## August 19, 2016

### Compact Closed Bicategories

#### Posted by John Baez

I’m happy to announce that this paper has been published:

Abstract. A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual ‘zig-zag’ identities of a compact closed category only up to natural isomorphism, and the isomorphism is subject to a coherence law. We give several examples of compact closed bicategories, then review previous work. In particular, Day and Street defined compact closed bicategories indirectly via Gray monoids and then appealed to a coherence theorem to extend the concept to bicategories; we restate the definition directly.

We prove that given a 2-category $C$ with finite products and weak pullbacks, the bicategory of objects of $C$, spans, and isomorphism classes of maps of spans is compact closed. As corollaries, the bicategory of spans of sets and certain bicategories of ‘resistor networks” are compact closed.

This paper is dear to my heart because it forms part of Mike Stay’s thesis, for which I served as co-advisor. And it’s especially so because his proof that objects, spans, and maps-of-spans in a suitable 2-category forms a compact symmetric monoidal bicategory turned out to be much harder than either of us were prepared for!

A problem worthy of attack
Proves its worth by fighting back.

In a compact closed category every object comes with morphisms called the ‘cap’ and ‘cup’, obeying the ‘zig-zag identities’. For example, in the category where morphisms are 2d cobordisms, the zig-zag identities say this:

But in a compact closed bicategory the zig-zag identities hold only up to 2-morphisms, which in turn must obey equations of their own: the ‘swallowtail identities’. As the name hints, these are connected to the swallowtail singularity, which is part of René Thom’s classification of catastrophes. This in turn is part of a deep and not yet fully worked out connection between singularity theory and coherence laws for ‘$n$-categories with duals’.

But never mind that: my point is that proving the swallowtail identities for a bicategory of spans in a 2-category turned out to be much harder than expected. Luckily Mike rose to the challenge, as you’ll see in this paper!

This paper is also gaining a bit of popularity for its beautiful depictions of the coherence laws for a symmetric monoidal bicategory. And symmetric monoidal bicategories are starting to acquire interesting applications.

The most developed of these are in mathematical physics — for example, 3d topological quantum field theory! To understand 3d TQFTs, we need to understand the symmetric monoidal bicategory where objects are collections of circles, morphisms are 2d cobordisms, and 2-morphisms are 3d cobordisms-between-cobordisms. The whole business of ‘modular tensor categories’ is immensely clarified by this approach. And that’s what this series of papers, still underway, is all about:

Mike Stay, on the other hand, is working on applications to computer science. That’s always been his focus — indeed, his Ph.D. was not in math but computer science. You can get a little taste here:

But there’s a lot more coming soon from him and Greg Meredith.

As for me, I’ve been working on applied math lately, like bicategories where the morphisms are electrical circuits, or Markov processes, or chemical reaction networks. These are, in fact, also compact closed symmetric monoidal bicategories, and my student Kenny Courser is exploring that aspect.

Basically, whenever you have diagrams that you can stick together to form new diagrams, and processes that turn one diagram into another, there’s a good chance you’re dealing with a symmetric monoidal bicategory! And if you’re also allowed to ‘bend wires’ in your diagrams to turn inputs into outputs and vice versa, it’s probably compact closed. So these are fundamental structures — and it’s great that Mike’s paper on them is finally published.

Posted at August 19, 2016 2:10 AM UTC

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### Re: Compact Closed Bicategories

Congratulations, Mike!

…a deep and not yet fully worked out connection between singularity theory and coherence laws for ‘$n$-categories with duals’.

Anyone looking for hints about this, a wide-ranging discussion occurred on this blog almost 10 years ago.

Posted by: David Corfield on August 19, 2016 2:34 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

proving the swallowtail identities for a bicategory of spans in a 2-category turned out to be much harder than expected.

How is this related to (coworker and sometimes coauthor) Jamie Vicary’s Globular proof:

Swallowtail coherence (globular.science/1512.006). In a 3-category, an adjunction of 1-morphisms gives rise to a coherent adjunction satisfying the swallowtail equations.

Posted by: RodMcGuire on August 19, 2016 4:27 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

I think this calculation does indeed show that, in the definition of a compact closed monoidal 2-category, the swallowtail equations are redundant, in the sense that whenever all the structure except the swallowtail equations is present, some of the structures can be redefined to make the swallowtail equations also hold. It sounds like this result might have been useful for Mike’s work here, in which case I am sorry I have not done such a good job publicizing the proof (and others like it) which we’ve formalized in Globular.

This doesn’t at all detract from what Mike has done here. Firstly, this swallowtail coherence proof has not to my knowledge been published in a research paper. It is not due to me; I learned it from some combination of Chris Douglas, Chris Schommer-Pries and his student Piotr Pstragowski, who wrote about it in his Masters thesis. Secondly, I expect that the snakeorators have natural definitions in Mike’s setting, and it is interesting to know whether they already satisfy the swallowtail equations on the nose (as I expect they do), rather than needing to be redefined. Thirdly, when one categorifies, a very large number of additional equations (essentially, all associated with oriented 3-manifolds) will appear, and it is not known that they can all be automatically made to hold—so getting some practice in at this level is very useful. (Although we have already shown that, in a suitable semistrict setting, the butterfly equations can indeed be automatically satisfied.) Fourthly, it’s often better to give a direct proof of something, rather than relying on some flashy shortcut that may ultimately reduce understanding.

Posted by: Jamie Vicary on August 22, 2016 7:21 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Perhaps Homotopy coherent adjunctions and the formal theory of monads is relevant? Riehl and Verity define a notion of “fully coherent $\infty$-adjunction” and show that all of its structure can be obtained automatically from an “adjunction up to homotopy”. I would expect that that includes the swallowtail equations as well as higher-dimensional analogues.

Posted by: Mike Shulman on August 22, 2016 8:57 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Gosh, that looks interesting. OK, so my “thirdly” above is nullified :).

Posted by: Jamie Vicary on August 22, 2016 11:25 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

I wonder lazily to what extent this Riehl and Verity result is constructive; i.e. giving you an algorithm to actually perform the promotion on a particular structure, in a traditional globular $n$-category setting.

Posted by: Jamie Vicary on August 22, 2016 11:28 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

It’s been a while since I looked at it, but my memory is that everything is pretty explicit. Maybe Emily will show up and enlighten us.

Posted by: Mike Shulman on August 23, 2016 4:11 AM | Permalink | Reply to this

### Re: Compact Closed Bicategories

In our work http://dx.doi.org/10.1016/j.aim.2015.09.011 (http://arxiv.org/abs/1310.8279) we define a homotopy coherent adjunction of $\infty$-categories to be a simplicial functor from a certain simplicial category Adj to the simplicially enriched category of $\infty$-categories. The simplicial category Adj is itself constructed by applying the nerve construction to the hom-categories of the Street and Schanuel “walking adjunction” 2-category.

So what evidence do have that this notion does indeed encapsulate all of the higher homotopy coherences that might be expected of a fully coherent adjunction between $\infty$-categories?

1. The simplicial category Adj is cofibrant, so any homotopy coherent adjuction may be built up from one dimension to the next through the specification of new homotopies (simplices) at that higher dimension. In particular, this process does not require us to identify together any information already present at lower dimensions.

2. The simplicial category Adj is fibrant, that is its hom-spaces are all quasi-categories (nerves of categories). It follows therefore that all possible composities of the new homotopies added at each dimension are present in Adj. As we say in our paper:

Now observe that, as a 2-category, the hom-spaces of our simplicial category Adj are all quasi-categories, a “fibrancy” condition that indicates all possible composites of coherence data are present in Adj and thus picked out by a simplicial functor with this domain. But of course, these hom-spaces, as nerves of categories, have unique fillers for all inner horns, which says furthermore that this coherence data is “minimally chosen” or “maximally coherent” in some sense.

3. Every adjunction of $\infty$-categories, defined in terms of an appropriate $\infty$-categorical universal property, extends to a homotopy coherent adjunction. Indeed more is true, there is a space (Kan complex) of all such extensions and that space is contractible. In other words, every adjunction of $\infty$-categories extends “homotopically uniquely” to a homotopy coherent adjunction.

4. Let Mnd denote the full simplicial sub-category on Adj determined by one of its objects (which one?). We define a homotopy coherent monad to be a simplicial functor out of Mnd and into the simplicial category of $\infty$-categories. We can define the $\infty$-category of Eilenberg-Moore algebras for such a gadget (using a limit parameterised by a certain weight also derived from Adj) and prove that this construction enjoys (generalisations of) all of the properties one might expect from our experience of the E-M construction in category theory. In particular, a variant of the Beck Monadicity theorem characterises this construction, and thus ties our monad notion to that of Lurie (expressed in terms of $\infty$-operads).

Of course, it is reasonable to ask whether the process of extending an adjunction of $\infty$-categories to a homotopy coherent adjunction is constructive. The answer here is, well, yes and no. On the one hand, we give an explicit extension algorithm which proceeds from dimension to dimension by specifiying precisely what higher homotopies must be provided and in what order. The order of things is not uniquely determined by the structure of Adj, but any total order amongst generating higher homotopies will suffice so long as it satisfies certain constraints we describe. What is more, at each step the higher homotopy (simplex) required is not uniquely determined, this is afterall homotopy theory. What is important here is that these choices are ultimately immaterial, since however we make them we end up constructing a structure which is a point in a contractible space of extensions.

Finally, one might ask whether the swallowtail equations appear as part of the structure of this homotopy coherent adjunction? The answer here is that each one appears as a pair of 3-simplices within the hom-categories of Adj. The precise pair of 3-simplices in question is discussed in some detail in the introduction to our paper (end of page 3 beginning of page 4 of the arXiv version), and they are drawn later (end of page 20) in our string diagram inspired calculus for the simplices of Adj.

These latter pictures are pleasingly familiar:

Posted by: Dominic Verity on August 24, 2016 9:26 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

I’d also like to know what can be said using these (or other) methods about making equivalences coherent.

I’d also like to know what “homotopy coherent adjunction” means!

Posted by: Jamie Vicary on August 23, 2016 9:11 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

“Homotopy coherent adjunction” is defined in the Riehl-Verity paper. I think you should be able to apply the theory to an equivalence by first doing the usual “equivalence-to-adjoint-equivalence” at the bottom level to get an “adjunction up to homotopy” and then do their full coherentification.

Posted by: Mike Shulman on August 23, 2016 9:41 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Thanks. The term is used freely in the abstract in a way that suggests it is standard. I see now how they define it.

I would expect that ‘coherent’ here means that all equations hold. I’m not quite sure how to express that in the simplicial category terminology, however, and I’m having trouble finding a theorem in an appropriate early part of the paper that I can map to this suggestion.

Regarding equivalences: I understand one can do the usual trick. But I had in mind that an equivalence in fact gives rise to an ambidextrous adjunction, with a correspondingly stronger notion of coherence.

Posted by: Jamie Vicary on August 24, 2016 5:18 AM | Permalink | Reply to this

### Re: Compact Closed Bicategories

I would expect that ‘coherent’ here means that all equations hold. I’m not quite sure how to express that in the simplicial category terminology, however, and I’m having trouble finding a theorem in an appropriate early part of the paper that I can map to this suggestion.

According to the abstract,

The hom-spaces are … nerves of categories, which indicates that all of the expected coherence equations in each dimension are present.

In fact, not only are the hom-spaces nerves of categories, the whole simplicially enriched category is the “local nerve” of the “free adjunction” 2-category. Here the “hom-spaces” are being regarded as quasi-categories, so that a simplicially enriched category presents an $(\infty,2)$-category, and a quasicategory being the nerve of a category means that all possible equations between $k$-morphisms hold for $k\ge 2$. E.g. the swallowtail identity is an equation between 2-morphisms in these hom-spaces (i.e. 3-morphisms in the $(\infty,2)$-category). The triangle identities themselves are equations between 1-morphisms in the hom-categories, but not all such equations hold; saying that the free homotopy coherent adjunction coincides with (or “is the local nerve of”) the free ordinary adjunction says that exactly those equations between 1-morphisms hold that do for an ordinary adjunction.

Posted by: Mike Shulman on August 24, 2016 6:34 AM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Thanks for that explanation!

Posted by: Jamie Vicary on August 24, 2016 10:04 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Jamie wrote:

I would expect that ‘coherent’ here means that all equations hold.

In homotopy theory, ‘homotopy coherent’ means ‘obeying the desired equations up to homotopies which obey the desired equations up to homotopies which obey the desired equations up to…’

In any particular example of homotopy coherence, once has to either figure out what equations are ‘desired’ at each step, or, preferably, invent a machine that somehow takes care of them all with the flick of a switch.

Posted by: John Baez on August 24, 2016 8:17 AM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Jamie wrote:

This doesn’t at all detract from what Mike has done here. Firstly, this swallowtail coherence proof has not to my knowledge been published in a research paper. It is not due to me; I learned it from some combination of Chris Douglas, Chris Schommer-Pries and his student Piotr Pstragowski, who wrote about it in his Masters thesis.

When exactly did they come up with that proof? I’m interested, not for “priority dispute” reasons, but because I’m wondering if talking to you more could have spared us some months of agony! We would have been much happier if we’d known this result.

It seems Mike cracked his problem (proving the swallowtail identity in the bicategory of spans in a 2-category with finite products and iso comma objects) around June 4, 2015, when he wrote:

Here’s a draft of the main proof. That insight about omnitrucation let me choose pullbacks in such a way that all the 2-morphisms except the yankings were identities. Yesterday I came up with some topological notation for weak pullbacks that made the proof almost obvious that versions of the yankings with inserted identities around the edges were also identity 2-morphisms.

I see that on March 2, 2015 I had written:

Sounds like good progress! Please don’t forget, when you write this up, to say that ultimately someone should prove that regardless of how we choose our composites of spans, we get equivalent symmetric monoidal bicategories.

But I’m not even sure this concept - “equivalence” for symmetric monoidal bicategories - has been defined yet!

Chris Schommer-Pries has been doing a lot of work on symmetric monoidal bicategories lately [….]

Jamie Vicary should know a lot about this, because his massive paper with Bruce, Chris S-P and Chris Douglas has been held up by this problem.

Posted by: John Baez on August 23, 2016 2:47 AM | Permalink | Reply to this

### Re: Compact Closed Bicategories

I saw Piotr’s Masters thesis in August 2014, in which he works out some of the details. Chris Douglas and Chris Schommer-Pries were aware of it (considerably) earlier, I expect.

Posted by: Jamie Vicary on August 23, 2016 5:39 AM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Hi John,

It’s great that Mike and separately Emily and Dominic and separately Nick Gurski and collaborators have been doing significant work in this space. Obviously their work goes (in three separate directions!) far beyond the swallowtail issue we thought about, discussed below.

Like you, I’m not interested in priority discussions (much less disputes!) and would normally steer well clear of this, but since you asked directly and explicitly, I’ll say a bit about our related work.

Andre Henriques and I proved a swallowtail coherence, as Jamie refers to above, (that, for instance, you can modify cusps that don’t a priori satisfy swallowtail so that they do) in February 2008.

The reason we never published anything about that is the following. For us, this was a step in our program to construct 3-dimensional local field theories. Roughly speaking, in our approach, there were four steps to this: 1) build a 3-category (for us this was conformal nets — everything about conformal nets is joint work with Arthur Bartels), 2) prove some object there is in a certain sense (incoherently!) dualizable, 3) prove that such an incoherently dualizable object can be promoted to a coherently dualizable object, 4) prove that a coherently dualizable object gives a local field theory.

What we managed to do in Feb 08 is step (3) but only in the case of incoherently invertible objects. (We already understood step (4).) (The swallowtail coherence is part of step (3), and occurs even in the invertible case.) Now, the way we were thinking of it was that step (4) was a form of the 3-dimensional cobordism hypothesis. And in fact, the combination of steps (3) and (4) [as we conceived them] is quite close to the modern form of the cobordism hypothesis for the structure ‘stable framing’, ie for the structure group \Omega(O(infty)/O(3)).

When I gave a talk about these things at MIT in May of 08, I chatted to Mike and it started to become clear the remarkable approach he and Jacob had managed to figure out in this arena, and I think sometime that summer Jacob announced the proof of the cobordism hypothesis in all dimensions and for all structure groups. At that point, even though our approach was rather different (in particular it was constructive, which was both a feature and a liability), we felt it no longer made sense to pursue steps (3) and (4) and henceforth relied on Mike and Jacob’s cobordism hypothesis for that part of the story; in particular, everything about the coherence results in step (3) got excised from the conformal nets series of papers and put in a dusty drawer.

It’s nice to see some of these issues seeing the light of day.

Best, Chris

Posted by: Chris Douglas on August 24, 2016 8:26 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

I think this calculation does indeed show that, in the definition of a compact closed monoidal 2-category, the swallowtail equations are redundant, in the sense that whenever all the structure except the swallowtail equations is present, some of the structures can be redefined to make the swallowtail equations also hold….

This doesn’t at all detract from what Mike has done here. Firstly, this swallowtail coherence proof has not to my knowledge been published in a research paper.

As Jamie points out, the fact that the swallowtail equations are redundant in this sense is something that was known to the pre-history of weak 3-category theory. For example, this result appears in my own thesis (1992) http://www.tac.mta.ca/tac/reprints/articles/20/tr20abs.html, where it hides as an immediate consequence of Lemma 1.3.9(iii). This result is stated there for biadjunctions in any category strictly enriched in bicategories (in a suitable sense). Semi-strict 3-categories of this kind are closely related to, but weaker than, Gray-categories.

The point made there, in Example 1.1.7, is that for a biadjunction of bicategories $F\dashv_b U\colon\mathcal{A}\to\mathcal{B}$ the existence of triangle isomorphisms $\alpha\colon I_{F}\cong\epsilon F\cdot F\eta$ and $\beta\colon U\epsilon\cdot\eta U\cong I_{U}$ is equivalent to positing the existence of a (pseudo-)natural family of equivalences $\mathcal{A}(Fb,a)\simeq\mathcal{B}(b,Ua)$. The swallowtail equations, equation (1.37) in my thesis, are then exactly what we need to ensure that these derived natural equivalences of hom-categories are actually adjoint equivalences.

The fact that triangle isomorphisms may always be chosen to satisfy the swallowtail equations follows from this observation by chosing adjoint inverses to these hom-category equivalences and then passing back to triangle isomophisms. In that way, we replace $\alpha$ by a some isomorphism which satisfies swallowtail with respect to the original $\beta$.This proof extends representably from biadjunctions of bicategories to any biadjunction within a category enriched in bicategories.

In my thesis, these swallowtail equations become a vital part of the study of change of base between various enriched and internal category theories. In particular, they enable us to tie biadjunctions between bicategories of categories and functors to well behaved local adjunctions between associated bicategories of categories and profunctors.

Posted by: Dominic Verity on August 24, 2016 7:27 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

I’m glad to see this goes back further! We always imagined it might have been known before in some form. I suspect Dominic’s thesis is the earliest categorical formulation and proof.

I wonder if (if we are in an archeological mood) this could be uncovered in a geometric form back in the catastrophe theory literature? In a sense I won’t try to sort through right now, the very existence of the butterfly singularity guarantees that cusps can be made to satisfy swallowtail, and so as soon as Thom (?) saw the butterfly, in a sense the seed was there.

Similarly, the very existence of the wigwam tells us that swallowtails can be modified to satisfy butterfly. (Which Jamie has proven in Globular!) I suspect this ‘layered’ view is (secretly or not I’m not sure) part of what Emily and Dominic managed to show in their work on homotopy coherent adjunctions. So it all comes full circle.

Posted by: Chris Douglas on August 24, 2016 8:43 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

It’s good to see this interesting discussion developing.

Chris, what do you mean by the wigwam?

Posted by: Jamie Vicary on August 24, 2016 10:01 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

The wigwam is the simplest interesting coherence relation satisfied by the butterfly. (The cusp is what you get by looking at a plane curve with two morse critical points, the swallowtail with three, the butterfly with four, the wigwam with five, …) Won’t try to draw it here in ascii …

Posted by: Chris Douglas on August 25, 2016 3:22 AM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Chris wrote:

Similarly, the very existence of the wigwam tells us that swallowtails can be modified to satisfy butterfly.

Jamie wrote:

Chris, what do you mean by the wigwam?

Chris wrote:

The wigwam is the simplest interesting coherence relation satisfied by the butterfly.

Darn, I had hoped you meant that the very existence of this traditional native American dwelling relied implicitly on aspects of catastrophe theory that we’re only now understanding in an $n$-categorical way.

Posted by: John Baez on August 25, 2016 3:40 AM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Dominic, that proof method—promoting the equivalences of hom-categories to adjoint equivalences—makes a lot of sense, and it’s how we developed our proof. However, it doesn’t obviously work; it could be the case that in fact, these promotions can’t all be made simultaneously consistent with the requirement that they arise from the unit and counit of some adjunction. Do you have a clear sense of why this problem doesn’t actually bite?

Posted by: Jamie Vicary on August 24, 2016 11:08 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

One answer (but probably not the one Dominic had in mind (-: ) is that adjointification of equivalences can be done in the internal homotopy type theory of simplicial spaces, and therefore it can automatically be done “in any context”. Emily and I mostly worked this out last spring, as part of developing further the simplicial-spaces approach to “directed homotopy type theory” (blog post, talk).

This way of looking at it is really quite pretty, and has some interesting consequences. For instance, if you apply the “has both a left and right inverse” characterization to the equivalences of hom-spaces, it implies that if you have transformations $\eta:1\to G F$ and $\epsilon_1: F G \to 1$ and $\epsilon_2 : F G \to 1$ such that $\eta$ and $\epsilon_1$ satisfy one triangle identity and $\eta$ and $\epsilon_2$ satisfy the other triangle identity, then in fact $\epsilon_1 =\epsilon_2$ and you have an adjunction. I don’t think I knew that before; did anyone else?

Posted by: Mike Shulman on August 25, 2016 5:46 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

I don’t think I knew that, and it’s nice. But it might follow from the usual string diagram argument that if $\eta$ and $\epsilon_1$ obey both triangle identities and $\eta$ and $\epsilon_2$ obey both triangle identities then $\epsilon_1 = \epsilon_2$. In other words: do that string diagram proof and see which of the four assumptions you’re actually using.

Posted by: John Baez on August 26, 2016 4:48 AM | Permalink | Reply to this

### Re: Compact Closed Bicategories

if you have transformations $\eta\colon 1 \to GF$ and $\epsilon_1\colon FG \to 1$ and $\epsilon_2\colon FG \to 1$ such that $\eta$ and $\epsilon_1$ satisfy one triangle identity and $\eta$ and $\epsilon_2$ satisfy the other triangle identity, then in fact $\epsilon_1 = \epsilon_2$

That seems very similar to the fact that given an associative binary operation on a set, if an element $\eta$ is left inverse to $\epsilon_1$ and right inverse to $\epsilon_2$ then $\epsilon_1 = \epsilon_2$. I don’t know whether it’s literally an instance of it.

Posted by: Tom Leinster on August 26, 2016 1:05 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

I intended to say (but maybe was not sufficiently clear) that it essentially comes from applying that fact objectwise to the hom-set-bijection characterization of adjunctions.

Posted by: Mike Shulman on August 26, 2016 4:58 PM | Permalink | Reply to this

### Re: Compact Closed Bicategories

Posted by: Tom Leinster on August 27, 2016 5:13 AM | Permalink | Reply to this

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