So one of the things I’ve been doing recently is finishing up a version, and talking about, this paper which I’ve now put on the arXiv. While at it, I figured I should update a previous paper – the current version cuts out part of the original subject (cobordism categories) and expands on the category-theory side of things, giving more detailed proofs, etc. That part will then be out of the way when the topology side shows up in another paper, yet to appear, which will also use the stuff about 2-vector spaces and groupoids from the “new” paper.
Ironically, although I fixed the “issue” which arose when I was posting on the subject – and I’ll come back to that – I’ve already talked about most of what’s in the “new” paper, whereas I never got around to talking about what’s in the “old” one, updated version or not. That’s the one called “Double Bicategories and Double Cospans”, which is the most strictly category-theoretic thing I’ve produced: all the motivation from physics has been abstracted away. So when I have some time, I’ll write something about that one.
For now, I just wanted to link to this new stuff.
Theoretical Atlas 
October 15, 2008 at 7:40 pm
Am reading your article.
– last line of p. 1: the expression for the (i,j) component probably is the cardinality of the preimage set: the right vertical bar is missing
– second line after equation (2) on p. 2: C(X) should be C(Y)
– question concerning (5) on p. 4: I was wondering about that recently in a slightly different context: can one say anything useful in general about the relation between the notion of morphisms of spans that you give here and the other one where we first form the joint pullback over the two left legs and then have the resulting kite-shaped diagram filled by a 2-morphism?
– page 12, first line of section 3: I guess “functions” here should be “functors” (even though the domain category is discrete)
– page 17, equation (39): ah, this is interesting and useful
October 15, 2008 at 8:01 pm
I feel like advertizing in this context some thinking I did about how to make the relation between the general idea of groupoidification and of quantum mechanics and the path integral more concrete. I was getting pretty fond of the picture drawn in then entry An exercise in groupoidification: the path integra, elementary as it is.
There I am talking (for simplicity) in terms of bundles of sets over discrete groupoids and am pull-pushing these through spans. But, as indicated also somewhere further down the comment section where I am going through the details with Eric, you can of course feel free to reverse the perspective here equivalently to set-valued functors on the groupoid and pull-push those, which brings this discussion more manifestly into the context that you are considering in your article.
I particularly I liked in the discussion I gave there the way $V$-colored sets appear naturally from starting with a background field given by an associated connection expressed in terms of action groupoids, something that to me nicely solved the puzzle about how to think of the “ontological origin” of these in your previous categoriefied algebra and quantum mechanics since there are indications from Lie-infinity algebraic reaoning that this expression of associated connections in terms of action groupoids is the more “natural” one.
Be that as it may, my aim has been to understand groupoidily-done linear algebra as a way to understand at least for finite/discrete theory the quantization step as an operation that sends classical parallel transport (the action functional) to quantum parallel transport (the quantum propagator). At the mentioned entry this is spelled out for d=0 QFT = quantum mechanics, but the generalization to higher dimensions is obvious from that, in principle.
October 18, 2008 at 11:55 pm
Nice paper! I just blogged about it.
A few more typos:
page 1 – there’s just one vertical bar in the formula for the cardinality of the set that’s a matrix entry of T. It’s missing its mate.
page 3 – the “hat” symbol right under equation (4) doesn’t cover enough stuff.
Also: in the abstract, it’s a bit unorthodox to use Span(Gpd) for the bicategory whose 2-morphisms are (equivalence classes of) spans of spans of groupoids; most people would guess the 2-morphisms were just (equivalence classes of) maps of spans. So, maybe you should insert a word about ‘spans of spans’.
October 19, 2008 at 2:52 pm
Maybe my last comment comes across as being obnoxious. Sorry if so. What I want to express is that there is a close relation between the two big projects John initiated: higher classical parallel transport and higher quantum propagation (extended QFT). And that I feel we could profit from connecting our work on the two ends of the spectrum more. Or at least try to understand the relation better.