Re: Hecke eigensheaves and KV 2-vectors

["John Baez" <baez@math.ucr.edu>]

Hi -

> Now, the 2-category (bicategory) Vect-Mod contains that of
> Kapranov-Voevodsky 2-vector spaces, but is larger than that (isn't it?).

Yes.  Yetter's paper "Categorical Linear Algebra: a Setting for
Questions from Physics and Low-Dimensional Topology" characterizes
the Kapranov-Voevodsky 2-vector spaces among the Vect-modules.

(Here Vect stands for the category of finite-dimensional vector spaces
as a monoidal category with its usual tensor product.)

In Proposition 25, he says that the 2-category of 2-vector spaces
is isomorphic to the 2-category of "finitely semisimple" Vect-modules.
These are Vect-modules with a finite set of simple objects such that
every object is a finite direct sum of these.   We call this a "basis" of
simple objects.

(I would feel more comfortable if he had said these 2-categories were
2-equivalent, and this is all I really care about.)

There are a bunch of obvious Vect-modules that aren't 2-vector
spaces in Kapranov and Voevodsky's sense.

One is the category VECT of not-necessarily-finite-dimensional vector
spaces: not every object is a *finite* direct sum of copies of C.

Another is the category Vect^{infinity} for your favorite infinite
cardinal "infinity": the same problem, but for a different reason.

These counterexamples are a bit dull, in that one imagines there
could be some definition of an "infinite-dimensional" 2-vector
space that they might fit.  But, it seems that that there are two different
kinds of infinite-dimensionality at work here: VECT is "tall", while
Vect^{infinity} is "wide".  There's also VECT^{infinity}, which is
tall and wide.  So, somebody should straighten this stuff out.

Other counterexamples are neither "tall" nor "wide".  For example,
the category of finite-dimensional representations of a non-semisimple
algebra like the algebra of upper triangular matrices.  Or, the
category of representaions of a quiver.  (I think representations of
the A_n quiver are the same as representations of the algebra of
upper triangular matrices.)

I'd be interested to know if any of these sorts of counterexamples is
useful in Witten's work, or he can live with 2-vector spaces.

> Therefore I was wondering what people knowledgeable in (higher) category
> theory would think of this. Does my observation make sense?

It makes sense to me!  Other mathematicians often respond better to
more precisely phrased questions.  :-)

> If yes, has it been observed before?

Not that I know of, but I could be missing examples from other
contexts.

Certainly category theorists think about "weak fixed points" of
endofunctors T: C -> C, that is, objects c in C equipped with
isomorphisms Tc -> c.  The category of all these weak fixed points
is something people enjoy studying: it's an example of a "weak
equalizer" and on a good day I know lots of examples.

You're asking about something more general: "weak eigenobjects" of
endofunctors T: C -> C on *monoidal* categories C.  In other words: objects
c in C equipped with isomorphisms Tc -> a tensor c for some fixed object
a.

I bet some people have studied these in some context or other.
You may have scared these people away with all your Langlands,
branes and defect lines.  Category theorists are dual to ordinary
people: they often get more confused when you surround an abstract
concept with a lot of distacting specifics.  "Could you please not
give me an example, to help me understand what you're saying?"  :-)

Best,
jb