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This is to split off from the other discussion a talk about $C$-torsors for $C$ not a groupoid.
The third of Moerdijk’s conditions (at torsor with structure category) is stated thus:
(freeness) a parallel pair $u_1,u_2: c\to c'$ of morphisms in $C$, may induce coalescence $E(u_1)(\alpha)=E(u_2)(\alpha)$ for some $\alpha\in E(c)_x$ only if there is a morphism $w:b\to c$ and $\zeta\in E(b)_x$ such that $u_1\circ w = u_2\circ w$ and $E(w)(\zeta)=\alpha$.
the phrase ’may induce coalescence’ is a bit confusing. If it is not incorrect, I would change it to
(freeness) for a parallel pair $u_1,u_2: c\to c'$ of morphisms in $C$, $E(u_1)(\alpha)=E(u_2)(\alpha)$ for some $\alpha\in E(c)_x$ implies there is a morphism $w:b\to c$ and $\zeta\in E(b)_x$ such that $u_1\circ w = u_2\circ w$ and $E(w)(\zeta)=\alpha$.
It seems to me that the second and third conditions are a reformulation of the conditions for a flat functor.
NB This is a bit of a note to myself, I don’t have time to think about this now.
For Street’s definition (at torsor with structure category), in the diagram
,
what is the definition of the arrow $A\downarrow a \to E$? I’m guessing it exists because the span $A \leftarrow E \to U$ is a two-sided discrete fibration (of internal categories), but I haven’t checked. This is a bit different to the 2010 paper (definition 4.1), but again I haven’t checked.
For Street’s definition, in the diagram ,
I haven’t looked at their article. Is this by any chance just the model for the lax (comma-)pullback of the point, the way we discuss at generalized universal bundle, at Grothendieck construction and elsewhere?
Urs, this diagram is from Street’s article Combinatorial aspects of descent theory pdf, page 25 and is reproduced at torsor with structure groupoid with all the notation explained, but no further explanations. So you do not need to read the more technical paper Street-Verity to get the definition. I also wondered if it is related to the picture which you mention, but certainly not that directly. There is no universal bundle in the picture. $V\to U$ is a cover there. $A$ is an internal category. In the groupoid case, the insight is due Joyal and the following lines from Street-Verity sketch the gist of the argument ($B$ replaces $A$, $E$ replaces $C$):
André Joyal’s lectures in the Category Seminar at Macquarie University (22 October and 5 November 1980) were the other source of inspiration. André stressed the important case where $B$ is a groupoid in a fairly general category $E$ (a groupoid is a category in which every morphism is invertible; a group is a groupoid with one object). He defined a $B$-torsor to be a discrete fibration $E$ over $B$ which is locally representable and said that the relationship between $B$-torsors and cocycles could be explained in terms of the comprehensive factorization of a functor into a final functor followed by a discrete fibration.
Oh, from that article. Good, then it’s just our notion of generalized universal bundle:
So $U$ is the base space and $V \stackrel{}{\to} U$ the resolution. The anafunctor that gives the cocycle is
$\array{ V &\stackrel{a}{\to}& A \\ \downarrow \\ U } \,.$We then first form the universal $A$-bundle by the pullback
$\array{ \mathbf{E}A &\to& A_0 \\ \downarrow && \downarrow \\ A^I &\to& A }$and pull that back along our anafunctor to get the $A$-bundle that it classifies
$\array{ P &\to& \mathbf{E}A \\ \downarrow && \downarrow \\ V &\stackrel{a}{\to}& A \\ \downarrow \\ U } \,.$Observe that here
$P = \coprod_{x \in A} (const_x \downarrow a)$The remaining precomposition operation
$Mor(A) \times_{t, s} P \to P$is the “principal” $A$-action on the bundle. So Street’s $(Id_A \downarrow a)$ is the weak quotient $P//A$ . (That’s why his “total space” object has that map to $A$.)
If one wants to embed this into a good theory one cannot but demand that $V \to U$ is a weak equivalence. That means, as David R has been amplifying, that as long as the base space is just a space or an orbifold, $V$ will be groupoidal and the cocycle factor through the core of $A$
$a : V \to Core(A) \hookrightarrow A \,.$So then that total space $P$ is indeed precisely the associated bundle to the $Core(A)$-principal bundle.
This is an important special case of higher bundle theory. For instance the central extensions of groupoids which people like to call nonabelian bundle gerbes are precisely such pullbacks for $A = Grpd$ and $a$ factoring through the canonical inclusion $\mathbf{B}AUT(H) \hookrightarrow Grpd$ . (On the other hand the true total space of the underlying principal $AUT(H)$-2-bundle is that nonabelian bundle gerbe semidirectly multiplied with its band).
Now, a truly new step into unexplored territory is obtained as soon as base space $U$ is no longer groupoidal, but taken to be a genuine category. (Such as the fundametal category of a Lorentzian spacetime.) In that case something new will happen. A notion of fiber bundle where fibers may change non-isomorphically as time proceeds . I have no hint that the need for this has been observed secretly in physics anywhere, but it is a possibility.
see my comment on the directed spaces thread! I end up with almost the same question! I think it is linked to extensions of categories.
I think it is linked to extensions of categories.
Yes, exactly!
Principal $\infty$-bundles are, after all, exactly extensions of $\infty$-groupoids, namely homotopy fibers = $(\infty,1)$-fibers of maps in $\infty Grpd$ (or more generally in $(\infty,1)Sh(C)$).
This is evidently the beginning of a pattern. $(\infty,1)$-bundles should be the lax (or rather “comma”) $(\infty,2)$-fibers of morphisms in $(\infty,1)Cat$ (or more generally in $(\infty,2)Sh(C)$).
That’s precisely the point, yes, bundle theory is precisely extension theory. Also known as the theory of fiber sequences.
Now, the prize question is: can anyone think of a known phenomenon in theoretical physics, which would be a candidate for something that fundamentally is described by a kind of bundle on spacetime whose fibers change in time ? And non-invertibly so? That would be a smoking gun indication for a manifestation of spacetime in terms of its fundamental $(\infty,1)$-category. I can’t quite think of any such candidate, though. One problem of course is that fundamental physic has invertible time evolution. So I am not sure.
The following link related to the discussion above is temporary and the file not otherwise available online (djvu with Ocr) at this time
One problem of course is that fundamental physic has invertible time evolution.
Let’s play with entropy a bit :)
One problem of course is that fundamental physic has invertible time evolution.
Let’s play with entropy a bit :)
That’s a good point. Maybe something like this: ordinary quantum mechanics is encoded in a Hilbert space bundle with connection (= the Hamiltonian) on the worldline.
Now suppose we couple this to an environment. Then one of several ways to encode the resulting decoherence is that we take partial traces that map a large Hilbert space to a smaller one
$tr_{envir} : H_{sys} \otimes H_{envir} \to H_{sys} \,.$Maybe such a kind of operation is a potential candicate for a Hilbert space bundle over a directed space.
But on the other hand that tracing is a somewhat discrete operation.
Ah, here is another idea (will post this in the next comment…)
Here is a better idea: maybe we shouldn’t be thinking about bundles whose fibers change non-invertibly.
Rather we should be thinking of bundles with connection whose parallel transport is no longer with values in isomorphisms.
And that of course has immediate physics applications:
The operation $(t \to t') \mapsto (H_t \stackrel{\exp(i (t'-t) K)}{\to} H_{t'})$ is a parallel transport/ on the undirected worldline: unitary time evolution.
Now switch on dissipation and replace the worldline’s fundamental $groupoid$ with its fundamental category: then we may have also dissipative contributions
$(t \to t') \mapsto (H_t \stackrel{\exp((t'-t) (i K - Q))}{\to} H_{t'})$
(for $K$ and $Q$ hermitean operators, hence with real spectrum).
So the fibers $H_t$ are isomorphic for all $t$, but the parallel transport is no longer an isomorphism between them.
Wow, you are brave going into an unknown land…(the idea with the partial trace toward such bundles).
So you still call that case a connection ? (I am not complaining)
So you still call that case a connection
Well, I didn’t up to this point, but it would make sense to call this a connection with parallel transport along directed paths, yes.
I did think about connections and higher connections on directed spaces before. In my article on AQFT I show – slightly paraphrased in the terms we started using above – that every parallel 2-transport on the fundamental $(2,1)$-category of a causal 2d Lorentzian manifold induced a local net of observables.
For that statement it is required that the 2-transport takes values in invertible 2-morphisms. The directedness of the underlying Minkowski space affects not the notion of parallel transport, but is the extra structure that allows to extract a local net of observables from it.
But one could also consider parallel 2-transport on the fundamental $(2,1)$-category of a Minkowski space with values in non-invertible 2-morphisms. That would not give rise to a local net of observables – but that is good, because it would describe a “Euclidean” QFT and these do not come with local nets of observables (not in that sense of local, anyway).
So, yeah. I haven’t thought much about non-invertible parallel transport, but it certainly sounds like a sensible thing to consider in this context.
I have changed the ’freeness’ condition at torsor with structure category as I suggested in my #1 above.
@Urs,
I know $A\downarrow a$ comes with a map to $A$, the question was how it comes with a map to $E$.
Let me just write it out for my own explanation, I’ll put this into the entry when I’m happy with it. Let me reproduce the diagram
So $V\to U$ here is a map in the ambient category $S$ and $A$ is a category in $S$. $disc(U)\leftarrow E\to A$ is anafunctor-like, but I’m not sure how close it is to the ’usual’ notion. The ’locally trivial condition’ asks that for the cover $V\to U$ (for Street a regular epimorphism, hence an element of a subcanonical singleton pretopology), there is a map $disc(V)\to A$, i.e. a map $V\to A_0$ such that $E\times_{disc(U)} disc(V) \simeq A^I \times_A \disc(V)$ ($A^I$ = category of isomorphisms) - this much is clear to me, because $A \leftarrow A^I \to A$ is the (weak!) identity 1-arrow in the 2-category of saturated anafunctors, and so represents the trivial $A$-torsor. However, it is not immediate that $A^I \times_A \disc(V)$ comes with an arrow to $E$ so as to make it the pullback. Zoran tells me my guess is true that the lifting properties inherent in $E$ being discretely fibred over $U$ and $A$ gives this map (and it had better be unique), so I think I’ll have to sit down and convince myself how this works.
Then there is the problem of showing how the old definition and the new one (from the 2010 paper) line up.
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