All right. Looking forward to it!

]]>let be a 1-form on a manifold . then the action is a functor , and we can naturally extend it to . then path-integral quantization of should be (something related to) the extension of along . since is just the same thing as , if the extension of to has be done "correctly", the functor is a 1-dimensional TQFT (there's surely something to be fixed here, vector spaces arising seems not to be finite dimensional..) and

(a probable way of fixing things is the following: one adds as an external field on a Riemannian metric and considers euclidean bordisms. then one has a function of the real parameter given by integration on the space of loops of length )

ok, enough of this. next post will say "created page..." :-) ]]>

later this evening I'll add here

You should rather type this into a page on your web!

]]>don't worry about being busy: I'll be too, till the end of february. after that I'll enjoy investigating in detail the stuff which is emerging here and in the thread on -connections.

later this evening I'll add here something on path-integral quantization of the action given by integration of a 1-form along a 1-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.

]]>on the differential forms side, we may define th pushforward of along a smooth map between compact manifolds by the adjunction formula

for any differential form on .

in order to write this formula in the abstract functorial language of "integration without integration" we just need to identify its ingredients (exept the push-forward, which is what we aim to define). we have

i) differential forms of degree k on a manifold <----> k-functors from to

ii) integration <---> "integration without integration"

iii) pull-back <---> functoriality of

iv) wedge product of forms <---> this is the composition and uses in an essential way the abelianity of and the decompostion of into copies of 's (here is the topological realization of the k-simplex)

howevere there's also another point of view on push-forward of differential forms, which I prefer and which I'll now try to sketch below. ]]>

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.

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.

]]>how does this (very nice, in my opinion) abstract nonsense fit with ordinary integration of differential forms on a manifold ? where's the groupoid? since you're reading this forum, you already know the answer: the involved groupoid is the smooth path groupoid of , and one is taken back to the original naive notion of -differential form: an function measuring (infinitesimal) -volumes. having said "infinitesimal" we have moved from the world of groupids to the one of algebroids, and the algebroid here is the prototipical one: . from this point of view, differential forms on should rather be looked at as -connections with values in , a point of view stressed somewhere by urs in his Lab area (if I didn't misunderstood this). now we have all the ingredients: the groupoid and the functor and we can wonder about Kan extension along the groupoid morphism induced by . how is this related to integration of top-degree forms on ?

a closely realated instance is what happens when we have a group acting on . there the groupoid structure of becomes more complicated, since also the -action comes in. also it seems natural that in this context the push-forward along is replaced by push-forwarding along . so it is natural to wonder: is the natural (Kan extension type, I mean) integration on -manifolds equivariant integration? ]]>