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December 12, 2009

The Problem With Lax Functors

Posted by Mike Shulman

As many readers probably know, part of categorification involves replacing equalities by coherent isomorphisms or equivalences. This is basically unavoidable, since after replacing sets by categories, they’re now objects of a category, so it would be evil to ask them to be equal instead of isomorphic. However, there’s another side to categorification: once we turn equalities into isomorphisms, we can go one step further and allow them to be arbitrary morphisms, not even invertible.

Over at the nLab we’ve been discussing how to distinguish between these two terminologically. Whatever we call them, they’re both indisputably important. But the second one (“directification” or “laxification”) has some subtleties that can trap the unwary. In particular, the notion of lax functor between 2-categories (or nn-categories for n>2n\gt 2) is surprisingly ill-behaved. You can’t make a tricategory out of 2-categories, lax functors, and any of the usual sorts of transformations. In fact, lax functors are not even invariant under equivalence!

(Thanks to Chris Schommer-Pries for bringing up this question at mathoverflow and on SBS. I thought I’d take the liberty of elaborating here, since it gives me the opportunity to proselytize a bit more on one of my favorite topics.)

Different people seem to have different initial responses to the idea of lax things (where structure is only preserved up to not-necessarily-invertible morphisms). If you’re used to thinking only about “groupoidal categorification,” then allowing things not to be invertible may seem really dangerous. And you’re not wrong. But if you’re used to thinking about, say, monoidal functors, then requiring everything to be invertible seems far too restrictive. And you’re not wrong there either.

The “pseudo” (or “strong”) kind of functor between monoidal categories, which is the one you might first come up with if you sit down and think “hey, let’s categorify monoids and monoid homomorphisms!”, comes with a natural isomorphism F(ab)F(a)F(b)F(a\otimes b)\cong F(a)\otimes F(b). However, while these do occur, in practice one very often also sees lax monoidal functors which have only a morphism F(a)F(b)F(ab)F(a)\otimes F(b) \to F(a\otimes b). (Colax monoidal functors have a morphism in the other direction. One way to remember which is which is that lax monoidal functors preserve monoids—if I have a multiplication aaaa\otimes a\to a, then a lax monoidal functor gives me a multiplication F(a)F(a)F(aa)F(a)F(a)\otimes F(a)\to F(a\otimes a)\to F(a). And dually, colax monoidal functors preserve comonods.) One particularly important example of a lax monoidal functor that isn’t pseudo came up here just recently.

Lax monoidal functors are great things and behave just about as nicely as you could wish. There is a 2-category MonCat laxMonCat_{lax} of monoidal categories, lax monoidal functors, and monoidal transformations, which of course contains the 2-category MonCat strongMonCat_{strong} where we restrict the monoidal functors to be strong. And by some 2-categorical diagram chasing called “doctrinal adjunction,” any left adjoint in MonCat laxMonCat_{lax} is automatically strong monoidal. In particular, any equivalence in MonCat laxMonCat_{lax} consists of strong monoidal functors, since any equivalence in a 2-category can be improved to an adjoint equivalence. Thus, the notion of “equivalence of monoidal categories” doesn’t depend on what kind of functor you pick, and the notion of “lax monoidal functor” is invariant under this equivalence.

(There’s also, of course, a similar 2-category MonCat colaxMonCat_{colax} with dual properties. If, like me, you are unhappy with this situation, there’s also a double category that combines lax and colax monoidal functors.)

But now, of course, monoidal categories are just (weak) 2-categories with one object, so it’s natural to generalize lax monoidal functors to lax functors between 2-categories. And in fact there are plenty of important examples of these too. For instance, any lax monoidal functor VWV\to W induces a lax functor VProfWProfV Prof \to W Prof between 2-categories of profunctors; this sort of functor is important for “change of enrichment” arguments. If 11 denotes the terminal 2-category, then a lax functor 1C1\to C, for any 2-category CC, is the same as a monad in CC. And more generally, for any set XX, if X iX_{i} denotes the chaotic or indiscrete 2-category with set of objects XX, a unique 1-morphism between any two objects, and only identity 2-cells, then lax functors X iCX_i\to C are the same as “CC-enriched categories” with set of objects XX. (If the idea of categories enriched in a 2-category is unfamiliar, just think of the case when CC is a monoidal category, regarded as a 2-category with one object.)

But wait a minute! The 2-category X iX_i is equivalent to 11, but certainly not every enriched category is equivalent to one having only one object! So lax functors can’t be invariant under the usual notion of equivalence of 2-categories. That is, if CC and CC' are equivalent 2-categories, then I can’t expect to have equivalent 2-categories Fun lax(C,D)Fun_{lax}(C,D) and Fun lax(C,D)Fun_{lax}(C',D).

That already tells us that there’s no hope of a good situation like for monoidal categories, where equivalences in MonCat laxMonCat_{lax} are the same as equivalences in MonCat strongMonCat_{strong}. We might still hope that there’s some 3-category “2Cat lax2 Cat_{lax},” even if the equivalences in it aren’t the same as those in 2Cat strong2Cat_{strong}. But it turns out that fate basically conspires against us here too.

The problem hinges on what the 2-cells in our putative 3-category 2Cat lax2Cat_{lax} are going to be. Suppose F,G:CDF,G\colon C\to D are lax functors; what is a transformation between them? Clearly it should have components α x:F(x)G(x)\alpha_x\colon F(x) \to G(x), with some sort of naturality condition relating to squares like F(x) F(f) F(y) α x ?? α y G(x) G(f) G(y)\array{F(x) & \overset{F(f)}{\to} & F(y)\\ ^{\alpha_x}\downarrow & ?? & \downarrow^{\alpha_y}\\ G(x)& \underset{G(f)}{\to} & G(y)} We could choose to ask this square to commute up to isomorphism (a pseudo natural transformation), or merely up to a morphism in one direction or the other (a lax or colax transformation). All of these work up to a point: we get hom-2-categories 2Cat lax,???(C,D)2Cat_{lax,???}(C,D) consisting of lax functors, ??? transformations (could be pseudo, lax, or oplax), and “modifications.”

However, in no case do we get a 3-category. Why? Well, one of the things you can do in a 3-category is “whisker” a 2-cell α:FG\alpha\colon F\to G with a 1-morphism H:DEH\colon D\to E. But now suppose HH is a lax functor and α\alpha is one of the kinds of transformation above. For the sake of argument, suppose the 2-cell component of α\alpha goes in this direction: α yF(f)α fG(f)α x. \alpha_y \circ F(f) \overset{\alpha_f}{\to} G(f) \circ \alpha_x. We’d like to get a square HF(x) HF(f) HF(y) Hα x Hα y HG(x) HG(f) HG(y)\array{H F(x) & \overset{H F(f)}{\to} & H F(y)\\ ^{H\alpha_x}\downarrow & \Downarrow & \downarrow^{H\alpha_y}\\ H G(x)& \underset{H G(f)}{\to} & H G(y)} i.e. a 2-cell Hα yHF(f)"Hα f"HG(f)Hα x. H\alpha_y \circ H F(f) \overset{\text{"}H\alpha_f\text{"}}{\to} H G(f) \circ H\alpha_x. but how can we do that? The obvious thing to do is to apply HH to α f\alpha_f, but this gives us only a 2-cell H(α yF(f))H(G(f)α x). H\big(\alpha_y \circ F(f)\big) \to H\big(G(f) \circ \alpha_x\big). If HH were a strong functor, then we could compose with constraints like H(jk)H(j)H(k)H(j \circ k) \cong H(j) \circ H(k) on either side like so: Hα yHF(f)H(α yF(f))H(G(f)α x)HG(f)Hα x. H\alpha_y \circ H F(f) \overset{\cong}{\to} H\big(\alpha_y \circ F(f)\big) \to H\big(G(f) \circ \alpha_x\big) \overset{\cong}{\to} H G(f) \circ H\alpha_x. But since HH is only lax, one of these two transformations is going the wrong way, and we’re out of luck. Clearly it wouldn’t help matters if α\alpha were a strong or oplax transformation; we just can’t compose morphisms “head-to-head” or “tail-to-tail.”

So we’re stuck: there is no 3-category of 2-categories, lax functors, and any of the usual sort of transformation. Does this mean that lax functors are terrible things and we should throw them out the window? Well, maybe. There are a few remedies one can try. There is a 1-category of 2-categories and lax functors; it just so happens that lax functors compose strictly associatively. There is even a 2-category of 2-categories and lax functors, where the morphisms are a restricted sort of transformation called an icon. (One interesting thing is that when you regard monoidal categories as 2-categories with one object, lax monoidal functors get identified with lax functors of 2-categories, but monoidal transformations get identified, not with any of the usual sorts of transformation between 2-categories, but with icons! This “explains” why things work so well in the monoidal case but fail in the 2-categorical case, and also leads into some of the subtleties relating to the delooping hypothesis.)

Personally, however, I think the best solution is to consider double categories instead of 2-categories. Many of the 2-categories for which we care about lax functors, such as 2-categories of modules and profunctors, can be enhanced to double categories by adding extra “vertical” arrows. There is a good notion of lax functor between double categories, and of “vertical transformation” between such functors, and these things do arrange themselves into a 3-category DblCat lax,vertDblCat_{lax,vert}. The key is that the lax functors are laxly functorial on horizontal arrows (the arrows in our original 2-categories), but strict or strongly functorial on the new vertical arrows, while the components of the transformations are vertical and thus naturality is preserved by whiskering. Moreover, all the nice things about MonCat laxMonCat_{lax} are also true for DblCat lax,vertDblCat_{lax,vert}; for instance, any equivalence in DblCat lax,vertDblCat_{lax,vert} consists of strong functors, and lax functors of double categories are invariant under equivalence. There are also lots of nice adjunctions in DblCat lax,vertDblCat_{lax,vert} that are hard to see or describe from a purely 2-categorical setting. Some more on this point of view can be found here, although that only gets as far as a version of DblCat lax,vertDblCat_{lax,vert} that is a 2-category. If you want the full 3-categorical glory, the only place I know of to find it written down is Dominic Verity’s still (?) unpublished thesis.

So, what do we conclude from the poor behavior of lax functors between 2-categories? My personal conclusion is that whenever we start seeing lax functors appearing, it’s a good bet that our 2-categories are really double categories.

Posted at December 12, 2009 3:35 AM UTC

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27 Comments & 1 Trackback

Re: The Problem With Lax Functors

I’ve always disliked the idea of lax functors being actual functors, because they don’t behave like functors “should”. I feel like we want our notion of a functor to be a generalization of the classical idea of an algebraic homomorphism.

They always seemed too messy to me, like something was wrong in principle and the definition was unnatural.

Posted by: Harry Gindi on December 12, 2009 6:30 AM | Permalink | Reply to this

Re: The Problem With Lax Functors

Does your dislike extend to lax monoidal functors and lax functors of double categories? I think lax functors of double categories are very analogous to lax monoidal functors. For lax monoidal functors, you have algebraic structure on a category which is being preserved laxly, but the functor is an honest functor. A double category can be thought of as “algebraic structure” on a Cat-graph, which is therefore preserved laxly by a lax functor, but the underlying “functor” on Cat-graphs is again an honest functor. In addition, I forgot to mention that MonCat laxMonCat_{lax} is a full sub-2-category of DblCat lax,vertDblCat_{lax,vert}, when you view a monoidal category as a double category with one object and one vertical arrow.

Posted by: Mike Shulman on December 12, 2009 6:59 AM | Permalink | PGP Sig | Reply to this

Re: The Problem With Lax Functors

Yeah, that’s the impression I got from your post, we’re considering “additional structure”, so we can build another structure entirely on top, but this isn’t where categorification takes place, so everything behaves as it should.

Posted by: Harry Gindi on December 12, 2009 7:41 AM | Permalink | Reply to this

Re: The Problem With Lax Functors

Thanks — a very succinct presentation of the problem, and partial roundup of solutions!

I’ve never been happy with the definition of icons, though, since it seems to be uncategorical, requiring that things agree precisely on objects; and because of the problems of laxness, this (at least apparently) is unavoidable. On the other hand, I’ve never really worked with them, so perhaps when you do, a reason for this becomes clearer?

Posted by: Peter LeFanu Lumsdaine on December 12, 2009 12:19 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

I don’t regard icons as a “fundamental” notion, but rather as a technical one that is sometimes convenient. As you probably know if you’ve read the paper, one way in which icons arise “naturally” is that when you write down a 2-monad TT on the 2-category of Cat-enriched graphs for which TT-algebras are 2-categories, then pseudo, lax, and oplax functors are the pseudo, lax, and oplax TT-morphisms, and icons are the TT-transformations. This means that you can apply 2-monad theory.

For instance, there’s a general coherence theorem saying that if a 2-monad TT is well-behaved, then any pseudo TT-algebra is equivalent to a strict one. I don’t know a version of this theorem that applies to this 2-monad exactly, but it does apply if you consider only Cat-enriched graphs with a fixed set of objects. Then the general theorem tells you that any (unbiased) bicategory is equivalent to a strict 2-category, where “equivalence” means in the 2-category of bicategories, pseudo functors, and icons. This implies an equivalence of bicategories in the usual sense, but is stronger, and also contains the fact that the equivalence is bijective on objects.

Of course at the end of the day, we are mostly interested in the “real” notion of equivalence. But the icons provide a useful technical tool allowing us to apply general theory to this situation.

Another way to think of icons is that if you regard bicategories as double categories “horizontally” with no nonidentity vertical arrows, then icons get identified with vertical transformations. In fact that embeds the 2-category Bicat lax,iconBicat_{lax,icon} fully into DblCat lax,vertDblCat_{lax,vert}. So icons are what you get out of the sensible notion of “vertical transformation of double categories” if you insist (often, unreasonably) that your bicategories really don’t have any vertical arrows.

Posted by: Mike Shulman on December 12, 2009 5:09 PM | Permalink | PGP Sig | Reply to this

Re: The Problem With Lax Functors

Thanks Mike! This gives a good picture of the pitfalls of laxification. This also seems like another possible way to motivated double categories. I’ll have to think more about this double category point of view.

Posted by: Chris Schommer-Pries on December 12, 2009 2:00 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

This was actually precisely the reason I got interested in double categories. It’s just so nice that vertical transformations whisker compose with lax double functors, in contrast to (ordinary) lax transformations, which don’t whisker compose with (ordinary) lax functors between bicategories.

Of course, lax functors come up so naturally (in particular, as Mike mentions, when considering the change of base for enriched categories), that it’s important that they work well in the double categorical setting.

Posted by: Geoff Cruttwell on December 12, 2009 11:37 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

What version of string diagrams do you get as the diagram calculus for double categories?

Posted by: Noah Snyder on December 12, 2009 4:14 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

So I guess it’s something like each 2-morphism is a square with strings coming into various sides where when you isotope you’re not allowed to ever rotate squares. Does that sound right?

In particular, there’s probably not a good notion of double category with duals?

Posted by: Noah Snyder on December 12, 2009 6:29 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

I guess it’s something like each 2-morphism is a square with strings coming into various sides where when you isotope you’re not allowed to ever rotate squares.

Yes, that sounds right. Although if your double category is a framed bicategory, then horizontally running strings can be bent into vertical ones. So in that case, the string diagrams look just like those for a bicategory but with a certain class of (dualizable) strings singled out as coming from the vertical arrows.

In particular, there’s probably not a good notion of double category with duals?

Why not?

Posted by: Mike Shulman on December 12, 2009 8:37 PM | Permalink | PGP Sig | Reply to this

Re: The Problem With Lax Functors

If the failure of “2Cat_{lax}” to form a 3-category boils down to the failure of arrows to compose “head-to-head” and “tail-to-tail”, wouldn’t one solution be to work in a context where those arrows have duals? This is less restrictive than assuming they are invertable, but more restrictive than allowing arbitrary morphisms.

Posted by: Kevin Walker on December 13, 2009 6:34 AM | Permalink | Reply to this

Re: The Problem With Lax Functors

Assuming the existence of null morphisms shouldn’t be necessary for categorification.

Posted by: Harry Gindi on December 13, 2009 3:16 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

Well, to start with, the constraint cells of a lax 2-functor are 2-cells in a 2-category, so they are dualizable iff they are invertible. However, we could imagine considering lax functors between 3-categories, and ask whether they form a 4-category if their constraints have adjoints. Or we could try some other variation on lax 2-functors that comes with constraints in both directions, such as saying we have a functor that’s both lax and colax with the constraints satisfying some relationship other than adjointness, e.g. one being a retraction of the other, or maybe a “Frobenius” condition. Another possibility that some people sometimes think of is generalizing the notion of transformation so that α f\alpha_f can be a zigzag rather than just a morphism in one direction or the other.

Unfortunately, none of these ideas give you a 3-category either, although they come closer. All these ideas are aimed at allowing us to whisker transformations by functors. Notice that in all cases, for this to work the notion of “transformation” involved has to be some sort of lax, since the 2-cell components of the whisker-composite HαH\alpha involve the not-necessarily-invertible constraint cells of the functor HH. If we require these cells to be dualizable, or have retractions, then we must consider at least lax transformations whose 2-cell components are dualizable, or have retractions, etc.—pseudo natural transformations aren’t general enough.

However, there is no 3-category of 2-categories whose 2-cells are ??? transformations for any value of ??? weaker than “pseudo,” no matter what the 1-morphisms are. For although we might be able to perform whiskering, the interchange law now fails. Namely, suppose we have 2-categories C,D,EC,D,E, functors F,G:CDF,G\colon C\to D and H,K:DEH,K\colon D\to E, and ??? transformations α:FG\alpha\colon F\to G and β:HK\beta\colon H\to K. Then in a 3-category, the two composites (βG).(Hα)and(Kα).(βF) (\beta G) . (H\alpha) \qquad\text{and}\qquad (K\alpha) . (\beta F) must be isomorphic. (In a 2-category, they would be equal, while in a 4-category, they would be equivalent, and so on.) Now the components of these two transformations at an object xCx\in C are, respectively, β GxH(α x)andK(α x)β Fx \beta_{G x} \circ H(\alpha_x) \qquad\text{and}\qquad K(\alpha_x) \circ \beta_{F x} So for the interchange law to hold, we’d want an isomorphism in the following square: HFx H(α x) HGx β Fx ?? β Gx KFx K(α x) KGx\array{H F x & \overset{H(\alpha_x)}{\to} & H G x\\ ^{\beta_{F x}}\downarrow & ?? & \downarrow^{\beta_{G x}}\\ K F x& \underset{K(\alpha_x)}{\to} & K G x} This is precisely the naturality square for β\beta at α x\alpha_x, so the only thing we can put in there is the 2-cell component β α x\beta_{\alpha_x}. But of course, this 2-cell is not invertible, in general, unless β\beta is pseudo natural.

However, this is now the only thing that fails. That means that 2Cat pseudo,lax2Cat_{pseudo,lax}, for instance, while not a 3-category, is a very 3-category-like structure which merely has “lax interchange.” There’s even a nice way to say this formally: while (weak) 3-categories can be described as categories weakly enriched over 2Cat2Cat with the cartesian product, or equivalently the pseudo version of the Gray tensor product (the two being equivalent for weak enrichment, though not for strict enrichment), 2Cat pseudo,lax2Cat_{pseudo,lax} is a category weakly enriched over 2Cat2Cat with the lax version of the Gray tensor product. In fact, the lax Gray tensor product is closed on both sides (though not symmetric), and 2Cat pseudo,lax2Cat_{pseudo,lax} is essentially its canonical self-enrichment arising from this closed structure. I haven’t written it down, but it seems possible that the above modifications of lax functors might give you categories weakly enriched over 2Cat2Cat with other versions of the Gray tensor product. So although you don’t get a 3-category, you don’t get nonsense either.

Posted by: Mike Shulman on December 13, 2009 11:22 PM | Permalink | PGP Sig | Reply to this

Re: The Problem With Lax Functors

Well, to start with, the constraint cells of a lax 2-functor are 2-cells in a 2-category, so they are dualizable iff they are invertible.

Hmm… maybe we have different ideas of what “dualizable” means? In the Temperley-Lieb 2-category (1-morphisms are dots on an interval, 2-morphisms are linear combinations of curves in a disk, modulo some relations), I would say that all 2-morphisms are dualizable, though they are typically not invertible. Maybe my sense of “dualizable” is non-standard? If so, what term should I have used instead? I’m thinking “dual” in the sense that the dual of an inclusion of vector spaces is a projection. n-morphisms in n-categories arising from combinatorial topology usually have a “mirror image” operation that satisfies various identities. That’s the sense of “dual” I had it mind in my original comment.

Posted by: Kevin Walker on December 14, 2009 11:28 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

This reminds me of a question on MO.

Posted by: Chris Schommer-Pries on December 15, 2009 3:01 AM | Permalink | Reply to this

Re: The Problem With Lax Functors

Ah, I see. Yes, there are two kinds of “dual” floating around, and I’m sure there’s a Cafe post or a TWF somewhere that explains them better than I ever could; maybe someone who’s been around longer can help us out?

On the one hand, a “dual” for a kk-morphism f:abf\colon a\to b in an nn-category for 0<k<n0\lt k\lt n is the same as an adjoint, i.e. a map in the other direction f :baf^&#8727;\colon b\to a and a unit and counit relating ff f f^&#8727; and f ff^&#8727; f to identities. This also makes sense when k=0k=0 if the nn-category is monoidal. When k=nk=n, a morphism has an adjoint in this sense iff it is invertible, but it’s still true that in some cases all (or some) of the nn-morphisms f:abf\colon a\to b also come with specified morphisms f :baf^\dagger\colon b\to a in the other direction, which are also sometimes called “duals”. There’s also a proliferation of terminology like dagger category, compact closed category, dagger compact category, etc. etc. referring to all these kinds of duals.

I wouldn’t tend to refer to an nn-morphism with a specified f f^\dagger as dualizable, though, because unlike the situation for adjoints when 0k<n0\le k\lt n, the existence of such a “dual” is not a property of ff but rather structure imposed on the nn-category in question. But maybe other people would?

Posted by: Mike Shulman on December 15, 2009 3:38 AM | Permalink | Reply to this

Re: The Problem With Lax Functors

Like Mike, I wouldn’t apply the word ‘dualizable’ to the nn-morphisms in an nn-category — because this suggests you’re talking about some property of nn-morphisms, whereas in fact duality for nn-morphisms is an extra structure you can slap on your nn-category.

However, I’m a great fan of ‘nn-categories with duals’, where the kk-morphisms have adjoints for 0<k<n0 \lt k \lt n, and there’s also a duality structure for nn-morphisms. They’re conceptually a bit mysterious, but common in nature.

Posted by: John Baez on December 15, 2009 5:31 AM | Permalink | Reply to this

Re: The Problem With Lax Functors

I agree – “dualizable” is a not a good term for what I was talking about. Sorry to cause all the fuss by my initial poor choice of words.

While we’re on the subject of n-category terminology, it seems to me that there are (at least) two useful notions of “n-categories with duals”, each of which deserves its own name. Roughly speaking, the weaker (less restrictive) version of an n-cat with duals requires that every morphism have an adjoint/dagger. The stronger version further requires that these adjoints/daggers satisfy conditions related to homeomorphisms of k-balls. For example, starting with a 2-morphism from aa to b*cb*c, we can apply adjoints six times to rotate the 2-morphism 360 degrees. (i.e. a*c¯ba*\bar{c} \to b, then c¯a¯*b\bar{c} \to \bar{a}*b, then c¯*b¯a¯\bar{c}*\bar{b}\to\bar{a}, then b¯c*a¯\bar{b}\to c*\bar{a}, then b¯*ac\bar{b}*a\to c, then finally back to the original ab*ca\to b*c.) In the strong version of an n-cat with duals, we would require this “rotation” operation on 2-morphisms to be the identity (or maybe weakly the identity if n>2n \gt 2). Similarly, for each loop of (combinatorial) rotations of a k-ball we have a condition adjoints must satisfy in the strong version of an n-cat with duals.

As John points out, the stronger version is common in nature.

If I understand things correctly, the weaker n-cats with duals correspond to framed TQFTs, while the stronger n-cats with duals correspond to ordinary (unframed) TQFTs.

I think most people use “n-cat with duals” to refer to the weaker version. If that’s the case, then what should we call the stronger version? When I’ve given talks, I’ve made up terms like “n-category with strong duality” and “disk-like n-category”. But maybe there’s another name for this sort of n-cat that is already well-established?

Posted by: Kevin Walker on December 15, 2009 3:40 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

Recently, when I’ve talked informally about these different notions of dualizable I’ve been using the phrase “framed dualizable” to refer to the weaker notion that Kevin refers to and “oriented dualizable” to refer to the stronger version. The reason being that these sorts of duality, whether they are structures or, in the framed case, just properties, are intimately tied to TQFTs.

I think this terminology has the advantage that it is descriptive (unlike “strong”, sorry Kevin) and also suggests rightfully that there is a whole zoo of different kinds of duality, e.g. spin duality, p1-duality, string duality, stably framed duality, stably complex duality, etc.

Posted by: Chris Schommer-Pries on December 15, 2009 6:27 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

Kevin does bring up something interesting that I hadn’t thought about before. Namely, if your 2-categories have duals for 2-morphisms, i.e. they are enriched over dagger categories rather than merely categories, then it looks like every lax functor will also be oplax in a canonical way, and similarly for transformations. Maybe we should call them \dagger-lax.

Moreover, it seems like these “locally-\dagger” 2-categories together with \dagger-lax functors, \dagger-lax transformations, and modifications ought to form one of these “3-categories with lax interchange” like 2Cat pseudo,lax2Cat_{pseudo,lax}, or even better a “3-category with \dagger-lax interchange,” having a \dagger structure on the 3-cells.

Posted by: Mike Shulman on December 15, 2009 6:00 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

My personal conclusion is that whenever we start seeing lax functors appearing, it’s a good bet that our 2-categories are really double categories.

I was hoping for the stronger statement

whenever we start seeing lax functors appearing, it’s a good bet that our 2-categories are really framed bicategories

Because the notion of a bicategory BB that is a proarrow equipment in that there is a functor KBK \to B with certain properties

(links are for other readers, Mike is of course the expert on this stuff)

looks like it would straightforwardly generalize to whatever higher notion of bicategory we have, notably to the notions of (,2)(\infty,2)-categories that are available. On the other hand, the notion of a double category as such seems to be much less well adapted to useful straightforward generalization.

Posted by: Urs Schreiber on December 13, 2009 6:52 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

It’s certainly true that most of the double categories that arise in this way are framed bicategories, aka proarrow equipments on their vertical 2-categories. It’s also true that the “quick-and-dirty” definition of proarrow equipment as a certain kind of functor KBK\to B may be easier to write down for (,2)(\infty,2)-categories. However, I still believe that developing a theory of “\infty-double categories” will be important in order to really understand the notion of “framed (,2)(\infty,2)-category”.

Posted by: Mike Shulman on December 13, 2009 11:29 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

I still believe that developing a theory of “∞-double categories” will be important in order to really understand the notion of “framed (∞,2)-category”.

I see. But it worries me that next we’ll need \infty-triple categories and without a good systematic way to proceed the pattern that sounds like asking for headaches.

On the other hand, likely I am missing a good feeling for all the relevant structure involved here. Maybe you can help me with the following toy case:

given a monoidal category CC and with BC\mathbf{B}C its incarnation as a 1-object bicategory, I suppose it becomes a framed bicategory in the trivial way with the identity functor

BCBC \mathbf{B}C \to \mathbf{B}C

exhibiting the (trivial) proarrow equipment.

With monoidal categories regarded as such a bifunctor this way, how do we naturally see lax 2-functors BCBD\mathbf{B}C \to \mathbf{B}D arise, and hence lax monoidal functors CDC \to D? Is there a good way?

Posted by: Urs Schreiber on December 14, 2009 10:51 AM | Permalink | Reply to this

Re: The Problem With Lax Functors

But it worries me that next we’ll need ∞-triple categories and without a good systematic way to proceed the pattern that sounds like asking for headaches.

What are you thinking that the systematic categorification of KMK\to M would be? A suitable functor between nn-categories? Or a string of nn functors between bicategories? Or between kk-categories? The pattern “category \to double category \to triple category” seems just as systematic to me.

In general, though, I think it’s important in writing down categorical structures to be guided by what structure is actually present on the objects we are interested in, rather than some preordained notion of how patterns ought to go. As Sherlock Holmes would say, it is a capital mistake to theorize in advance of the facts. What I currently see appearing at the next level after framed bicategories are what I call (2×1)(2\times 1)-categories and (1×2)(1\times 2)-categories: internal 2-categories in CatCat and internal categories in 2Cat2Cat. (Chris Douglas likes to talk about these too, although I don’t think he uses my terminology. He even extends it to draw a distinction between what I would call (3×0)(3\times 0)-categories and (0×3)(0\times 3)-categories, although at a formal level those two have the same definitions.)

For instance, rings, two-sided algebras, and bimodules form a (2×1)(2\times 1)-category, while monoidal categories and bimodules between them form a (1×2)(1\times 2)-category. The “fibrant” (2×1)(2\times 1)- and (1×2)(1\times 2)-categories are both natural categorifications of framed bicategories. Fibrant (1×2)(1\times 2)-categories can probably be written as proarrow-equipment-like functors KMK\to M where KK and MM are 3-categories, but it’s not immediately clear to me how to describe fibrant (2×1)(2\times 1)-categories from that point of view.

given a monoidal category CC and with BC\mathbf{B}C its incarnation as a 1-object bicategory, I suppose it becomes a framed bicategory in the trivial way with the identity functor BCBC\mathbf{B}C\to \mathbf{B}C

Not quite. Remember that the arrows in the image of KMK\to M are supposed to have right adjoints. So BCBC\mathbf{B}C\to \mathbf{B}C is not even a proarrow equipment unless every object of CC has a dual.

I think the “right” way to regard a monoidal category as a framed bicategory is with the vertical category being entirely trivial, i.e. the proarrow equipment 1BC1\to \mathbf{B}C. In this way, lax monoidal functors CDC\to D are identical to lax functors BCBD\mathbf{B}C \to \mathbf{B}D, in both the bicategorical and the framed-bicategorical cases.

Posted by: Mike Shulman on December 14, 2009 4:12 PM | Permalink | PGP Sig | Reply to this

Re: The Problem With Lax Functors

I think the “right” way to regard a monoidal category as a framed bicategory is with the vertical category being entirely trivial, i.e. the proarrow equipment 1BC1 \to \mathbf{B}C

Ah, right, of course. I did think of this first but then worried that this violates some of the conditions.

This nicely matches with the observation that after delooping it is the pointed objects that are equivalent to the undelooped ones.

So, generally, with KBK \to B a proarrow equipment, what are the morphisms such that the lax bifunctors enter at the right point? Is it squares

K B K B \array{ K &\to& B \\ \downarrow && \downarrow \\ K' &\to& B' }

such that the right vertical bifuntor is allowed to be lax?

That’s the type of natural picture that I am looking for. If it exists, this kind of condition would have a natural generalization. If not then the other approach is better, Mr Holmes.

Posted by: Urs Schreiber on December 14, 2009 6:41 PM | Permalink | Reply to this

Re: The Problem With Lax Functors

Well, you undoubtedly can express lax functors between framed bicategories in terms of the proarrow equipments KMK\to M, since the latter contain all the data of the former, but I don’t know of any way to say it that makes it look natural. There is a square K M F F K M\array{K & \overset{}{\to} & M\\ ^F\downarrow && \downarrow^F\\ K'& \underset{}{\to} & M'} where the left vertical functor is strong or strict, and the right vertical functor is lax, and the square commutes at the level of objects, but it doesn’t in general commute at the level of morphisms. In the notation used here, we have a globular 2-cell FB(1,Ff)F(B(1,f))F B(1,F f) \to F(B(1,f)), but it might not be an isomorphism. (See section 6 of the framed bicategories paper for where this 2-cell comes from.) I think this means that the above square comes with an icon sitting in the middle of it.

However, I still don’t think this is enough to recover the action of your lax functor on squares. There’s also a dually defined globular 2-cell FB(Ff,1)F(B(f,1))F B(F f,1) \to F(B(f,1)), which I don’t see how to construct from the previous one. That means we also have an icon sitting in the other square K coop M F F (K) coop M\array{K^{coop} & \overset{}{\to} & M\\ ^F\downarrow && \downarrow^F\\ (K')^{coop}& \underset{}{\to} & M'} where the horizontal arrows send each morphism ff to B(f,1)B(f,1) instead of B(1,f)B(1,f). If we have both of these icons, then for any square A H B f g C L D\array{A & \overset{H}{\to} & B\\ ^f\downarrow & \Downarrow & \downarrow^g\\ C& \underset{L}{\to} & D} in the source, we can use the “central lemma” of framed bicategories to replace it by a globular 2-cell C(f,1)HD(1,g)LC(f,1) \odot H \odot D(1,g) \to L, hence a 2-cell in MM, and now consider the composite FC(Ff,1)FHFD(1,Fg)F(C(f,1))FHF(D(1,g))F(C(f,1)HD(1,g))FL F C(F f,1) \odot F H \odot F D(1,F g) \to F(C(f,1)) \odot F H \odot F(D(1,g)) \to F(C(f,1) \odot H \odot D(1,g)) \to F L (using both of the icons and the lax structure map) which represents a square FA FH FB Ff Fg FC FL FD\array{F A & \overset{F H}{\to} & F B\\ ^{F f}\downarrow & \Downarrow & \downarrow^{F g}\\ F C& \underset{F L}{\to} & F D} in the target. Maybe we need to assume some additional compatibility between the above two icons in order that this operation on 2-cells satisfies the axioms of a lax functor between double categories; I don’t know.

I do know that this all looks very ad hoc to me, especially by contrast with the simple definition of (horizontally) lax functors between double categories, which to me is obvious and obviously correct. And I think things are only going to get worse when you start to think about transformations between such functors.

Posted by: Mike Shulman on December 14, 2009 8:07 PM | Permalink | PGP Sig | Reply to this

Re: The Problem With Lax Functors

Thanks. I was afraid that this would be the case. Too bad. Would have been nice if there’d been a way to look at monoidal categories with lax monoidal functors as pointed bicategories in a way that generalizes to framed bicategories as “generalized pointed bicatgeories”.

Posted by: Urs Schreiber on December 15, 2009 10:09 AM | Permalink | Reply to this
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