Re: This Week's Finds in Mathematical Physics (Week 210)

From: Urs Schreiber (Urs.Schreiber_at_uni-essen.de)
Date: 01/27/05 

Date: Thu, 27 Jan 2005 21:30:04 +0000 (UTC)

"John Baez" <baez@math.removethis.ucr.andthis.edu> schrieb im Newsbeitrag
news:ctas57$1p2$1@news.ks.uiuc.edu...

> 2-bundles are meant to be an alternative to gerbes: although I've done my
> best to hide it above, a gerbe is really more like a categorified *sheaf*
> than a bundle. And, just as a bundle has a sheaf of sections, we're
> hoping
> that a 2-bundle has a stack of sections, which in certain cases will be a
> gerbe. That's one of the things we need to figure out, though.

Maybe somebody can tell me/us if the following has already been considered
in some context in the literature:

First, by definition, a gerbe is a stack is a fibered category. The concept
"fibered category" is a categorification of the concept "presheaf". But not
its full categorification. I am wondering if the full categorification of
the concept "presheaf" has been studied before, and under which name.

More precisely, what I am talking about is this:

A presheaf over a topological space X is a morphism in Cat, namely
  a (contravariant) functor
  from the category O(X) of open subsets of X
  to Set.

The full categorification of this should be called a 2-presheaf over a
2-space S. 2-spaces are "smooth categories" and were defined (at least) by
Toby Bartels, these are categories internalized in Diff. So they consist of
a smooth space of points together with a smooth space of morphisms between
these points, which composition etc. all being smooth.

By the technique of categorification by internalization
a 2-presheaf over S should hence be
  a (contravariant) 2-functor
  from the 2-category of open 2-subspaces O(S)
  to Cat.

(Here O(S) has objects being open sub-2-spaces, morphisms being "injective"
functors between these in an obvious sense and 2-morphisms being natural
transformations between the latter.)

I believe that a 2-presheaf over S is the same as a fibered category over X
precisely if the space S is categorically discrete (has only identity
morphisms) and we identify its point space with X.

Hence the concept "fibered category" is only "half the categorification" of
the concept "presheaf", in that it only categorifies the target, not the
source of the presheaf. A fibered category does know about space, but not
about 2-space. It captures the configuration space of a particle, but not
that of a string.

My question is if what I called 2-presheafs here has been studied before.
And if yes, under which name.

Because, as John Baez mentioned, while it remains to be checked if a
2-bundle with trivial base 2-space has a gerbe of 2-sections, it is clear
that every 2-bundle (with in general nontrivial base 2-space) has a
2-presheaf of 2-sections.