The String Coffee Table

[Update:

What I am talking about here is mostly the result presented first in

Aaron Lauda
Frobenius algebras and ambidextrous adjunctions
math.CT/0502550

which is a strengthening of a result found in

M. Müger
From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories.
J. Pure Appl. Algebra, 180 (2003)
math/0111204

amplified by John Baez in TWF 174.

]

As a motivation, consider the following fact.

Every (direct sum of) matrix algebra(s) can be regarded as a Frobenius algebra. (Using Wedderburn’s theorem.) The product is ‘index contraction’, the coproduct is ‘insertion of a unit’.

Phrase this in more highbrow terms. Let $V$ be some vector space and $\mathrm{End}(V)\simeq V\otimes V^{*}$ its space of endomorphisms. Let’s regard the monoidal category $\mathrm{Vect}$ of vector spaces as a 2-category with a single object, whose 1-morphisms are vector spaces, 2-morphisms are linear maps and horizontal composition is the tensor product. For instance $V\otimes V^{*}$ looks like

‘Index contraction’ (the evaluation map of covectors on vectors) is a linear map, hence a 2-morphism, of the kind

where $K$ is the ground field which we are working on, regarded as a 1-D vector space. ‘Insertion of a unit’ is the map

which sends $1\in K$ to ${\mathrm{Id}}_V\in \mathrm{End}(V)$. This is the archetypical example of what is called an adjunction in a 2-category. In fact, since $\mathrm{Vect}$ is symmetric, the order of $V$ and $V^{*}$ in the above is not really essential and hence we really have something Aaron Lauda calls an ambidextrous adjunction. The way I have approached adjunctions here it is clear that they are related to algebras. It is a simple exercise to show that the product and coproduct coming from $e$ and $i$ above are such that we really have a Frobenius algebra.

Aaron Lauda turns this into a stronger and more precise theorem (his theorem 8) which essentially says that every Frobenius algebra comes from an adjunction along the lines sketched above.

Building on that, the second part of the paper is concerned with categorifying this situation. If you know about the main idea of categorification, know how Frobenius algebras are related to topological strings and know what a topological membrane is supposed to be it should not come as a surprise that there is a way to relate categorified Frobenius algebras with topological membranes. I don’t want to say any more about this part right now.

What I do want to point out is, how the above mentioned result on adjunctions and Frobenius algebras is based on the same general mechanism which relates 2-trivializations of 2-functors with Frobenius algebra.

Namely given a 2-functor (or just a 2-bundle without connection, if you prefer) which I shall call

from some 2-category of surface elements in $M$ to some target 2-category, we can restrict it to open subset $U_i\subset M$ and demand that there it looks like a ‘more trivial’ 2-functor ${tra}_i$ in some sense.

Hence we demand there to be morphisms of 2-functors

and

which relate the two.

If these were 1-functors, we would define the transition between ${\mathrm{tra}}_i$ and ${\mathrm{tra}}_j$ as

But since 2-functor live in a 2-category, we should really replace the equality here with 2-morphisms

and

This way one gets something rather similar to an adjunction. (It’s actually a little more general.) By precisely the same underlying mechanism which Aaron Lauda uses to show the relation between adjunctions and Frobenius algebras, one finds that a 2-trivialization of the above sort gives rise to something like a Frobenius algebra structure.

There would be more to say, and I did before it was erased. Right now this shall be it.