When we have an algebraic concept described as a set with extra structure, the morphisms between such structured sets are usually structure-preserving functions between the underlying sets. This gives us a “forgetful” functor which returns the underlying sets and functions. Then as we saw, we often have a left adjoint to this forgetful functor giving the “free” structure generated by a set.
But now that we’re talking about monoid objects we’re trying not to think about sets. A monoid object in is a monoidal functor from to , and a “homomorphism” of such monoid objects is a monoidal natural transformation. But the object part of such a functor is specified by one object of — the image of — which we can reasonably call the “underlying object” of the monoid object. Similarly, a natural transformation will be specified by a morphism between the underlying objects (subject to naturality conditions, of course). That is, we have a “forgetful functor” from monoid objects in to itself. And a reasonable notion of a “free” monoid object will be a left adjoint to this functor.
Now, if the monoidal category has coproducts indexed by the natural numbers, and if the functors and preserve these coproducts for all objects , then the forgetful functor above will have a left adjoint. To say that the monoidal structure preserves these coproducts is to say that the following “distributive laws” hold:
For any object we can define the “free monoid object on ” to be , equipped with certain multiplication and unit morphisms. For the unit, we will use the inclusion morphism that comes for free with the coproduct. The multiplication will take a bit more work.
Given any natural numbers and , the object is canonically isomorphic to , which then includes into using the coproduct morphisms. But this object also includes into , which is isomorphic to . Thus by the universal property of coproducts there is a unique morphism . This is our multiplication.
Proving that these two morphisms satisfy the associativity and identity relations is straightforward, though somewhat tedious. Thus we have a monoid object in . The inclusion defines a universal arrow from to the forgetful functor, and so we have an adjunction.
So what does this look like in ? The free monoid object on a set will consist of the coproduct (disjoint union) of the sets of ordered -tuples of elements of . The unit will be the unique -tuple , and I’ll leave it to you to verify that the multiplication defined above becomes concatenation in this context. And thus we recover the usual notion of a free monoid.
One thing I slightly glossed over is showing that satisfies the hypotheses of our construction. It works here for the same reason it will work in many other contexts: is a closed category. Given any closed category with countable coproducts, the functor has a right adjoint by definition. And thus it preserves all colimits which might exist. In particular, it preserves the countable coproducts, which is what the construction requires. The other functor preserves these coproducts as well because the category is symmetric — tensoring by on the left and tensoring by on the right are naturally isomorphic. Thus we have free monoid objects in any closed category with countable coproducts.
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.