It’s August now, and the fall semester approaches apace. That can mean only one thing: it’s time to sell my
- Functors extending the Kauffman Bracket
The Kauffman Bracket is a family of invariants of knots and links up to regular isotopy taking their values in commutative rings, and defined by a “skein theory”. We want to find monoidal functors defined on the category of framed tangles so that if we restrict the functors to knots and links we recover (essentially) the old invariants. This approach highlights the fact that “skein theories” are actually just generating sets for monoidal categorical ideals, and that the skein-theoretic approach to knot invariants is another branch of representation theory.
We thus study the representation theory of -linearizations of the category of framed tangles, and of the Temperley-Lieb categories . We show that the representation theory of these categories is equivalent to the theory of (non-symmetric) nondegenerate bilinear forms over .
- Spans: A (Braided) (Monoidal) Bicategory (with Duals)
The most famous braided monoidal bicategory with duals is the “universal” such category: that of tangles, or of 2-tangles up to isotopy. Slightly less well-known are the bicategories arising from knot homology theories. Still, the question is asked, “where are the braided monoidal bicategories with duals?”
One way to produce a bicategory with such extra structure is to consider the bicategory of spans on a category with the analogous such structures. In this talk, we discuss the span construction (and its dual, cospans) and show how well-behaved monoidal structures, braidings, and dualities lift to the bicategory of spans.
- The Tangle Group
The group of a knot or link is a well-known invariant of ambient isotopy. We would like to extend this invariant to a monoidal functor on the category of tangles in such a way that when we restrict to knots and links we recover (essentially) the old knot group.
Here, we define a monoidal bifunctor from the bicategory of (tangles, isotopies) to the bicategory of cospans of groups, and show how the restriction of the decategorification of this bifunctor to knots and links reproduces the knot group. We also indicate how the use of cospans immediately applies to generalize the fundamental quandle of a link, the fundamental biquandle of a virtual link, and other such invariants.
- A Categorification of Quandle Coloring Numbers by Anafunctors
The number of colorings of a link by a given quandle is a classical invariant of links up to ambient isotopy. We would like to categorify and extend this invariant to the category of tangles.
Here, we show how to associate, functorially, to each tangle an anafunctor between two comma categories of quandles. When we restrict this assignment to knots and links and specify a quandle of colors we recover -coloring invariant. If we first decategorify and specify a quandle of colors we recover the -coloring matrix of a given tangle.
This approach can be significantly generalized. We indicate the existence of a similar “-coloring” invariant for any co- object in the category of pointed topological pairs up to homotopy.
And now some comments. Generally, these abstracts apply to the highest-level version of each talk. I can tweak any of them down a bit, mostly to adjust for familiarity of the audience with categories and with knot theory.
The Kauffman Bracket talk is probably the most straightforward. It clearly highlights the relationship between skein theory and representation theory. Its primary interest is in this connection, and in the fact that it lays the groundwork for parallel categorifications of the Kauffman Bracket to Khovanov homology.
The talk on spans is, strictly speaking, spun off of my work on tangle groups. More explicitly, once you set things up in terms of cospans, extending the knot group to the tangle group becomes effortless. This talk, though, focuses on how spans and cospans are good tools for moving “up the ladder” of categorical structures, bringing lower-level structures along with them. This talk, it should be noted, is very preliminary, as I’m in the process of writing some of these things down.
The knot group talk should be clear to an algebraic topology audience. It’s really the genesis of the use of cospans in the study of tangles For audiences more familiar with knot theory in particular, I can do the whole thing from the get-go in quandles.
The quandle talk really isn’t that abstract when it comes down to it, but it uses a number of tools possibly unfamiliar to the general mathematical audience. In fact, a good part of it is devoted to getting the definitions down straight. Once they’re in place, the whole structure just sort of builds itself, which is how I really like my mathematics to go. The caveat, then, is that the audience really does need to either be interested in knot theory already, or somewhat familiar with and friendly towards categories. Otherwise it’s really tough to motivate the material and to cover it within the usual microcentury.
I could possibly put the latter two together in a pair of lectures, since the quandle coloring invariant is a direct outgrowth of the fundamental quandle of a tangle. That would also make it a bit easier to motivate the second half, so it may well go more smoothly as a pair to a more general audience.
So, if your department is looking to fill a slot in an algebraic topology (or “quantum topology”, as they’re calling this stuff now) or a category theory seminar, let’s talk. Clearly the easier it is for me to get there from New Orleans the easier it will be to make arrangements. Also, though I’ve gotten used to paying out of pocket for these things, assistance in travel would also be helpful.
I am particularly looking for an engagement in the Baltimore/Washington D.C. area around the weekend of October 6, so that gets high priority.
A closed category is a symmetric monoidal category where each functor has a specified right adjoint called an “exponential”: . By what we said yesterday, this means that there is a unique way to sew all these adjoints parametrized by together into a functor.
The canonical example of such a category is the category of sets with the cartesian product as its monoidal structure. For each set we need an adjunction . That is, functions taking a pair of elements — one from and one from — to an element of should be in bijection with functions from to . And indeed we have such an adjunction: is the set of all functions from to .
Let’s say this a bit more concretely. If we consider the set of natural numbers we have the function , which takes two numbers and gives back a third: if we stick in and we get back . But what if we just stick in the first number of the pair? Then what we get back is a function that will take the second number and give us the sum: if we just feed in we get back the function . That is, we can see addition either as a function taking a pair of numbers to a number, or we can see it as a function , taking a number to a function from numbers to numbers.
More generally, if we can define , which is defined by . That is, takes an element of and gives back a function from to . We call this process of turning functions of many variables into functions of a single variable at a time “currying”, after Haskell Curry. It turns out to be phenomenally useful for discussing theories of computation, and forms part of the basis of functional programming.
Closed categories are an attempt to mirror this currying procedure in other categories. In general, if the monoidal structure in question is the categorical product (which is always symmetric) then we say the category is “cartesian closed”. Most such categories still look a lot like this example in sets, with morphisms given by functions preserving structure and the exponential given by an appropriate analogue of the functor.
Here’s an example, though, of a cartesian closed category that looks rather different. It requires the notion of a “predicate calculus”, but not very much of it. Basically, if you have a rough idea of what such a thing is you’ll be fine.
Okay, there’s a category whose objects are well-formed formulas in the calculus, and whose arrows are proofs. The product in this category is , read “and”. If we know then we can derive and we can derive , so there are two arrows. On the other hand, if implies both and , then implies . This is just a rough sketch that is the categorical product, but it’s good enough for now.
Anyhow, the exponential is , read “implies”. This is subtler than it seems. In the last paragraph I was saying things like “ implies “, but this is not what I mean here. The formula is the statement within the calculus that there exists a proof starting from and ending with . Writing out “implies” as I’m discussing the structure is a high-level view from outside the system. Writing in the structure itself is a low-level view, “internal” to the category. “ implies ” is a statement about the category, while “” is a statement within the category.
Now if implies we have (by definition) a proof starting from and ending with . Then we can use this proof to devise a proof starting from and ending with . That is, implies if and only if implies . This shows that is right adjoint to , as we claimed.