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Just in case anyone is wondering:
I am back from a small vacation. Will probably not have much time to catch up with accumulated discussion here, but maybe a bit.
What is this thing called "vacation" of which you speak? :)
Welcome back :)
relaxed and eager to add a new bit to structures in an (oo,1)-topos.
Okay, good! :-)
today's challenge is.. cobordims! I feel we have discussed path groupoids enough to be ready for this further step. we've been discussing something related here but remaining somehow halfway.
There are some loose ends that me personally I need to tie up, and be it only to write them up. Right this moment I am polishing differential cohomology in an (oo,1)-topos.
Concerning cobordisms:
I am interested in the following slight variant of the usual lore:
we know that dual to cobordism representations is factorization algebras on cobordisms. I happen to know from my work on AQFT that functors out of the path oo-groupoid of paths in a given cobordism give rise, if the cobordism is equipped with a Lorentzian structure, to a "Lorentzian factorization algebra" (a local net of algebras).
The way this works suggests that one wants to do the following: consider a Lorrentzian manifold as a smooth poset, namely an -category valued presheaf on CartSp whose plots in degree 0 are smooth spacelike maps , and whose plots in degree 1 are U-families of future-directed paths.
Then the definition of path oo-groupoid of an oo-stack/-sheaf generalizes to this -sheaf situation, and we find a smooth oo-category whose k-morphisms are something like these causal subsets used everywhere in AQFT.
I would like eventually to think about how to produce representations of n-categories of Lorentzian cobordisms from representations of these Lorentzian path oo-groupoids by taking some colimit or something.
The finite version of this, of course, being -diamonds :)
I'll be curious to see if you just need "future" directed paths or if "piecewise lightlike" directed paths somehow work out nicer. My gut tells me the latter...
I'll be curious to see if you just need "future" directed paths or if "piecewise lightlike" directed paths somehow work out nicer. My gut tells me the latter...
Right, I am still experimenting. But I think what I really meant to refer to above is really just morphisms in the poset that underlies the (globally hyperbolic, say) Lorentzain spacetime, i.e. just pairs of points with one in the future of the other.
Somehow the causal diamond should appear from this by some universal construction. Because every open subset in a (globally hyperbolic) Lorentzian spacetime has this universal causal hull, which is the intersection of all the complements of the lightcones of all its points. Somehow this must have a good and simple poset-categorical description...
Somehow the causal diamond should appear from this by some universal construction.
Sorry for being so vague, i was distracted by something else. Of course given any set of points in the Lorentzian spacetime, regarded as a poset whose morphism are precisely the pairs of points where one is in the future of the other, the causal subsets are precisely the limit and colimit cones (joins and meets) over that set.
Domenico,
yes, I know what you mean. There is a bit of discussion of what one might expect here, precisely along the lines you just indicated, in the query box here.
I am still interested in that approach, too.It just occurred to me that I might be able to say more interesting things about that variant that I mentioned. What is interesting about that variant is that it differs from most (all) of the other discussion of higher categorical QFT in that it does make sense for Lorentzian signature. And in fact crucially depends on it. This is an interesting aspect all by itself that eventually deserves to be expounded on more.
something like is the free tensor category generated by .
That's an interesting thought. Maybe it's not the free tensor category, but the free tensor category with duals for objects. Because it is these duals that give the basic morphisms missing in . For instance the path category cannot regard a path as a morphism from the empty object to the union of its two endpoints (with orientation). It's precisely the step of giving every point a dual object that makes this work.
Notice that something like this is correct for the case that . Because in that case is itself the point, and by the cobordism hypothesis-theorem we know that is something like the free -category with some such properties.
Hm, so that's an interesting thoought, then. There must be still a bit more fine-print, since in the case with we know that not every parallel transport -- which is just a choice of a single object -- extends to a cobordism representation, but it needs to be a "fully dualizable" object, e.g. a finite dimensional veector space. So it can't be quite true that is simply the image of a left adjoint to the functor that forgets just monoidal structure. But something close might be true. Hm...
not the free tensor category, but the free tensor category with duals for objects
I have this half-dormant project to think about oo-stacks with values in dendroidal sets, i.e. dendroidal set-valued presheaves satisfying some descent condition.
Together with Thomas Nikolaus we have pretty much convinced ourselves that a dendroidal set that fills all inner horns and all outer corolla horns is precisely one coming from an symmetric monoidal (oo,1)-category. This should be useful for speaking about including both its smooth structure as well as its symmetric monoidal structure: the smooth structure is encoded in the presheaf of dendroidal sets, the monoidal structure in the fact that these dendroidal sets are objectwise symmetric monoidal (oo,1)-categories.
So it can't be quite true that is simply the image of a left adjoint to the functor that forgets just monoidal structure. But something close might be true.
@Domenico, no 11
Welcome back (and you too, Urs).
the extended functor , instead, is a tensor functor. so, by saying that the datum of the latter representation is the same thing as the datum of the representation , we are giving a universal property, and we are saying something like is the free tensor category generated by .
I was discussing precisely this yesterday with a fellow student at uni. But I'm not sure that " is the free tensor category generated by ", as the arrows in are thin homotopy classes and include back-trackings and the like. Actually, I'm not sure I know the definition of , so I can't make too much comment, but I would imagine it is something like paths mod reparameterisations. Help, someone!
Yes, there is some fine-tuning necessary here to make this anywhere close to being precise. But notice that the -category itself contains difeomorphisms as its higher morphisms, so should similarly have cells that divide out reparameterizations.
What exactly we mean by in this context is also a question. I would think a natural guess is to take the simplicial presheaf or similar (details at path oo-groupoid) and then pass to its fibrant replacement not in the model category for oo-groupoid valued stacks on CartSp, but in that for (oo,1)-category valued stacks (the left Bousfield localization of the model structures of presheaves on with values in the model structure for Cartesian fibrations over the point).
That will produce the smooth path (oo,1)-category of X. I imagine that this would be the gadget that is most directly related to whatever precise definition for one uses.
Well I was thinking of having as a 1-category for a start ;-) (and would be the usual thing, paths mod thin homotopy).
Here's a trial definition of (the 1-cat) - I don't claim this is new, I'm just writing down what comes to mind: Objects are pairs where is an object of (i.e. a finite collection of oriented points) and is a map . Arrows are equivalence classes of pairs where is a representative from of an arrow from (a 1-dim oriented manifold with boundary) an is a map from to . Two of these , are equivalent if there is an orientation preserving diffeomorphism commuting with and . Composition is the usual composition of cobordisms. The monoidal product is the usual one.
Sure, something like this. Possibly with some extra discussion of what you do about collars and how exactly to glue/compose.
two thin homotopic maps \Delta^1\to X need not be related by a reparametrization
That's why I suggested above to use the path category instead of the path groupoid (only that I said -category...). This is obtained by dividing out just reparameterization-preserving diffeos of the interval.
That matches then pretty literally the quotient that David recalled above is used to define the morphisms in .
In fact, for all matters of parallel transport with values in groups (groupoids), functors out of the path category are just as well as functors out of the path groupoid: a smooth functor from the path category to will factor through the path groupoid. (This is not meant to be tautologically obvious.)
So, to sketch the scene, I am imagining that the general statement might be that the -category (or or whatever) is something like the free sheaf of symmetric monoidal -categories with all duals on something lilke the sheaf , where denotes fibrant replacement in -category valued stacks and is -truncation.
Hm, well, this needs further qualification to make good sense, but something like this seems to be what we are conjuring here.
One more comment:
if we take the path n-category whose (equivalence classes of) n-morphisms are homotopy classes of n-dimensional paths, then this n-path -category should have a very intrinsic characterization:
namely: we ought to want to be saying that -sheaves on something like CartSp form a locally contractible -topos , for which the constant -stack functor has a left adjoint .
This would be our bare path -category. It would generally be very interesting to understand what its free completion to monoidal thing with duals would be.
oh, fine! I had missed the distinction :-( so the 1-path category seems to be what I called above, and then what we are saying perfectly agrees :-)That's why I suggested above to use the path category instead of the path groupoid
In fact, for all matters of parallel transport with values in groups (groupoids), functors out of the path category [...] obvious.)
and then what we are saying perfectly agrees :-)
Right!
but can you address me to a reference?
An argument is sketched in appendix B of
Freed, Classical Chern-Simons theory, Part I
that smooth parallel transport with values in over paths has to come from integrating a 1-form, even without using quotients by thin homotopy.
It seems that also Dumitrescu's Connections and parallel transport does not use thin-invariance.
I had never much looked into this variant of the question, since I had always concentrated on path groupoids, but when I look at the proof in our Parallel transport and functors again, I guess it is true that where the 1-form is derived from the parallel transport, in fact the thin-invariance is not used. This just comes in because by decree this works with groupoids, not with categories.
(Incidentally, this is an example of a discussion with a - now - completely inappropriate title!)
Just a quick remark: Taking the path category (called by Domenico above) and a functor for some groupoid , then this factors through the category of fractions . I'll have a think about this today and see what it looks like. I agree with Andrew, and let us continue this discussion in this thread.
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