## August 26, 2010

### The Geometry of Monoidal Fibrations?

#### Posted by Mike Shulman

In my paper framed bicategories and monoidal fibrations I wrote down a definition of a “monoidal fibration,” which you can think of as a family of monoidal categories indexed by a (usually cartesian) monoidal category. (I certainly wasn’t the first person to write down such a thing.) A canonical example is that if $V$ is any monoidal category, then the categories $V^X$, for $X$ a set, are each monoidal and are indexed by the cartesian monoidal category $\mathrm{Set}$.

What I want to know now is – is there a string diagram calculus for these things? Never mind proving anything about it; I can’t even visualize what it ought to look like!

That’s all I want to ask, but in case you don’t want to search through the long paper looking for the definition, let me give it to you, in two different ways.

If $B$ is a cartesian monoidal category, then a $B$-indexed monoidal category, is just a pseudofunctor $B^{op} \to MonCat$ where $MonCat$ is the 2-category of monoidal categories, strong monoidal functors, and monoidal transformations.

On the other hand, if $B$ is any monoidal category, then a monoidal fibration over $B$ is a fibration $p:E\to B$ such that $E$ is also a monoidal category, $p$ is a strict monoidal functor, and the tensor product of $E$ preserves cartesian arrows. If you convert this fibration to a pseudofunctor $B^{op}\to Cat$ sending $x\mapsto E_x$, then what the extra structure gives you is a collection of functors $\boxtimes: E_x \times E_y \to E_{x\otimes y}$ with suitably coherent associativity and unit isomorphisms. Note that in general, the individual fiber categories $E_x$ are not monoidal categories in their own right.

However, when $B$ is cartesian monoidal, we can pass back and forth between the two definitions as follows. Given a monoidal fibration $E\to B$, we define a monoidal structure on the fiber $E_x$ by $a \otimes_x b = \Delta_x^\ast (a\boxtimes b)$ where $\Delta_x \colon x\to x\times x$ is the diagonal in $B$. Conversely, given a $B$-indexed monoidal category, we make it into a monoidal fibration by defining, for $a\in E_x$ and $b\in E_y$, $a\boxtimes b = \pi_y^\ast(a) \otimes_{x\times y} \pi_x^\ast(b)$ where $\pi_y$ and $\pi_x$ are the two projections $x\times y \to x$ and $x\times y \to y$, respectively.

I’d be ecstatic with a string diagram calculus including the case when $B$ is not cartesian, but I’d be pretty happy with one that only works when $B$ is cartesian. There are a bunch of monoidal categories around, and also (in the second case) a bunch of monoidal functors; we know how string diagrams for monoidal categories work, and Micah McCurdy recently showed us one nice way to do string diagrams for monoidal functors. Can we put those together somehow and incorporate the monoidal structure of $B$?

Posted at August 26, 2010 4:52 AM UTC

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### Re: The Geometry of Monoidal Fibrations?

Micah McCurdy recently showed us one nice way to do string diagrams for monoidal functors

Nice, thanks. I’ve used similar technique:

(sorry for Russian subscripts, but I hope the principle is clear).

It was originally placed in my blog record.

Btw I made all calculations in Appendix A of that paper using this technique.

Posted by: osman on August 26, 2010 5:27 PM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Yes, that does look like essentially the same idea. Doing it in three dimensions also seems to make it easier to see what’s happening when composing functors, although it’s probably harder to draw that way in the computer.

Posted by: Mike Shulman on September 3, 2010 5:58 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Do you know of Charles Peirce’s existential graph approach to logic? Your modus ponens diagram looks very like how you would render his version (see, e.g., half way down here) in 3D.

From your mention of ‘Laws of Form’ perhaps you have looked more at Spencer-Brown. If I recall correctly, his system was very close to Peirce’s, but juxtaposition of proposition letters signified ‘or’ rather than ‘and’.

Todd Trimble would be able to tell us more. See his papers with Geraldine Brady.

Posted by: David Corfield on September 3, 2010 9:29 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Yes, I’ve read exactly that wikipedia article about Spencer-Brown. This is why I called my post “laws-of-form alive”.

But what I mean is the similarity of principles only, not exact correspondence of calculi. And I don’t know about Charles Peirce’s existential graph, thank you very much for that link.

Actually I would like to treat tensor product as “AND”, but how to treat my $\varepsilon_A$ in this case? I think the tensor unit $I$ should be understood as “Unknown” and then everything looks good.

Posted by: osman on September 3, 2010 9:50 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

I think the tensor unit $I$ should be understood as “Unknown” and then everything looks good.

Or as “IgnorableInformation”. In everyday logic we usually conclude from “A AND NOT A” to “IgnorableInformation”, if both sources of information are equivalently reliable. We don’t conclude something like FALSE, because “A and IgnorableInformation” is again “A”, not FALSE.

So I suggest to interprete the “white space” as “ignorable information”.

Posted by: osman on September 3, 2010 3:07 PM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

I presented another version of graphical calculus for proofs, based on game semantics. Maybe it was proposed by others before, but I don’t know about it.

I think that monoidal game semantics can be even more rosettic then monoidal Rosetta, since any dynamics (even in physics) can be presented as a game.

Posted by: osman on January 13, 2012 12:35 PM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

What I want to know now is – is there a string diagram calculus for these things?

It seems that when monoidal fibration is defined as fibration between monoidal categories (with additional restriction), it’s easy to extend McCurdy’s (or my “braided”) graphical calculi to express it.

Let $\Phi: E\to B$ is monoidal fibraion. If i’m not mistaken, it means that we can transform any (McCurdy’s or my) diagram of the form $f: A\to \Phi(M)$ to the form $\Phi(\phi_{f, M}): \Phi(f^*M)\to \Phi(M)$. So we need just add some rules of such transformations, taken from axioms of fibration and monoidal fibration.

The rules of fibrations are easy to imagine. The only additional rule in monoidal case is that we can transform the diagram of $f\otimes g: A\otimes B \to \Phi(M)\otimes \Phi(N)$ directly to $\Phi(\phi(f, M)\otimes \phi(g, N))$

Or it’s different from what you want? Then sorry.

Posted by: osman on August 27, 2010 3:28 PM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Thanks for taking a stab at this! Sorry for the delayed response.

The rules of fibrations are easy to imagine.

Not to me; I don’t think I understand what you have in mind. I gather that you want to apply the graphical calculus of monoidal functors to the fibration $\Phi\colon E\to B$, which is great, but I don’t see how to notate the operation of pullback along cartesian arrows. For instance, how would you notate the object $\Delta_x^*(a\boxtimes b)$ which defines the “fiberwise” tensor product?

Posted by: Mike Shulman on September 3, 2010 6:04 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

For $E$ and $B$ monoidal categories, it seems natural to declare that a monoidal fibration between them is a 2-fibration $\mathbf{B}E \to \mathbf{B}B$ between their delooping 2-categories.

How does that definition relate to yours?

Posted by: Urs Schreiber on September 2, 2010 10:47 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

I think it’s very different, for the same reasons that MonCat is not a full sub-3-category of Bicat. A monoidal functor between monoidal categories is the same as a 2-functor between their deloopings, but saying that the monoidal functor is a fibration is about liftings of 1-morphisms, which becomes lifting of 2-morphisms in the delooping. But saying that a 2-functor is a 2-fibration entails not only lifting of 2-morphisms, but also lifting of 1-morphisms.

Thus, if $\Phi\colon E\to B$ is a monoidal functor, then for its delooping to be a 2-fibration it would have to be the case that for any object $x\in B$, there is an object $a\in E$ with $\Phi(a)=x$ (or $\cong x$) such that for any $b\in E$ with $\Phi(b) = y\otimes x$ (or $\cong$), there exists a unique (maybe up to isomorphism) object $c\in E$ with $\Phi(c) = y$ and $b = c\otimes a$. This is kind of a weird property, and I don’t think it holds in any of the examples I’m interested in.

Posted by: Mike Shulman on September 3, 2010 6:21 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Mike, might work on representing the predicate calculus help? Fibred over a set, we have the collection of well-formed formulas in that set of variables. With implication and ‘and’, each fibre has the structure of a monoidal category. Then we need your collection of functors

$\boxtimes: E_x \times E_y \to E_{x\otimes y}.$

This again can be conjunction, taking a formula over $X$ and another over $Y$ to one over $X \union Y$.

If this form of the predicate calculus is an example of a monoidal fibration, then Charles Peirce’s system Beta of his existential graphs might give you some clues for a diagrammatic calculus. Fortunately, Todd Trimble has already put the system into a category theoretic system with Geraldine Brady in A string diagram calculus for predicate logic.

Posted by: David Corfield on September 3, 2010 11:33 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

That’s a good idea; that expression of predicate calculus is a very good example of a monoidal fibration. (Actually, once you have monoidal fibers, the functors $\boxtimes$ come for free as I sketched above; they end up being conjuction with contexts made disjoint.) And now that you link to it, I dimly remember skimming through the Brady-Trimble paper a long time ago. Actually, it probably influenced my early thinking about monoidal fibrations and framed bicategories, although I had forgotten it until now.

It seems as though their string diagrams consist essentially of first making the monoidal fibration into a locally-posetal compact-closed 2-category (which is the horizontal 2-category of the double category that can be constructed from any suitable monoidal fibration), and then using ordinary-ish monoidal-category string diagrams there (with surgeries on string diagrams to represent the 2-cells). Is that accurate? (I’m ignoring things like sep lines which seem specific to classical predicate calculus, versus general monoidal fibrations.)

If my understanding is right, then it is sort of an answer to my question, but I’m not sure how I feel about it. I’d have to try working with it for a bit to see how convenient it is. One problem is that in the non-posetal case, surgeries don’t seem quite as convenient a way to represent the 2-cells, since you need to label them with which 2-cell they represent and give rules for when two surgeries should be considered equal. (Probably that is an indication that we really need to move up a dimension and use real surface diagrams for monoidal 2-categories.)

I’d also kind of hoped for a more “native” string diagram calculus for monoidal fibrations, which one could use (among other things) to prove things like that you can construct a compact-closed double category from any monoidal fibration, rather than building that fact into the string diagram calculus. In particular, it’s not immediately clear whether these string diagrams could be generalized to monoidal fibrations lacking pushforwards (existential quantification), since pushforwards are required to get a double category (although you can get a virtual double category in general…).

Regardless, this is a really interesting approach, and I need to think about it some more.

Posted by: Mike Shulman on September 6, 2010 6:09 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Coincidentally, in a discussion of string diagrams for closed monoidal categories on the categories list, Peter Selinger recently concluded

I guess the point is that one can save some time by exploiting what logicians have already done, using the connections between logic, category theory, and string diagrams, rather than re-inventing the wheel.

Thinking about this overnight, I realized that there are “native” string diagrams for monoidal fibrations sitting inside those for the monoidal 2-category, and these can be described directly even when pushforwards are missing or the base is not cartesian. These are the string diagrams with no strings coming out the bottom, i.e. whose target is the unit object. Reading carefully, I see that Brady-Trimble singled these out as well, calling them “positive geometric formulas” (p17) — I guess that indicates that those are also (closer to) the ones that Pierce originally described.

So in the graphical calculus that I’m now seeing for a monoidal fibration $\Phi:E\to B$, an object of $E$ is represented by a string diagram, with no strings coming out the bottom, with strings labeled by objects of $B$, and with two types of nodes: (1) objects of $E$, which have empty target and a source labeled with their image under $\Phi$, and (2) morphisms of $B$, which have source and target just as in ordinary string diagrams for $B$. The “value” of this string diagram is given by tensoring together all the objects of $E$ which occur in it, then pulling back along a suitable morphism of $B$ built from all those occurring in the diagram. The morphisms of $E$ are then represented by surface diagrams relating these string diagrams.

This version seems to solve all my initial complaints! I need to spend a while working with it to get used to it, but it may be exactly what I was looking for. I’m a little scared by the appearance of surface diagrams, but probably we’re up high enough in dimensions here that something of that sort is inevitable.

Posted by: Mike Shulman on September 7, 2010 7:04 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Well Todd mentioned to me that he had been considering free cartesian bicategories in terms of surface diagrams. You two ought to confer.

By the way, it’s Peirce not Pierce (and pronounced like ‘purse’). It derives from a Dutch surname.

Posted by: David Corfield on September 7, 2010 9:00 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

We have, a bit, and probably will more in the future. I’ve been hoping that he would join this conversation, but maybe he hasn’t noticed it yet. Thanks for the spelling/pronunciation tip.

Posted by: Mike Shulman on September 7, 2010 6:45 PM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Hi Mike –

I’m here, but I have not had a chance to look at your paper, and haven’t studied this thread with care. But maybe we could “talk” off-line a bit?

(Thanks to David for giving me a heads-up.)

Posted by: Todd Trimble on September 7, 2010 9:21 PM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

And once you’ve sorted out Peirce’s beta system, I look forward to a category theoretic rendition of the gamma system. As I mentioned here, these use tincturing (in heraldic azure, jules, argent, etc.) to represent modalities.

Todd once handed me a pile of photocopies of Peirce’s notes on the gamma system, as we discussed here.

Posted by: David Corfield on September 8, 2010 8:10 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

I didn’t remember this post from (now over 5) years ago, nor do I remember handing you a pile of photocopied notes on Gamma. (I guess that was in London, in 1999?) But I am very interested to learn of this now, as I haven’t laid eyes on those notes in a long time, and don’t know where to look for them – if I have them at all still – in my house.

If they are the same notes as the ones I have in mind, then I remember a particular spot in them that used to obsess me, because it looked just as if Peirce were rediscovering power allegories (a form of topos theory but in terms of relations as opposed to functions), judging by the looks of his graphs. I can’t make up my mind whether it was coincidence or not, or how much one is to make of it. But at the time it made me wonder whether we should (also) be reading into Gamma an attempt at a higher-order logical calculus of relations, extending Beta which is a first-order logical calculus of relations.

Posted by: Todd Trimble on September 26, 2015 2:27 PM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Yes, it must have been around that time, in an apartment in London, somewhere around The City or further east near the river?

I believe you gave those notes to me to keep as you couldn’t see yourself continuing to work on them. I should still have them somewhere. I’ll have a look when I have a moment.

Posted by: David Corfield on September 29, 2015 8:51 AM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Sounds exciting! But don’t hold your breath. (-:

Posted by: Mike Shulman on September 8, 2010 8:17 PM | Permalink | Reply to this

### Re: The Geometry of Monoidal Fibrations?

Thinking about this some more, I think it is actually almost a special case of surface diagrams for monoidal bicategories, even for monoidal fibrations that are not “frameable” (i.e. are not cartesian, or lack pushforwards). An arbitrary fibration $E\to B$ is equivalent to a pseudofunctor $B^{op}\to Cat$, which we can consider as a 2-profunctor $1\to B$ and take its cograph. This cograph contains $B$ as a full subcategory, and one more object, call it $\star$. If $E\to B$ is a monoidal fibration, then its cograph is almost a monoidal bicategory, although its tensor product is only partially defined: we can’t tensor objects of $B$ by $\star$. But if we blithely draw monoidal-bicategory surface diagrams for it anyhow, and also ignore the surfaces labeled by $\star$ as redundant information, then I think we get essentially the same graphical calculus as above.

Posted by: Mike Shulman on September 30, 2010 8:53 PM | Permalink | Reply to this

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