February 22, 2009

Monoidal Closed Categories And Their Deviant Relatives

Posted by John Baez

Almost a year ago, Mike Stay and I proudly announced the completion of our Rosetta Stone paper, which explains how symmetric monoidal closed categories show up in physics, topology, logic and computation. But then Theo and Todd helped us spot a serious mistake in our description of a programming language that was supposed to be suitable for work inside any symmetric monoidal closed category — and hence, good for both classical and quantum computation.

We’re almost done fixing that problem now, and Mike is starting to write his thesis on related issues: classical and quantum computation and how they relate to ‘linear’ and also ‘categorified’ versions of the lambda-calculus.

But as a small side-effect, we’ve stumbled upon a deviant definition of ‘monoidal closed category’. I haven’t proved it’s equivalent to the usual definition. I bet it’s not. But I don’t know counterexamples. I wonder if anyone here has thought about such entities before.

Since several experts on monoidal closed categories read this blog, I’m optimistic.

A monoidal category $M$ is closed, or more precisely right closed, if for each object $a\in M$, the functor ‘tensoring on the right by $a$’ has a right adjoint.

This right adjoint sends any object $y\in M$ to an object $a⇒y$ of $M$ called the internal hom. It’s not always the same as the set of morphisms from $a$ to $y$, which is denoted $\mathrm{hom}\left(a,y\right)$ It’s kind of like what a guy living inside $M$ would use as a replacement for this set!

Thanks to the definition of right adjoint, we have a natural isomorphism of sets

$\mathrm{hom}\left(x\otimes a,y\right)\cong \mathrm{hom}\left(x,a⇒y\right)$

There’s a nice proof that starting with this, we can get a natural isomorphism of objects

$\left(x\otimes a\right)⇒y\cong x⇒\left(a⇒y\right)$

This is a kind of ‘internal’ version of the previous formula!

Now for my deviant definition. Call a monoidal category $M$ internally closed if it’s equipped with a functor

$⇒:{M}^{\mathrm{op}}×M\to M$

and a natural isomorphism

$\left(x\otimes a\right)⇒y\cong x⇒\left(a⇒y\right)$

where both sides here are shorthand for functors from ${M}^{\mathrm{op}}×{M}^{\mathrm{op}}×M$ to $M$.

Is every internally closed monoidal category actually closed? If not, what are some nice counterexamples? Does anyone think about these things? Is there any beautiful way to add axioms to get a definition equivalent to the usual definition of closed monoidal category?

Posted at February 22, 2009 11:30 PM UTC

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Re: Monoidal Closed Categories And Their Deviant Relatives

Let’s see if I understand the question. You have a category $M$, a functor $\otimes :M×M\to M$ satisfying all the axioms of a monoidal category, and moreover you have a functor $⇒:{M}^{\mathrm{op}}×M\to M$ and a natural isomorphism of functors ${M}^{\mathrm{op}}×{M}^{\mathrm{op}}×M\to M$:

(1)$\left(x\otimes a\right)⇒y\cong x⇒\left(a⇒y\right)$

This would be a right-closed monoidal category if there is a natural isomorphism of functors to SET:

(2)$\mathrm{hom}\left(x\otimes a,y\right)\cong \mathrm{hom}\left(x,a⇒y\right)$

We remark that $M$ is closed-monoidal exactly if we have an isomorphism of functors ${M}^{\mathrm{op}}×M\to \mathrm{SET}$ given by $\mathrm{hom}\left(a,y\right)\cong \mathrm{hom}\left(1,a⇒y\right)$, where $1$ is the monoidal unit.

In any case, let’s break the example. Take $M$ monoidal abelian, and let $a⇒y=0$ for any $a,y\in M$, where $0$ is the zero-object in the abelian category, and moreover let’s say that $⇒$ also kills any morphisms. For that matter, we could pick any constant functor for $⇒$ (all objects go to our particular choice of object, and all maps go to the identity on that object). Then the natural isomorphism is trivially satisfied — the two sides are in fact _equal_ as functors — and $⇒$ is not the internal hom of a closed-monoidal category.

Posted by: Theo on February 23, 2009 2:27 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

(where by “break the example” I meant “present a trivial example breaking the claim”)

I don’t know of good axioms that win other than asking for global sections of internal hom to return the true hom, which is very strong.

Posted by: Theo on February 23, 2009 2:29 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Are you aware of the Eilenberg-Kelly definition of closed category, which has only an internal-hom but no tensor product? Given that notion, an alternate equivalent definition of “closed monoidal category” is as a closed category whose internal-hom has a left adjoint; the associativity and unit laws of the monoidal product then follow.

However, one of the axioms in the Eilenberg-Kelly definition says essentially that “global sections of internal-hom return the true hom,” so if that is thought undesirable, this isn’t much help.

Posted by: Mike Shulman on February 23, 2009 2:53 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Mike wrote:

Are you aware of the Eilenberg–Kelly definition of closed category, which has only an internal-hom but no tensor product?

Yes, thanks.

Our desire was to give a definition of ‘closed monoidal category’ that involves only a category, functors from products of this category and its opposite to products of this category and its opposite, natural isomorphisms between such functors, and equations between such natural isomorphisms. I’m now getting the strong impression that this is impossible.

In particular, it’s against our rules to explicitly mention the hom-functor

$\mathrm{hom}:{C}^{\mathrm{op}}×C\to \mathrm{Set}$

which would make the job easy, but it’s okay to mention the internal hom

$⇒:{C}^{\mathrm{op}}×C\to C$

Also, while Owen Biesel kindly presents the Eilenberg–Kelly definition of closed monoidal category in a way that comes close to satisfying our rules, he also uses dinatural transformations:

$I\to \left(A⇒A\right)$ $\left(A⇒B\right)\to \left(C⇒A\right)⇒\left(C⇒B\right)$

So, we may have to bite the bullet and use these.

Posted by: John Baez on February 23, 2009 6:12 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

I doubt that you could get this to work without dinatural (or extraordinary-natural) transformations, but you could probably do it with those. Some possibly relevant facts are:

• Monoidal closed categories are monadic over the 1-category Cat.
• Monoidal closed categories are not 2-monadic over Cat, but they are 2-monadic over the 2-category ${\mathrm{Cat}}_{g}$ of categories, functors, and natural isomorphisms.
• (Non-monoidal) closed categories are not monadic over the 1-category Cat.

Cf. Kelly, “Examples of nonmonadic structures on categories,” and also section 5.8 of Steve Lack’s 2-categories companion. I don’t think that any of these facts really answers the question, but they are somewhat suggestive.

Posted by: Mike Shulman on February 23, 2009 6:18 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

One more quick comment, which is probably already obvious to at least half of the people reading this: if you’re willing to countenance dinatural transformations, then you can simply give the internal presentation of the tensor-hom adjunction.

Another way you could relax your requirements to make an internal presentation possible would be to allow families of functors indexed by the objects of $M$: if you define $A⊸B$ to be a family of functors $A⊸-$, so define it just for objects and not arrows in the contravariant position, then there’s no need to demand dinaturality of the unit and counit.

Posted by: Robin Houston on February 23, 2009 11:54 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Our desire was to give a definition of ‘closed monoidal category’ that involves only a category, functors from products of this category and its opposite to products of this category and its opposite, natural isomorphisms between such functors, and equations between such natural isomorphisms. I’m now getting the strong impression that this is impossible.

It seems to me that, just as you allow contravariant functors rather than (as one might naïvely expect) only covariant ones, so you should allow dinatural transformations rather than only natural ones. In other words, once you start using the duality features of $Cat$ (which is after all more than just another $2$-category), you should allow yourself to take full advantage of them.

Posted by: Toby Bartels on February 25, 2009 8:08 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

I’ve been thinking a lot about this recently as well, and I think the answer is in a paper from the Proceedings of the Conference on Categorical Algebra in La Jolla, 1965, called “Closed Categories”. The upshot is that to define a closed category in absence of any monoidal structure, you need the following data: $⇒:{M}^{\mathrm{op}}×M\to M$ $I:1\to M\phantom{\rule{1em}{0ex}}\left(\mathrm{an}\mathrm{object}\mathrm{of}M\right),$

as well as a natural isomorphism $A\equiv \left(I⇒A\right)$, and (di)natural maps $I\to \left(A⇒A\right)$ $\left(A⇒B\right)\to \left(C⇒A\right)⇒\left(C⇒B\right),$ something of an “internal unit” and an”internal Yoneda embedding”. [A nice exercise is to show that a discrete closed category is the same as a group, in the ordinary set+structure sense.]

What remains is to specify the axioms: There’s an apparent axiom requiring that the unit be compatible with the Yoneda map, as well as a more complicated “pentagon identity” expressing the idea of associativity: $\left(C⇒D\right)\to \left(B⇒C\right)⇒\left(B⇒D\right)$$\to \left(B⇒C\right)⇒\left(\left(A⇒B\right)⇒\left(A⇒D\right)\right)$

must equal

$\left(C⇒D\right)\to \left(\left(A⇒C\right)⇒\left(A⇒D\right)\right)$$\to \left(\left(\left(A⇒B\right)⇒\left(A⇒C\right)\right)⇒\left(\left(A⇒B\right)⇒\left(A⇒D\right)\right)\right)⇒\left(\left(B⇒C\right)\left(\left(A⇒B\right)⇒\left(A⇒D\right)\right)\right)$$\to \left(B⇒C\right)⇒\left(\left(A⇒B\right)⇒\left(A⇒D\right)\right)$

(If anyone wants to clean up my iTeX, please do! I’m not sure what carries over from LaTeX.)

I think there may be more axioms, but these are the only two I can remember offhand. My guess is that giving a monoidal category an internally closed structure causes it to automatically satisfy some of the axioms, and the rest may need to be imposed by hand to produce an “externally closed” structure.

Next time I get my hands on that paper, I’ll bring back more of what it has to say, and in the meantime I’ll try demonstrating the “nice proof” you mentioned that goes from closed to internally closed.

[Whoops, it looks like I’ve been beaten to the punch. I’ll close by noting that dinaturality of the unit map always gives you, for any arrow $A\to B$, a global element of $A⇒B$, so it isn’t altogether unreasonable to ask that this be a one-to-one correspondence.]

Posted by: Owen Biesel on February 23, 2009 3:13 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Thanks for taking the trouble to list those axioms! I remember being awed by Eilenberg and Kelly’s original definition of closed categories that aren’t necessarily monoidal — but I can never remember the definition. Now that I’ve studied the lambda calculus it doesn’t look quite so crazy. When I get time, I’ll try to add it to the $n$Lab page on closed categories. It deserves to be better known!

Posted by: John Baez on February 24, 2009 10:22 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Sorry, in a rush, so this is going to be a “Have you looked at my paper?” kind of comment, meaning that it’s possibly not overly useful.

In A diagrammatic approach to Hopf monads I introduced a diagrammatic calculus for monoidal categories with duals rather than with internal homs – this used a description with dinatural transformations. I did think a little about doing it with internal homs instead of duals, mainly as I wanted a diagrammatic description of some of the results in the lovely paper of Fausk, Hu and May, but I never got very far with it.

Posted by: Simon Willerton on February 23, 2009 10:37 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Thanks! I’ve looked at this paper, but I’ll check it out again.

Do you think you know a general diagrammatic calculus for dinatural transformations? Or could you at least invent one if forced to do so at gunpoint?

Posted by: John Baez on February 24, 2009 11:05 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Well I’m not sure what you mean by that. But possibly the answer is yes.

Posted by: Simon Willerton on February 24, 2009 11:15 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Do you think you know a general diagrammatic calculus for dinatural transformations?

I think you know this yourself, at least in certain forms. But your “bite the bullet” comment a while back makes me suspect that there’s something about the concept of dinatural transformation that bothers you. In fact, a lot of people seem to be bothered by this notion.

I am unfortunately not very capable with drawing diagrams in this medium, but let me say that you are surely already familiar with how dinatural transformations work diagrammatically – at least the type of dinatural transformations which are relevant to closed category theory. (Please mark my words here, because a lot of people, who have read the definition of dinatural transformation in say Categories for the Working Mathematician, are put off and develop a mental block about these things, but: the types of dinatural transformations which are relevant to closed category theory are the simplest, most (um) natural thing in the world.)

For these purposes, one should just forget the hexagon identity in Categories Work, and instead keep in mind two basic examples: the evaluation map

${\mathrm{ev}}_{x,y}:{x}^{y}\otimes y\to x$

which is dinatural in $y$, and the coevaluation map

${\mathrm{coev}}_{x,y}:x\to \left(x\otimes y{\right)}^{y},$

also dinatural in $y$. [I’m using exponential notation for the internal hom, because it’s going to look a lot less cluttered than the notation which uses $⇒$, and it saves me a lot of typing.]

Let’s look first at evaluation. It is gotten by applying the adjunction $\left(-\otimes y\right)⊣\left(-{\right)}^{y}$ to the identity

${x}^{y}\to {x}^{y}$

which is of course natural in $y$. That is, we are “decurrying” the identity, where decurrying shifts the instance of $y$ in an arrow $z\to {x}^{y}$ over to the other side: $z\otimes y\to x$. Now: all dinaturality of the evaluation map means is that it is the decurryed analogue of what ordinary naturality means for the identity.

That is, if we stare at the naturality equation

$\left({x}^{z}\stackrel{{x}^{g}}{\to }{x}^{y}\stackrel{{\mathrm{id}}_{{x}^{y}}}{\to }{x}^{y}\right)=\left({x}^{z}\stackrel{{\mathrm{id}}_{{x}^{z}}}{\to }{x}^{z}\stackrel{{x}^{g}}{\to }{x}^{y}\right)$

and decurry each side of this equation in turn, we arrive at the equation

$\left({x}^{z}\otimes y\stackrel{{x}^{g}\otimes y}{\to }{x}^{y}\otimes y\stackrel{{\mathrm{ev}}_{x,y}}{\to }x\right)=\left({x}^{z}\otimes y\stackrel{{x}^{z}\otimes g}{\to }{x}^{z}\otimes z\stackrel{{\mathrm{ev}}_{x,z}}{\to }x\right)$

That’s all dinaturality means here. You can easily write up this equation in the diagrammatic calculus you and Mike have written about.

The general type of dinaturality we’re talking about here is contained in the following definition:

Definition: Let $F:{C}^{\mathrm{op}}×C\to C$ be a functor, and let $x$ be an object of $C$. A dinatural transformation from $F$ to $x$ consists of a family of morphisms

${\theta }_{y}:F\left(y,y\right)\to x$

such that for each morphism $g:y\to z$, the following equation holds:

$\left(F\left(z,y\right)\stackrel{F\left(g,y\right)}{\to }F\left(y,y\right)\stackrel{{\theta }_{y}}{\to }x\right)=\left(F\left(z,y\right)\stackrel{F\left(z,g\right)}{\to }F\left(z,z\right)\stackrel{{\theta }_{z}}{\to }x\right)$

Written string-diagrammatically, this is saying something very simple. Hint: it looks like you are sliding a ‘$g$’ over a hump, from a covariant application of $g$ to a contravariant one.

The general point for closed category theory is that currying or decurrying, by applying an adjunction $\left(-\otimes y\right)⊣\left(-{\right)}^{y}$, transfers instances of naturality in $y$ over to instances of dinaturality of $y$, and vice-versa. For example, we get the family of morphisms

${\mathrm{coev}}_{x,y}:x\to \left(x\otimes y{\right)}^{y}$

by currying the identity $x\otimes y\to x\otimes y$. The naturality in $y$ of the identity map is thereby transferred to dinaturality of ${\mathrm{coev}}_{x,y}$ in the variable $y$, in the sense that the following equation holds:

$\left(x\stackrel{{\mathrm{coev}}_{x,y}}{\to }\left(x\otimes y{\right)}^{y}\stackrel{\left(x\otimes g{\right)}^{y}}{\to }\left(x\otimes z{\right)}^{y}\right)=\left(x\stackrel{{\mathrm{coev}}_{x,z}}{\to }\left(x\otimes z{\right)}^{z}\stackrel{\left(x\otimes z{\right)}^{g}}{\to }\left(x\otimes z{\right)}^{y}\right)$

and this leads to a second form of dinaturality:

Definition: Let $x$ be an object of $C$ and let $F:{C}^{\mathrm{op}}×C\to C$ be a functor. A dinatural transformation from $x$ to $F$ consists of a family of morphisms

${\eta }_{y}:x\to F\left(y,y\right)$

such that (you fill in the rest).

Giving a diagrammatic presentation of this again makes it look like something very easy and obvious.

The cute thing about the definition of dinatural transformations as given by the hexagon identity in Mac Lane’s book is that it takes care of both these definitions in one fell swoop, without case-splitting. But, that sort of slick definition tends to obscure the true provenance of dinatural transformations, as they occur in nature. In nature, they practically always can be explained in terms of one of the two definitions I gave above. (Aside: that’s even true for the case of ‘Church numerals’.) And that is certainly the case for how they come up in closed category theory.

Moral: dinatural transformations are nothing to be afraid of. And they are perfectly natural (pun intended) from the standpoint of diagrammatic calculus.

Posted by: Todd Trimble on February 25, 2009 3:45 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Why not link to the nLab entry you created for dinatural transformations?

Posted by: David Corfield on February 25, 2009 8:48 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Todd, do you do anything geometric with dinatural transformations in your mythical Geometry of Gray categories paper? Nudge, nudge.

Posted by: Simon Willerton on February 25, 2009 11:08 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

I’ve thought about it, but the mythical paper was more about higher-dimensional analogues of progressive string diagrams.

I have a private corner in the nLab where I intend to write up stuff that’s never been made public, and the surface diagrams work is on the list. I’d love to have your reactions to it at some point, Simon.

Posted by: Todd Trimble on February 25, 2009 9:35 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Todd wrote:

But your “bite the bullet” comment a while back makes me suspect that there’s something about the concept of dinatural transformation that bothers you. In fact, a lot of people seem to be bothered by this notion.

At first I was bothered by it because it seemed weird and mysterious. But then Mike Stay convinced me that we need it to explain ‘inference rules’ in our Rosetta paper.

(This isn’t news to you, but people who don’t know what I’m talking about can see an explanation starting around page 45; dinatural transformations come to the rescue on page 48.)

So by now I accept the importance of dinatural transformations. I know how to work with some using string diagrams that describe morphisms in a given category. And your latest post makes me even more inclined to love them.

So here’s why they still bother me! I’m bothered because I don’t know the answer to this question:

What type of algebraic gadget is this an example of: the 2-category of all categories, functors and natural transformations along with finite products, the ability to take ‘opposites’, and dinatural transformations.

It’s more than just a 2-category $\mathrm{Cat}$ with finite products and a product-preserving ‘op’ 2-functor $\mathrm{op}:\mathrm{Cat}\to {\mathrm{Cat}}^{\mathrm{co}}$, because of those dinatural transformations! They’re 2-cells of some different type than the natural transformations!

I want to hear some answer that sounds sort of like ‘it’s a globular 2-operad with duals’… except that it makes sense after you explain it.

And, I would like an extension of the usual graphical calculus for working in a monoidal 2-category, which handles this kind of algebraic gadget.

(See, I want to work not in a particular category, but in $\mathrm{Cat}$.)

Posted by: John Baez on February 26, 2009 12:15 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

This is a stab in the dark. But an ordinary adjoint relation is a zig-zag move such as the deformation:
x^3-x goes to x^3 + x via the deformation
x^3- t x. As such it is a cusp on the projection of the surface (x, x^3-tx, t) to the (y,z) plane.

A cusp in the presence of a saddle can do strange things. As JB knows, this is the “horizontal cusp” move. Which I’ll describe in a minute.

Saddles and optima also appear as evaluations and co-evaluations in some contexts. So my *guess* would be that the transformations that are being sought are horizontal cusps: (Forgive the lack of TeXing):
[(cap ox |) =(add cusp)=> ( | ox cap)(cup ox 1)(cap ox |)
==(3 equiv) ==>
[(cap ox |) =(saddle)=> (cap ox |) (| ox cup)(| ox cap) =(remove cusp)=> (| ox cap)]

My reading of the situation is hampered by not having all the fonts loaded (so I don’t see exponents as such) and only a vague understanding of n-cats.

On the other hand, the issue that I helped Bruce with also involved the horizontal cusp move, and this is an identity that is hard to miss from the singularity point of view, but easy to miss from other points of view.

In some sense, the other evaluation/coevaluation identities would be cancelations of critical points.

In general, adjoint type relations are cusps, and related matters are the horizontal cusp, the swallowtail and the critical cancelations.

Sorry I can’t put this into the correct context, but it feels right…

Posted by: Scott Carter on February 26, 2009 2:52 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

You’re definitely on the right track, Scott — I discussed some related stuff, complete with pictures of 2-tangles, back in week 18 of my spring 200 seminar on computation. See both the blog entry and the class notes.

Right now Mike and I are trying to go further than that…

Posted by: John Baez on February 26, 2009 7:17 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Well, one answer is that it’s an autonomous monoidal bicategory — as long as by “it” you are willing to mean the bicategory of categories and distributors rather than functors. See for instance:

• Street, “Functorial calculus in monoidal bicategories.”

where he also mentions a graphical calculus.

As is usual in Australia, he identifies functors with the distributors they represent, thereby working “modulo Cauchy completion.” But you can retain the functors as separate data by working in an autonomous framed bicategory (or equivalently, an autonomous equipment). Writing down something about autonomous framed bicategories (which I used to call “anchored bicategories”) has been on my to-do list for a while.

This answer has good points, but it’s also not completely satisfying; I also would like a more “operadic” description. I think it ought to be some sort of categorification of a modular operad. Possibly recent speculations about hyperstructures are also relevant.

Posted by: Mike Shulman on February 27, 2009 7:09 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

The string diagram calculus useful for monoidal categories has a version which is moderately appropriate here, where a term built from object symbols and the hom-functor $\left[,\right]:{C}^{\mathrm{op}}×C\to C$ has an associated sequence of strings, each marked by a variable name and oriented up or down depending on the overall co- or contravariantness in that variable.

The idea is to interpret each hom-object $\left[A,B\right]$ as if the closed category were monoidal compact, rewriting it as ${A}^{*}\otimes B$, where $\otimes$ is the fictional tensor product. For example, the term $\left[\left[\left[A,B\right],C\right],D\right]$ becomes $\left(\left(\left({A}^{*}\otimes B{\right)}^{*}\otimes C{\right)}^{*}\otimes D\right)={C}^{*}\otimes {A}^{*}\otimes B\otimes D$. Then the ordinary string diagram calculus applies, and in particular the map $I\to \left[A,A\right]$ becomes a map $I\to {A}^{*}\otimes A$, which we know to denote by a “cup” appropriately labeled and oriented. The Yoneda map $\left[A,B\right]\to \left[\left[C,A\right],\left[C,B\right]\right]$ similarly becomes a map ${A}^{*}\otimes B\to {A}^{*}\otimes C\otimes {C}^{*}\otimes B$, which is another “cup” (not a cap!) but oriented the other way and fenced in on either side by an identity string. Dinaturality is then as clear as Todd Trimble has just described it: morphisms on one side of a cup can swoop around to the opposite side.

Unfortunately, this “free compactification” loses a lot of necessary information. For example, $\left[\left[\left[A,B\right],C\right],D\right]$ and $\left[C,\left[\left[B,A\right],D\right]$ are in general not isomorphic, but they have the same presentation as ${C}^{*}\otimes {A}^{*}\otimes B\otimes D$; more description is required to determine which term from the closed, noncompact category is desired. Parentheses will do, but to save clutter these can be stored as a binary tree off to the side, whose leaves represent the variables and whose other nodes the intermediate stages of computation. The unit and Yoneda maps would then require rules for how they transform the trees so that the leaves continue to match the variables in the “compactified” expression.

Now, if you wanted, you could move into a three-dimensional picture, with the compact-style string diagram on a vertical plane running from bottom to top, and for each stage, a horizontal slice would be given by the binary tree, whose leaves are on the vertical plane at their respective locations. Stack all those horizontal slices together, and you’ve got a branching manifold behind the scenes of the original string diagram.

This works out to be what you get when you employ the picture notation developed in the above paper of Simon Willerton’s. As a tool for understanding the basic axioms for closed categories, it’s a bit messy, but it becomes clearer if you view the axioms applied to the vertical slices (the compact-style string diagrams) as being ordinary topological manipulations (which they are), and the branching behind as making sure you don’t accidentally try to compose maps that only appear to match (such as ${1}_{\left[\left[\left[A,B\right],C\right],D\right]}$ and ${1}_{\left[C,\left[\left[B,A\right],D\right]}$).

Now, when you work out what the string diagrams are for the two sides of the associativity-expressing axiom, you get something that looks coincidentally like dinaturality of Yoneda, only not quite. I think you’ll have to produce the diagrams yourself to see what I mean: one cup follows the orientation of the other until it has bent around completely, thickening the cup (actually thickening one cup and creating another) and adding a contravariant cap to the opposite side. Is there something deep going on here, with the topological motion mimicking the interpretation of dinaturality, or is it just a coincidence that that’s what this motion appears to be?

Posted by: Owen Biesel on February 25, 2009 5:11 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

It’s possible to get from “internally closed” to “closed” by adding two things:

1. A natural isomorphism $M\left(I,A⊸B\right)\cong M\left(A,B\right)$. Theo’s example shows that this isomorphism does not necessarily exist in general.
2. A coherence condition. If we write the components of the internal currying isomorphism as
(1)${\psi }_{A,B,C}:\left(A\otimes B\right)⊸C\to A⊸\left(B⊸C\right),$
then the condition is that the diagram below must commute for all $A$, $B$, $C$, and $X\in M$:
(2)$\begin{array}{ccc}\left(A\otimes \left(B\otimes C\right)\right)⊸X& \stackrel{{\psi }_{A,B\otimes C,X}}{⟶}& A⊸\left(\left(B\otimes C\right)⊸X\right)\\ \mid & & ↓A⊸{\psi }_{B,C,X}\\ {\alpha }_{A,B,C}⊸X& & A⊸\left(B⊸\left(C⊸X\right)\right)\\ ↓& & ↑{\psi }_{A,B,C⊸X}\\ \left(\left(A\otimes B\right)\otimes C\right)⊸X& \underset{{\psi }_{A\otimes B,C,X}}{⟶}& \left(A\otimes B\right)⊸\left(C⊸X\right)\end{array}$

The second one is a bit more subtle: without it, you do of course have a closed monoidal category (since the required “external currying” adjunction can be constructed from the internal currying and the isomorphism of condition 1) but then the canonical internal currying map of that closed category is not necessarily equal to the $\psi$ you started with! So the definitions are not actually equivalent since one carries additional structure.

This condition is one of the axioms in Eilenberg and Kelly’s paper on closed categories. I think it’s possible to concoct an example showing it is necessary by starting with a monoidal closed category that admits two different associators, and using one of the associators to derive $\psi$ (using the external currying map, which exists by the first condition) and taking the other associator as the actual associator. The condition above would then not hold, unless I’m very confused.

If we have both these things, then (at least in the symmetric case) the category is closed in the usual sense. What’s going on is something like the following: the tensor and the internal hom both define promonoidal structures on the category $M$, but these do not necessarily coincide unless you force them to. (Section 8.4 of my thesis has more on this, though it’s about a slightly more general situation.)
Posted by: Robin Houston on February 23, 2009 11:44 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Here is a counter example. Let $C$ be your favorite closed (symmetric) monoidal category in the classical sense, and let $D$ be a left Bousfield localization of $C$, i.e. we have an adjunction $F:C⇄D:U$ (with $F$ the left adjoint). Assume that $D$ is endowed with a closed (symmetric) monoidal structure in the usual sense, and that $F$ is a (symmetric) monoidal functor (a simple example consists to take $C$ to be the category of abelian groups, and $D$ the category of rational vector spaces; you can also take for $C$ the category of simplicial sets (or your favorite kind of spaces), and for $D$ the category of sets, with $F={\pi }_{0}$). Then there is a funny internal Hom on $C$, defined by the formula: $\mathrm{Hom}\left(X,Y\right)=U\left(\mathrm{Hom}\left(F\left(X\right),F\left(Y\right)\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$ Unless if F is an equivalence of categories, this internal Hom will satisfy quite a few coherences, in particular, we will get that $\mathrm{Hom}\left(X\otimes Y,Z\right)\simeq \mathrm{Hom}\left(X,\mathrm{Hom}\left(Y,Z\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$ However, this internal Hom won’t be an internal Hom in the usual sense: for instance, the identity $X\simeq \mathrm{Hom}\left(1,X\right)$ will fail, because, instead, we will get $UF\left(X\right)\simeq \mathrm{Hom}\left(1,X\right)\phantom{\rule{thinmathspace}{0ex}}.$ However, not asking at least for some relationship between the identity and the functor $\mathrm{Hom}\left(1,-\right)$ (like being adjoints) seems very weird to me. I mean, in a closed monoidal category $C$, I would like the functor $\mathrm{Hom}\left(1,-\right)$ to be at least a lax monoidal functor: otherwise, how can you hope to get composition maps of shape $\mathrm{Hom}\left(X,Y\right)\otimes \mathrm{Hom}\left(Y,Z\right)\to \mathrm{Hom}\left(X,Z\right)?$ The example I gave still satisfies this property. If you don’t put any relation, you might take for your internal Hom the constant functor with value your favorite weird object, and still have your ‘internal closedness’…

Posted by: Denis-Charles Cisinski on February 24, 2009 1:23 AM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

I think a nice definition would be the following:

A monoidal structure on a category $M$ is given by a monad structure on the 2-functor $\left(M×-\right)$. Via $\left(A×B{\right)}^{\mathrm{op}}={A}^{\mathrm{op}}×{B}^{\mathrm{op}}$, this also gives us a monad structure on $\left({M}^{\mathrm{op}}×-\right)$.

On the other hand we have a functor ${\mathrm{hom}}_{M}:{M}^{\mathrm{op}}×M\to \mathrm{Ens}$. Using this functor and the unit of our monad ${M}^{\mathrm{op}}$ we get the “underlying set” Functor $V:M\to \mathrm{Ens}$, namely $X↦{\mathrm{hom}}_{M}\left(1,X\right)$.

Definition: An internally closed monoidal Category is a monoidal category $M$ together with a structure of an ${M}^{\mathrm{op}}$-Algebra on $M$ such that the operation ${M}^{\mathrm{op}}×M\to M$ is actually a $V$-lift of ${\mathrm{hom}}_{M}$.

Thus, we get the following (I will denote the operation by $\left[-,-\right]$): An isomorphism $\left[a\otimes b,c\right]\to \left[a,\left[b,c\right]\right]$ with $\left[1,a\right]=a$ and $V\left[a,b\right]={\mathrm{hom}}_{M}\left(a,b\right)$.

I’m not sure about what weakness and coherence conditions we need (i’m a bit new to this), but i guess this definition should give the general idea. Furthermore we should be able lift this definition to the case where we are in a monoidal closed category with duals, right?

Posted by: Gerrit Begher on February 25, 2009 8:26 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

An interesting fact, though perhaps not wholly relevant, is that there is a monoidal $2$-category wherein monoidal closed categories are pseudomonoids. It’s “$2$-Chu”.

Objects are triples $\left(U,X,e\right)$ where $U$ and $X$ are categories and $p:U×X\to \mathrm{Set}$. Morphisms $\left(U,X,p\right)\to \left(V,Y,q\right)$ are triples $\left(f,g,t\right)$ where $f:U\to V$, $g:Y\to X$ and $t$ is a natural isomorphism between $q\left(f-,?\right)$ and $p\left(-,g?\right)$. $2$-cells are the obvious thing.

The tensor product of $\left(U,X,p\right)$ and $\left(V,Y,q\right)$ is $\left(U×V,Z,r\right)$, where $Z$ is the category whose objects are triples $\left(a,b,s\right)$ with $a:U\to Y$, $b:V\to X$ and $s$ a natural isomorphism between $p\left(-,b?\right)$ and $q\left(?,a-\right)$, and $r$ sends $\left(u,v,a,b,s\right)$ to $p\left(u,bv\right)$ (or $q\left(v,au\right)$). The tensor unit is $\left(1,\mathrm{Set},{\pi }_{2}\right)$ (there are obviously some size issues I’m glossing over here).

Now for any locally small category $C$ there is an object $\left(C,{C}^{\mathrm{op}},\mathrm{Hom}\right)$ of this $2$-category, and it’s easy to calculate that a pseudomonoid structure on this is precisely a monoidal biclosed structure on $C$.

Posted by: Richard Garner on March 11, 2009 2:03 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

But presumably there are pseudomonoids in 2-Chu that are not of that form?

Posted by: Mike Shulman on March 11, 2009 6:10 PM | Permalink | Reply to this

Re: Monoidal Closed Categories And Their Deviant Relatives

Yes, plenty.

I got it back-to-front above – you have to start with $\left({C}^{\mathrm{op}},C,\mathrm{Hom}\right)$ to get a monoidal closed structure on $C$.

A general pseudomonoid looks like a triple $\left(C,E,p\right)$ where $C$ is a monoidal category, $E$ is a category, $p$ is a profunctor ${C}^{\mathrm{op}}×E\to \mathrm{Set}$ and there are left and right actions

$\left[-,?{\right]}_{l}:{C}^{\mathrm{op}}×E\to E$ and $\left[-,?{\right]}_{r}:{C}^{\mathrm{op}}×E\to E$

—equipped with the usual coherence isomorphisms for a left and right action by a monoidal category—such that

$p\left(c\otimes d,e\right)\cong p\left(c,\left[d,e{\right]}_{l}\right)\cong p\left(d,\left[c,e{\right]}_{r}\right)$

naturally in $c$, $d$ and $e$, and in a way which interacts nicely with the coherence isomorphisms for the actions.

(Note that in the case where $E=C$ and $p$ is the Hom-functor, the coherence isomorphisms for the actions come for free from the Yoneda lemma. In general, this will happen whenever the transpose of $p$ to a functor $E\to \left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ is fully faithful.)

Posted by: Richard Garner on March 12, 2009 6:24 PM | Permalink | Reply to this

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