Leaving aside the issue of thorny question of the correct plural form of canopolis, lets try and get the definition right!
The motivating idea here is that a planar algebra is a nice (the right?) formalism for tensor categories with duals, and that there’s a reasonable hope that the definition will extend nicely to something like an n-planar algebra (a noodle algebra, a foam algebra?), which we should think of, very roughly, as an n-category with lots of duals. A canopolis is a first step along the way – it adds an extra categorical dimension to a planar algebra, but without demanding any more duals.
The term canopolis was introduced by Bar-Natan, although there called a canopoly, then used by Ben and by me and Ari Nieh. It still doesn’t have a good definition written down anywhere. The idea is simple; a canopolis is meant to be a planar algebra of categories. I’m still not too sure how much devil there is in the details though, and would love some help.
When planar algebras were first introduced, they had various features specific to their application to subfactors. It’s not too hard to abstract these away, however, so let’s do that here. Pick your favourite monoidal category (pick Vect if you’re old fashioned, and want to recover the original definition). On this first attempt, let’s not worry about allowing labels on anything (in particular, no black and white regions like in the original definition of subfactor planar algebras, even though thinking about labels carefully is fun and illuminating, especially if you want to start think about higher dimensional versions of planar algebras).
A planar algebra over is then
- for each ‘marked circle’ S, an object of
- for each ‘spaghetti and meatballs diagram’ T, with internal marked circles and external marked circle , a morphism in ,
- for any marked circle, the corresponding ‘radial diagram’ is sent to the identity morphism
- isotopic diagrams are sent to the same morphism
- a composite diagram is sent to the corresponding composition of morphisms.
We better define one or two phrases there. A marked circle is just a pair , with just the standard circle, and a finite subset. A spaghetti and meatballs diagram is a
- a collection of disjoint round discs inside the standard disc,
- a ‘base point’ on the boundary of each of those discs,
- embedded arcs in the complement of those discs, forming closed loops, or meeting the external or internal boundaries transversely.
The ‘external marked circle’ of a diagram is simply the boundary of the standard disc, and the associated finite set is just the collection of arc endpoints. The ‘internal marked circles’ are just a little trickier; for each boundary of one of the round discs, consider the (unique!) isometry to the standard circle, taking the base point to your favourite base point of the standard circle. Now the finite set is just the image of the collection of arc endpoints under this isometry.
Okay, that’s enough of that sort of nonsense for now; time to move onto the next sort. What if our ground category were not just a category, but a 2-category? How would we want to change the definition?
Let’s just begin by trying to describe one of the planar algebras of 2-categories we’re really interested in, and seeing what goes wrong. We’d really like the canopolis of 2-tangles, which associates to each marked circle the category with
- objects the set of tangles embedded in (not up to isotopy, just embedded) , with boundary points lying on the equatorial $S^1$, coinciding with the marked points and
- morphisms cobordisms between tangles in , fixing the boundary points, modulo isotopies of the cobordisms. (Note that composition is associative on the nose, because the cobordisms are only up-to-isotopy.)
The action of spaghetti and meatball diagrams is fairly obvious; we can fatten up such a diagram to a , with the spaghetti lying in the equatorial plane, and then just glue smaller s in. Similarly, we can fatten up a diagram to , and get an action on the morphisms as well. This all fits together as a functor, as we need.
However axioms simply aren’t satisfied. Looking at the numbered axioms above, #3 does still work; composite diagrams are sent on the nose to the corresponding composite functors. However, each of #1 and #2 fail. Two isotopic diagrams define slightly different functors, simply because the tangle objects of our categories are not defined up to isotopy. This isn’t really a problem of course, because there is a natural isomorphism relating the different tangles we obtain — the cobordism which is in fact just an isotopy, the isotopy relating the two original spaghetti and meatball diagrams! So we should replace the axioms above with
- for any marked circle, the corresponding ‘radial diagram’ is sent to a morphism which is 2-isomorphic to the identity morphism
- isotopic diagrams are sent to morphisms which are 2-isomorpic
- a composite diagram is sent, on the nose, to the corresponding composition of morphisms.
Well, that was kinda dull, perhaps. This weakening of the axioms seems fine, to me, to allow us to talk about the canopolis of 2-tangles. It also seems like there’s no useful stricter version of the axioms, and hence it seems doubtful that there are any interesting strictification theorems out there, for those who like such things.
Implicitly, I guess, I’m proposing in this post that we relax the definition of a canopolis slightly (almost trivially) — we’ve all been using ‘canopolis’ previously as ‘a planar algebra of categories’ (and ignoring the need to weaken the planar algebra axioms), but I’d suggest we change our mindset so ‘canopolis’ is shorthand for ‘a planar algebra defined over a 2-category’, as above, knowing that the only examples of those 2-categories we ever care about are certain categories of categories.
Finally, let me finish up with the ‘todo-list’ for the nascent “Tale of planar-algebra-ification” (sorry John!):
- Explain how labelling strands and regions fit into the picture, by specifying the freely generated category of labels.
- Explain, and discover, the higher dimensional story. So far there are two good bits of progress towards this:
- That “freely generated category of labels” is secretly a 0-planar algebra (or a “linear algebra”, according to Noah!), and this fact is the hint that tells us that part of the data of an n-planar algebra is a free (n-1) planar algebra “of labels”.
- Khovanov homology can be dressed up as a 3-planar algebra (I know, I know, a definition doesn’t exist — but this example is my motivating one), and this nicely packages up everything I know about the categorical structure of Khovanov homology — it’s a braided tensor 2-category, and it has all sorts of duals. Even better, the 3-planar algebra statement is much easier to state and verify than the enormous list of axioms we need to think about to say the same things about Khovanov homology in the “old” langauge. A fair bit of this is described in my slides from the recent Oporto meeting in Faro.