thanks for correcting that. I had one 2-cell going one way where I should have had two 2-cells facing each other at the tip of the cospan diagram, as you describe on that page at “lax pullback”.

Posted by: Urs Schreiber on January 29, 2009 6:53 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

John said:

Jamie: the way you’ve phrased your question, few people will know enough to understand it. We can phrase it in a way that more algebraists will appreciate.

OK — thanks!

You list several 2-categories and bicategories into which 2Vect embeds, and Urs also suggests Vect-Mod, which I suppose is the Mother of them all. A difference appears between this bicategory and Hilb-Mod, where we have the option of taking the full sub-bicategory of H *H^*-categories, which are much better-behaved than general Hilb-modules. So, that’s another one to consider.

John also lists some 2-categories which are biequivalent to 2Vect. (Urs suggested the bicategory of Vect-modules with basis — this is biequivalent to 2Vect as usually defined, right?) It’s important to consider these because, as I said above, there’s this coherence theorem that says, if we choose cleverly among biequivalent bicategories, bilimits should turn into the more-manageable pie-limits. Surely, of all the possible choices, the most manageable bicategory biequivalent to 2Vect is the 2-category for which objects are skeletal 2-vector spaces, and we only allow a single 2-vector space of each dimension. But even then we don’t get iso-inserters, as far as I can tell! So it seems likely to me that this approach is doomed.

However, what Tom said struck a chime: Cauchy completion, and hence enrichment, is important, and I suppose I should really be considering 2Hilb as a 2Hilb-category, and trying to calculate 2Hilb-weighted limits, not Cat-limits. I’ll have a go at this.

Actally, I’m a bit surprised how readily John and Urs have listed other categories. Until I really needed to know, I always lazily assumed 2Hilb would have nice bilimit properties. After all, like I said in the post, there are good reasons to think of it as the categorification of Hilb, and that has good completeness properties. If 2Hilb really isn’t adequately complete, that does knock it off its pedestal somewhat. Maybe I shouldn’t have put it up there in the first place!

Posted by: Jamie Vicary on January 27, 2009 12:11 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Urs suggested the bicategory of Vect\mathbf{Vect}-modules with basis — this is biequivalent to 2Vect\mathbf{2Vect} as usually defined, right?

You mean the bicategory with objects Vect nVect^n, i.e. those VectVect-modules whose basis is a set should be biequivalent to the bicategory whose objects are VectVect-modules whose basis is an arbitrary VectVect-enriched category?

I’d be surprised if this were the case. I should maybe emphasize that I am using the term “basis” of a 2-vector space in what is probably a more general way than has been done elsewhere:

I suggested that given a VectVect-module WW we think of equipping it with a basis by finding an algebroid CC (a VectVect-enriched category) and an equivalence W[C,Vect]W \simeq [C,Vect]. The VectVect-modules Vect nVect^n arise this way by choosing CC to be a discrete category.

Posted by: Urs Schreiber on January 27, 2009 9:55 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Right, I see what you mean now.

I suppose there are plenty of other types of algebra, other than discrete ones, for which the category of representations is Vect n{}^n for some nn. Quasitriangular Hopf algebras, for example. Are there any others? It would surprise me if this were the only class of algebra with this property.

Posted by: Jamie Vicary on January 27, 2009 4:43 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

I suppose there are plenty of other types of algebra, other than discrete ones, for which the category of representations is Vect n\mathbf{Vect}^n for some nn

Yes. All algebras which are Morita equivalent have (by definition of Morita equivalence!) equivalent module categories and hence are, from this perspective, just different choices of bases for the same 2-vector space.

In particular all algebras of block-diagonal matrices with nn blocks are Morita equivalent to k nk^n.

Posted by: Urs Schreiber on January 27, 2009 7:31 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Hi Jamie!

I sit here in my new office, in that no-man’s land one finds oneself in when waiting to move into an apartment, with a getting-serious clothes washing conundrum on my hands.

Bravo for ‘you have to be quite brave to wheel them onto the battlefield’, that’s a lovely turn of phrase. Also, does the word ‘pie-limit’ refer to some kind of amalgam of ‘pie-in-the-sky’ and ‘bilimit’?

Ok, I’m a bit confused by the iso-inserters. Let’s take the example of trying to find the iso-inserter of the pair of identity maps on HilbHilb. Since this situation is so ‘symmetric’, our 2-Hilbert space SS, the 1-cell s:SHilbs : S \rightarrow Hilb and the invertible 2-cell σ:ss\sigma : s \Rightarrow s must also be of a very symmetric, canonical type, i.e. very ‘bland’. It seems impossible then right from the outset to consider that we could construct such a triple (S,s,σ)(S, s, \sigma) which is able to satisfy requirement 1. How could we ever reproduce that ζ\zeta if our starting data is so bland? I guess I’m asking: can you give me a quick example of a 2-category which does have all iso-inserters, just so I can get an intuition for how they work.

As far as 2-vector spaces go, another thing to consider is Lurie’s recent notes/sketch of a proof of the cobordism hypothesis (!). On page 9 he introduces the 2-category Vect 2Vect_2, whose objects are ‘cocomplete’ linear categories, that is Vect-enriched categories which are closed under the formation of direct sums and cokernels. At one point I convinced myself that this was the same as being Cauchy complete, but now suddenly I’m not so sure. In that case it is something to add to John’s list, and to consider.

Posted by: Bruce Bartlett on January 27, 2009 8:59 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Bruce said:

I sit here in my new office, in that no-man’s land one finds oneself in when waiting to move into an apartment, with a getting-serious clothes washing conundrum on my hands.

Great! I hope you’re keeping your office window open ;).

does the word ‘pie-limit’ refer to some kind of amalgam of ‘pie-in-the-sky’ and ‘bilimit’?

That would be fun! But, I think it actually refers to the fact that they’re defined as any limit formed from products, inserters and equifiers.

can you give me a quick example of a 2-category which does have all iso-inserters, just so I can get an intuition for how they work.

The 2-category Gpd is a good example, I think. The iso-inserter of id 2id _{\mathbb{Z}_2} with itself is the groupoid 2+ 2\mathbb{Z}_2 + \mathbb{Z}_2. It’s got two objects, because there are two natural isomorphisms id 2id 2id _{\mathbb{Z}_2} \Rightarrow id _{\mathbb{Z}_2}.

In cat2Hilb\cat{2Hilb}, every 1-cell has an infinite number of endo-isomorphisms, and this causes the problem. But it probably makes more sense to think of them not as an infinite set of discrete isomorphisms, but a finite-dimensional vector space of isomorphisms — in other words, I think the way we choose to express the enrichment is important.

[Jacob Lurie] introduces the 2-category Vect 2{}_2, whose objects are ‘cocomplete’ linear categories, that is Vect-enriched categories which are closed under the formation of direct sums and cokernels. At one point I convinced myself that this was the same as being Cauchy complete, but now suddenly I’m not so sure. In that case it is something to add to John’s list, and to consider.

If a category is preadditive, then it’s Cauchy complete iff it’s got direct sums (finite biproducts) and all idempotents split. In the case you mention, split idempotents follow from the cokernels, so I think his categories are all Cauchy complete.

But still, wouldn’t this still give a category inequivalent to 2Hilb? After all, that’s not the 2-category of Cauchy-complete Hilb-categories, but the category of Cauchy-complete H *H^*-categories.

Posted by: Jamie Vicary on January 27, 2009 4:06 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Ok, I think I sort of understand the 2\mathbb{Z}_2 example, but the whole thing seems so strange that I have no idea why anyone would want to study ‘iso-inserters’.

Thanks for clearing up the Cauchy completeness question. As for the difference between:

(a) the 2-category 2Hilb, whose objects are Cauchy-complete H *H^*-categories (that is, categories whose hom-sets are finite-dimensional Hilbert spaces and carrying a compatible **-operation), whose morphisms are linear **-functors and whose 2-morphisms are natural transformations, and

(b) the 2-category Vect 2Vect_2, whose objects are Cauchy-complete categories enriched over VectVect, whose morphisms are linear functors and whose 2-morphisms are natural transformations,

I would say that yes, certainly they are equivalent 2-categories, but that is missing the point, in the same way that one misses the point if one thinks of Vect and Hilb as equivalent categories. This is the point John tried to stress in his original Quantum Quandaries notes, and which it seems category theorists have been loath to accept :-) (unlike you of course!) Namely, Hilb should be thought of as carrying more structure than Vect, because the hom-sets carry a **-structure (you can take the adjoint of a linear map between Hilbert spaces).

That is, the difference between Hilb and Vect lies not in the structure of Hilb as a category, but rather in the fact that Hilb is not just a category but a category-with-extra-structure. The trouble is, once you start talking that way, the purist category theorists will say “Ah, but then you are no longer doing category theory, you are now doing categories-with-extra-bells-and-whistles theory, which is different. Therefore you are no longer a category theorist and you are expelled from the conference!”

Posted by: Bruce Bartlett on January 28, 2009 5:57 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

When I first spoke about dagger compact categories at the Montreal category theory seminar Joyal indeed mentioned that the dagger was not “categorical”. I never really cared about that but recently I started to think that maybe we shouldn’t be looking at Hilb but at “semi”-Hilb, that is, both considering linear and anti-linear maps as morphisms. The extra bit of structure in Hilb as compared to Vect is now represented by having extra anti-linear morphisms. This resonates particularly well with one of the key structural theorems of QM, namely Wigner’s theorem.

Posted by: bob on January 28, 2009 7:18 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

the dagger was not “categorical”

Is it the condition that the dagger functor :C opC\dagger : C^{op} \to C has to be the identity on objects which is meant to be “not ‘categorical’”?

Independently of how one thinks about this, there may be a point hidden here: to some extent the dagger compact structure on a category “tries to make the category forget the difference between source and target”.

I think this is an old point made by Nils Baas (at least this is my understanding of his point): categories and higher categories with their built-in source-target dichotomy are unnatural in (topological) quantum (field) theory. More natural would be a concept which does not have this distinction in the first place.

I tried to summarize this point at the entry nnLab: hyperstructure, which also states that one formalization coming close to this idea is that of multispans.

And indeed, as you all know, spans, after you force them to form a category by putting an ordering on their legs try to forget this violation by being dagger compact in turn.

Posted by: Urs Schreiber on January 28, 2009 7:41 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

I once (or twice) spent a while trying to write down a formal structure that would encode all the structure one sees not only on spans, but on modules, distributors (in the symmetric case), cobordisms, parametrized spectra, etc. In all these cases not only do you have a bicategory, but a framed bicategory (you have “vertical arrows” that are functions, ring homomorphisms, functors, diffeomorphisms, continuous maps, etc. that act on the spans, modules, etc.), and moreover there is this “nondirectionality.” Just as a span from BB to A×CA\times C can be identified with a span from B×CB\times C to AA, likewise an (A,BC)(A,B\otimes C)-bimodule can be identified with an (AC op,B)(A\otimes C^{op}, B)-bimodule, and so on. And once you have this nondirectionality, the “horizontal composition” in the bicategory is seen as a special case of a more general “gluing” or “cancellation” operation. In a way it feels like a categorification of a modular operad. But there’s also a sense in which it is a “fibration with extraordinary-naturality added”. But there was just so much structure that I gave up trying to encapsulate it all in a way that anyone other than a higher category theorist would be willing to think about.

Maybe I should take a look at Baas’ hyperstructures.

Posted by: Mike Shulman on January 28, 2009 8:43 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

In all these cases not only do you have a bicategory, but a framed bicategory

Or hopefully a framed weak nn-category at least in the case of cobordisms and multispans.

Maybe it is noteworthy that such a framing (or at least something pretty close) does naturally arise for multi(co)spans, as remarked by Marco Grandis at the beginning of part 1 of his series on Cospans in Algebraic Topology:

let DD be the poset which encodes the shape of our multi(co)spans (for Grandis we have D= ×nD = \vee^{\times n} where ={abc}\vee = \{a \to b \leftarrow c\}). Then a multi(co)span in some category C is of course functor

s:DC. s : D \to C \,.

Pullback/pushforward on these produces weak composition operations. But on top of all this, there is one “direction” in which we have strict morphisms: that’s the natural transformations sss \Rightarrow s'.

Now that I think about it, it may be precisely the words “framed weak cubical nn-category” that Grandis is “looking for” at the beginning of his “Cospans in Algebraic Topology”.

Posted by: Urs Schreiber on January 28, 2009 9:16 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

but recently I started to think that maybe we shouldn’t be looking at Hilb but at “semi”-Hilb, that is, both considering linear and anti-linear maps as morphisms. The extra bit of structure in Hilb as compared to Vect is now represented by having extra anti-linear morphisms. This resonates particularly well with one of the key structural theorems of QM, namely Wigner’s theorem.

Bob, how do you capture the structure of those antilinear maps? One way to capture antilinear maps in a structural way is to use the ‘bar’ functor approach as John used in his 2-Hilbert spaces paper,

(1)bar:HilbHilb bar : Hilb \rightarrow Hilb
(2)VV¯ V \mapsto \overline{V}

I’m interested to know if you have reservations about this way of doing things.

Posted by: Bruce Bartlett on January 29, 2009 8:30 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Sorry for the late reply; overlooked your posting. Having a dagger on a compact closed category is of course a way for accounting for anti-linearity. You think of the linear map f:A*\to B as the canonically corresponding anti-linear map f: A\to B. Unfortunately, while the transposed is whinessed explicitly by the cup’s and cap’s, conjugates aren’t. They are just defined by a functor. A way around this is by considering the canonical anitlinear maps s_A:A*\to A. They allow you to construct conjugates as s_B^dag o f o s_A. These s_A’s constitute a natural transformation betwen the identity and the conjugate functor. I am not sure to which extend this can be seen as a defining property for conjugates in `semi’FHilb.

Posted by: bob on February 5, 2009 6:16 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Bruce said:

Thanks for clearing up the Cauchy completeness question. As for the difference between:

(a) the 2-category 2Hilb … and

(b) the 2-category Vect 2{}_2

I would say that yes, certainly they are equivalent 2-categories

Could you explain why they’re equivalent as 2-categories? I don’t see why they should be.

Posted by: Jamie Vicary on January 29, 2009 10:15 AM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Could you explain why they’re equivalent as 2-categories?

Well, I don’t mean anything deep. I just mean that, firstly, anything enriched over HilbHilb is, just on the pure-category-theory level, always equivalent to the same gismo enriched over VectVect. (I guess both of us don’t like talking that way but now we are putting on our ‘pure-category-theorist’ hats).

Secondly, from the viewpoint of pure category theory, we can ignore the **-structure stuff because any category equivalent to Vect nVect^n can be given a **-structure. That’s because VectVect is equivalent to HilbHilb, so we can just pull the **-structure over from Hilb nHilb^n.

Finally, on the level of the 1-morphisms, requiring that a functor between 2-Hilbert spaces be ‘linear’ and requiring it to be ‘linear and preserve the **-operation’ makes no essential difference (again, at least from a pure category theory level. In practice of course, it should be regarded as making a world of difference!). A linear functor F:H 1H 2F : H_1 \rightarrow H_2 and a linear **-functor F:H 1H 2F : H_1 \rightarrow H_2 between 2-Hilbert spaces are both simply characterized by the nonnegative integers

(1)dimHom(e μ,Fe i) dim Hom(e_\mu, Fe_i)

where e μe_\mu runs over a representative set of simple objects for H 2H_2, and e ie_i runs over a representative set of simple objects for H 1H_1. (I once was unsure of this fact, but I checked it in thesis, it’s Lemma 3.8).

So basically at all levels they’re equivalentm, i.e. if we want to be brutal, we can just drop all pretense at being sophisticated and just say that they are the 2-category whose objects are Vect nVect^n for some nn.

Posted by: Bruce Bartlett on January 29, 2009 12:13 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

OK, I see what you’re getting at. But a H *H^*-category isn’t just a Hilb-category with **-structure, the **-structure also has to get along with the enrichment.

You’re the one who first pointed out to me how crucial this is! We can use the nice properties of H *H^*-categories to prove that 2-Hilbert spaces are semisimple, whereas for KV-style 2-vector spaces, it has to be required artificially. I wouldn’t have thought that the categories in Lurie’s Vect 2{}_2 are semisimple, and I would have thought that this would break biequivalence with 2Hilb, even when both are regarded as mere Cat-categories.

Posted by: Jamie Vicary on January 29, 2009 12:55 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

Mmm, I might be confused. I was thinking that if a Vect-enriched category is Cauchy complete (i.e. every pair of objects has a direct sum and all idempotents split) then it must be semisimple. I was worried that this might be a misconception of mine, so I guess I’d better clear it up.

Posted by: Bruce Bartlett on January 29, 2009 1:05 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

I created nnLab: dagger category and nnLab: dagger compact category and started filling in some material. Certainly not comprehensive yet. Your help is appreciated.

Posted by: Urs Schreiber on February 4, 2009 12:09 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

That’s great, Urs. I have changed one little thing already, I’ll have another thorough look at it later and maybe edit some more.

Posted by: Jamie Vicary on February 4, 2009 6:48 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

I have changed one little thing already

Yes, I noticed. Thanks!

I insert little mistakes and omissions here on purpose of course, because it is a great way to cheat you all helping out!

(Since we are on the internet I should admit that this last sentence is meant as a joke. But it is true that there is nothing like small mistakes to get an interaction started… ;-)

Posted by: Urs Schreiber on February 4, 2009 10:22 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

I would say that yes, certainly they are equivalent 2-categories, but that is missing the point, in the same way that one misses the point if one thinks of Vect and Hilb as equivalent categories. This is the point John tried to stress in his original Quantum Quandaries notes, and which it seems category theorists have been loath to accept :-) (unlike you of course!) Namely, Hilb should be thought of as carrying more structure than Vect, because the hom-sets carry a *-structure (you can take the adjoint of a linear map between Hilbert spaces).

In my opinion, having witnessed their objections on the categories mailing list (although getting sidetracked myself in a discussion of monoidal structures on the category of Banach spaces), the key reason for the objection is that they thought of categories as fundamental, when in fact groupoids (rather, \infty-groupoids) are fundamental. Of course, ‘fundamental’ is a pretty subjective term, but what I mean by it (here) is that the concepts of As and of Bs are ways of talking about (precisely) the same things if and only if the \infty-groupoid of As and the \infty-groupoid of Bs are equivalent; this is when terms about As (at least if not evil) can be translated into terms about Bs (and vice versa, compatibly), and the direction of arrows can be simply translated along with everything else.

So in the case at hand, both categories and \dagger-categories should be seen as groupoids with extra structure. A Hilbert space is not the same thing as a vector space (not even in finite dimensions); it has extra structure. So if you think of categories as fundamental, you will reject out of hand any alleged category of (finite-dimensional) Hilbert spaces that's equivalent to the category of (still finite-dimensional) vector spaces. However, the underlying groupoids are not equivalent, since the underlying groupoid of a \dagger-category is not the groupoid of invertible morphisms (interesting as that groupoid may sometimes be to consider) but rather the groupoid of unitary morphisms. And as the forgetful functor from the groupoid of Hilbert spaces (and unitary morphisms) to the groupoid of vector spaces (and invertible morphisms) is faithful but not full, we still get the correct result that Hilbert spaces are vector spaces with (nontrivial) extra structure.

That is, the difference between Hilb and Vect lies not in the structure of Hilb as a category, but rather in the fact that Hilb is not just a category but a category-with-extra-structure. The trouble is, once you start talking that way, the purist category theorists will say “Ah, but then you are no longer doing category theory, you are now doing categories-with-extra-bells-and-whistles theory, which is different. Therefore you are no longer a category theorist and you are expelled from the conference!”

Indeed, I would say that if you consider the (mere) category of Hilbert spaces, you may only include as morphisms those linear transformations with norm bounded by 11 (and perhaps not even all of those). Only the \dagger-category of Hilbert spaces is allowed to include all (bounded) linear transformations.

Posted by: Toby Bartels on February 6, 2009 6:25 PM | Permalink | Reply to this

Re: Limits in the 2-Category of 2-Hilbert Spaces

I created an entry nnLab: 2-vector space.

Please add further details as desired.

Posted by: Urs Schreiber on January 28, 2009 10:00 AM | Permalink | Reply to this

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