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created stub for n-connected object in an (infinity,1)-category
--( made connected redirect to it, so that one can easily link to this entry by saying xyz is 4-connected)
Also added to homotopy group a section on connected and truncated objects.
I'm remarking this since fibraton sequences in Whitehead towers (at least for topologicla spaces) precisely involve this kind of spaces:
Yes, thanks, good point.
I have typed now what I think is the fully general statement of this at Whitehead tower in an (oo,1)-topos -- Properties
I added a line in the Examples to say that when X is a pointed topological space, is an Eilenberg-MacLane space.
thanks! I added a further proposition here that asserts that and how this works generally
I will now add more details elsewhere.
By the way, I find the classical article
a good source of classical statements for the kind of statemements that we are after here.
Domenico,
here a thought on curvatures as obstructions to lifts through Whitehead/Postnikov towers:
classical obstruction theory in Top tells us that to compute iteratively the obstruction to extending a morphism through a morphism we are to form the Postnikov tower of and then lift through that step-by-step, each time picking up the corresponding k-invariant characteristic class as the obstruction.
But this story by itself fails to work sensibly for us in our oo-stack topos : the k-invariants are not useful here by themselves. For instance if is the coefficient object for complex bundles, the the corresponding (oo,1)-categorical Postnikov tower has essentially just a single term, namely itself. That's because is an ordinary group, so from the point of view of the categorical homotopy groups of , is a 1-type. Of course that just means that it is a Lie 1-groupoid, so it makes sense, but it makes the Postnikov tower non-useful for the obstruction theory.
The thing is that we rather want a Postnikov filtration with respect to the geometric homotopy groups of . These are by definition those of (non-boldface here!). As discussed elsewhere, is in fact the geometric realization of the groupoid , so is the correct topological classifying space for the unitary group.
We want the Postnikov filtration of and want to see its real cohomology groups appear as obstructions, which are precisely the Chern classes.
So I am thinking: we may have to redo the Postnikov theory of HTT for geometric homotopy groups.
Meaning: we let be the full subcategory of on those objects whose geometric homotopy groups vanish above degree . Then we need to check if the inclusion
has a left adjoint, the geometric truncation
If we could get this, then the rest would go through verbatim as before, but would now produce a Postnikov tower obstruction theory that encodes the desired information. We would decompose into a geometric Postnikov tower and then lift from step-by-step through that. In the nth step we should pick up the nth Chern character there (up to issues of how to count :-).
I guess that is in principle the right plan. I am just not sure if I am up to the task of finding the required localization left adjoints.
I guess that is in principle the right plan. I am just not sure if I am up to the task of finding the required localization left adjoints.
On a second thought, it should be clear how it works: we know the localization is specified by the collection of morphisms that the left adjoint sends to equivalences. These morphisms in should be those morphisms in that induce isomorphisms on geometric homotopy groups below degree . Probably.
Probably.
Right, so one would have to mimic the proof of HTT, Prop. 5.5.6.18 for another class of morphisms than used there. Hm...
We want the Postnikov filtration of and...
I think actually the "delooped" sequences are the fundamental ones. The other ones are "looped" :-)
In applications it's the "delooped" sequences that matter for the obstruction theory, as you say, and whose homotopy fibers we are actually interested in
By the way, discussion along these lines that you indicate is for instance at string structure and related entries.
Yes, right, both are interesting I think what I mean is that we start with a coefficient object and are interested in its decompositions. In applications that is and not
I think in Top, whatever statement about Whitehead towers of an object one has, they will induce corresponding statements about the Whitehead towers of the looping of . That's because the construction of Whitehead towers as well as that of the looping involves only (homotopy) limits which hence commute with each other, and the coskeleton functor which is right adjoint and hence also commutes with them.
In a general oo-topos I need to think about how to adapt this statement, as there everything is the same except that at one point we apply , which is left adjoint.
Hm...
a notion of deloopable morphism seems to be lacking.
Typically we would pass to the over-oo-category over a given object and call the looping/delooping on objects there the looping/delooping of the corresponding morphisms.
Do you have concretely an example in mind where you want to be delooping a morphism?
.
so that one has a topological 1-groupoid, but also 1-hom spaces were contractible, so that one could in some sense think of this 1-groupoid as a 0-groupoid (but I cannot say I am confident in this argument)
Suppose the pointed space X is already connected and simply connected and compute the homotopy fiber of . This is the ordinary pullback of the "tangent 2-groupoid" of : objects are paths in X from the basepoint. Morphisms are homotopy classes of fillings of triangles of paths, and 2-morphisms are "paper cup" diagrams of such fillings.
Pulling this back to X gives you the bundle of topological groupoids which over each point x in X has: objects the path from the basepoint to X, morphsism the homotopy classes of surfaces between these paths. This is equivalent to .
... which is just the special case of the formula that we derived at Whitehead tower in an (infinity,1)-topos: the topological groupoid sits in a fibration sequence , hence is a - bundle over .
The ordinary higher covering space (as a topological space) that you expect to see is the geometric realization of this.
I finally see it! and so what makes the difference between the topological and the Lie case is that in the latter there is (above first step) no "Lie realization". so the Whitehead tower of a Lie group is really made of n-Lie groups. however,
Yes! The internal Whitehead tower, internal to the Lie oo-topos. Yes. It also has an underlying topological tower, tough, in Top. Under the non-boldface .
however, one has a partial Lie realization: ,
Yes. I would put it like this: if we want to refine ordinary cohomology in Top and the extensions it classifies to the Lie context, then it matters for the smooth structure whether we use or .
for intance, the 3-connected cover String(n) is a topological group or a Lie 2-group, and the 7-connected cover Fivebrane(n) is a topological group or a Lie 6-group.
Yes!
(in case this should happen to be correct, please, be extremely cautious in telling me so: I could stand up, run to the window and cry I see the light! :-) )
Oops, i wasn't being cautious... :--)
All right!
So let's put some of this light into the entry.
I started polishing the Examples-section at Whitehead tower in an (infinity,1)-topos a bit
I moved the discussion of Postnikov towers by coskeletons and the Duskin reference to Postnikov tower in an (infinity,1)-category
I expanded on the remarks involving the use of the decalage construction to compute the homotopy pullback that defines the Whitehead tower elements in sSet.
David Roberts should have a look if he agrees with what I did so far.
@Domenico
I got confused about the 0th step: taking the connected component of a point (e.g., of the identity in a topological group). from the categorical abstract nonsense that seems to be a topological (-1)-groupoid, but I would have expected it to be a honest topological space
I was trying to think about this step in the context of the classical Whitehead construction recently and I got confused in a different way. But giving it another go, I came up with this: The zeroth stage in the Postnikov tower of (as constructed using topological n-groupoids) should be the Cech groupoid of the map (a unique arrow between any two points in a path component - we assume that is locally well-behaved for components and path-components to be the same ). Thus this is a topological groupoid which is equivalent (internally!) to a set, the 0-type of . Then, as usual, is pointed so that is a pointed groupoid. To construct the homotopy fibre we can take the source fibre of over the basepoint (this is just a shortcut description of the usual tangent category construction), which is a space over by the restriction of the target map. This space is precisely the connected component of the basepoint of , if I am not mistaken.
Looks good, Urs.
One thing I would like to edit, but the spam filter is blocking me, is in the section construction using topological groupoids, where I say
As the space is locally contracible, in particular semi-locally -connected, the space is semi-locally simply-connected (I have a fragment of a paper saying this is true for the 'absolute case' - that is, locally -connected implies the mapping space locally simply connected, but I expect it to be true for the relative case -DMR)
This is essentially now Corollary 5.9 in my thesis, but taking the space of pointed maps instead of all maps. The proof (a new one, not from the paper mention) is given in detail.
Concerning the bundles of (-1)-groupoids: yes, exactly: bundles whose fiber either is a point or empty.
David wrote:
I am reminded of Grothendieck's thoughts in the early parts of Pursuring Stacks, where he says the most general n-groupoids will be parameterised by some site i.e. bundles/stacks of groupoids.
Sure, we start with oo-groupoids parameterized by all topological spaces, these are topological oo-groupoids. Then if you have a particular one over a topological space, you can pull back from that gros site to the petit site of that space, to get one parameterized by that space.
This is a general principle: for C a category of test spaces, is the gros topos of all oo-groupoids modeled on them. Given any object in there, the overcategory is the "petit oo-topos" of .
Domenico wrote:
second, the bundle is the flat higher-bundle given by the natural representation given by .
Let me seee: that looks sensible, but can you see this formally? From that diagram in the proof at Whitehead tower in an (infinity,1)-topos we kow that the bundle is classified by the map which is the pullback of along .
Now I'd need to use and the adjunction to manipulate this to get it into the form you mentioned. Ahm, I can almost see this. But not quite actually. Hm..
Yes, that looks right.
added a liitle bit more details to n-connected object of an (infinity,1)-topos
At n-connected object of an (infinity,1)-topos I have
added the recursive characterization of $n$-connected morphisms;
added an Examples section “In Grpd” where I state the fact that 0-connected morphisms between groupoids are precisely the eso+full functors, and then spell out the proof.
(Probably there is a quicker proof using something or other. I just check it explicity.)
Probably there is a quicker proof
Of course in $\infty Grpd$ one can check connectedness of morphisms by checking connectedness of homotopy fibers over each connected component. It is immediate that an eso+full functor of groupoids has connected (homotopy) fibers.
But that argument does not seem to generalize well to stacks, unless I am missing something. I believe the argument that I wrote up does:
A morphism between two groupoid-valued presheaves that is objectwise an eso+full functor is a 0-connected morphism of stacks.
I felt like making explicit in this subsection that in a hypercomplete $\infty$-topos the notions of $n$-connected / $n$-truncated form not just a linear hierarchy, but a “clock”, which wraps around at
and at
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