The String Coffee Table

If you think like a physicist, a good heuristic picture to keep in mind for all of the following is the elementary open string interaction as realized, roughly, in open string field theory.

Think of an open string which stretches between a brane $A$ and a brane $B$ as composed of two halfs, one labelled by $A$, the other by $\bar B$

This may interact with a string streching from $B$ to $C$

The merging of the two strings corresponds to an elementary process which cancels the right half $B$ with the left half $\bar B$

Conversely, a single string

may split in two by the insertion of $\overset{\bar B}{\to}\overset{B}{\to}$

(This is supposed to make you think of adjunctions and/or of tensor products of bimodules. The corresponding precise statements will be discussed below. None of the symbols above is supposed to have any precise meaning without further qualification.)

I’ll begin by summarizing some key insights obtained by Aaron Lauda and/or collected from various sources. You should have a look at the most recent version of his paper on ambidextrous adjunctions, which right now – as Aaron has kindly pointed out to me – is not the arxive version but the one on his website. I’ll don’t make any effort to attribute any of the statements, some of which are classical results, some of which slight generalizations of classical results. For any further details please consult Aaron’s text and the references he gives.

In terms of the above heuristic pictures, the question could with some justification be phrased like this: Given some ‘algebra of string’, i.e. some way to merge two open string states by means of a ‘pair-of-pants’-diagram to obtain a new open string state, is there always a way to realize this operation by assigning labels to the left and right half of the string and implementing interactions by cancelling or introducing conjugate pairs of labels?

(Note again that this is just a heuristic motivation. It makes essentially the right identification of formalism with formal physics, but the details will have to be discussed elsewhere. The rough idea, however, is useful.)

More precisely, the issue is the following.

Consider a monad in some (weak) 2-category $C$. This here plays the role of a sophisticated version of an internal algebra. If $C$ has just a single object, then a monad in $C$ is nothing but an algebra object in $C$, when $C$ is regarded as a monoidal category. (I am not going to distinguish between monoidal categories and (weak) 2-categories with a single object.) More generally, a monad is a 2-functor from the trivial 2-category with a single object to $C$, i.e. an endo-1-morphism $B \overset{T}{\to} B$ of an object $B$ of $C$ together with a coherent multiplication 2-morphisms

It is clear what a (right, say) module of such a monad should be. It is just any old morphism on which $T$ ‘acts’, i.e. a morphism

together with an right-action $\eta$

For some strange (?) reason $s$ is called a ‘$T$-algebra’ in the context of monad theory, but we should think of it as a $T$-module.

The point now is that there is a certain completeness relation on $C$ which can be expressed in terms of such monad modules.

There is a functor

which assigns to every object $A \in C$ the category $T-\mathrm{Alg}$ of right $T$-modules starting at $A$ in the above sense. One says that the monad $T$ has an Eilenberg-Moore object $B^T$ if this functor is represented by $B^T$.

This means that the functor is equivalent to the $\mathrm{Hom}$-functor to $B^T$:

The point is, a $T$-module is a morphism on which $B \overset{T}{\to} B$ acts essentially by mere composition, mediated by the action 2-morphism $\eta$ above. If the Eilenberg-Moore object $B^T$ exists, then we can really essentially identify every $T$-module with a morphism in $C$. This is a phenomenon which will reappear in a different guise below in the context of internal $\mathrm{Hom}$s.

What we are really interested in here are not any old monads $T$, but those that are really ‘Frobenius monads’. I.e. those which not only have a product

but also a coproduct

satisfying some obvious properties. These are the monads that we would like to ‘split into left and right halfs’ according to the above heuristic imagary.

Now, an object $B^T$ with the desired property that it represents the $T-\mathrm{Alg}$ functor need not exist. Only if it does will be able to ‘split $B \overset{T}{\to} B$ in two halfs’!

Luckily, if the Eilenberg-Moore object $B^T$ of the monad $T$ does not exist yet, we can just enlarge $C$ by including a suitable new object $B^T$. More generally, we can form what is called the free Eilenberg-Moore completion $\mathrm{EM}(C)$ of $C$ which enlarges $C$ such that it contains all Eilenberg-Moore objects.

This Eilenberg-Moore completion $\mathrm{EM}(C)$ of a 2-category $C$ is a general abstract nonsense construction which can be found discussed here:

S. Lack & R. Street
The formal theory of monads II
available here.

The main point is, it can always be done. Then the theorem is the following:

1) $C$ embeds fully and faithfully into $EM(C)$ (meaning that it makes sense to regard $EM(C)$ as an enlargement of $C$)

2) If $C$ is any monoidal category, then any Frobenius algebra object in $C$ arises from a left and right adjunction in $EM(C)$ (meaning that we can ‘split the string in two halfs’).

I haven’t absorbed all details of the general construction yet. Luckily, Aaron Lauda in his paper spells it out for the case where $C = \mathrm{Vect}$, and I think essentially the same construction should hold for about any other monoidal category (?).

With all the abstract machinery involved, it is nice to see that $\mathrm{EM}(\mathrm{Vect})$ is an old friend. It is almost nothing but $\mathrm{Bimod}(\mathrm{Vect})$, the (weak) 2-category of all bimodules of algebras (which are just ordinary algebras for $C = \mathrm{Vect}$). More precisely, $\mathrm{EM}(\mathrm{Vect})$ is a little smaller than that, namely

This means that objects in $\mathrm{EM}(\mathrm{Vect})$ are algebras in $C = \mathrm{Vect}$, morphisms $A \overset{M}{\to} B$ are $A-B$-bimodules and 2-morphisms are bimodule homomorphisms (‘intertwiners’).

A ‘left-free’ $A-B$-bimodule is one which looks like

Here the right hand side denotes the object (vector space) $A \otimes V$ for some $V$. On this object the algebra $A$ acts from the left by the ordinary action of $A$ on itself, while $B$ acts from the right by first swapping $B$ and $V$ using the morphism $\phi$ and then acting with $A$ on itself from the right.

And it’s easy to see how any Frobenius algebra $A$ is now represented as an adjunction. There is an obvious way in which $A$ itself can be regarded as a left-free $k-A$-bimodule (here $k$ denotes the ground field regarded as a 1D vector space over itself)

and as a left-free $A-k$-bimodule

There is an obvious bimodule homomorphism

from the right and left ‘half’ to $A$ itself, regarded as the trivial bimodule over itself. Since $A \overset{A}{\to} A$ is the identity morphism on $A$ in the 2-category of bimodule, this realizes the ‘evaluation map’ $e$ of an adjunction. And one checks that

is really nothing but the product on $A$. Hence we have succeeded in realizing this product by ‘splitting the string in two halfs’ and contracting matching halfs away.

There is a subtle and somewhate self-referential interplay here of algebras, their left- right- and bimodules, of product operations of algebras and tensor products of bimodules.

More has to be said about this. In particular, I would like to compare all this to a very similar but subtly different story as told in

V. Ostrik
Module Categories, weak Hopf Algebras and Modular Invariants
math.QA/0111139.

But this has to wait until tomorrow.