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Some old thoughts on this are at Integration without integration (blog, pdf).
I think you need to first transgress, and then push forward.
For instance: take be the parallel transport of a bundle with connection. We may transgress this to loop space by homming the categorical sphere into it
.
The value of this functor over a loop is the parallel transport around that loop. If then the original functor encodes a 1-form on X, and the value of the transgressed functor on a loop is the integral of the 1-form around the loop and the functor is locally constant.
One thing I would like to understand rather sooner than later is how the well-developed theory of push-forward in ordinary differential cohomology -- which is indeed well known to be related to integration -- is expressed in terms of our functorial Yoga. It is a shame that I haven't understood this long ago. But I haven't. I suspect that there is a very simple answer, though, once one has it.
well, yes, so has
as objects smooth loops in X with a chosen point on them
as morphisms path in X between the chosen points of two loops, such that one is the result of conjufating the other by that path.
Hey Domenico,
nice, good point. Yes. I hadn't really thought about it this way. But I agree, yes, this business may require that we extend along first.
Very good point. And that then also makes the connection to quantization become more manifest, because as said elsewhere before, that should somehow be extension along .
We should follow up on this, this looks promising.
Right now I am still a bit busy with finalizing some aspects on path oo-groupoids. Tomorrow I'll be busy with a seminar on oo-category theory. But then maybe after this I can try to invest some more time into the stuff you are discussing here. Very nice.
later this evening I'll add here
You should rather type this into a page on your web!
All right. Looking forward to it!
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