I may still be confused. We have the construction in #28
and my next task would be to find from these fiberwise flat differential forms a globally defined differential form on . From looking at the model one expects this not to be necessarily flat anymore, as the horizontal part of the differential will see how the fiberwise flat forms change as one moves along the fibers. My expectation was that I’d have to first pass from fiberwise flat to fiberwise general -valued forms and then observe the vanishing of some cocycle in order to find a section of a trivial -bundle over which would be the globally defined form in question.
But what confuses me now is that I seem to get all this for free and for flat forms: namely the map in the above diagram is equivalently a section of the pullback of the bundle to . But by the commutativity of the diagram, that is equivalently the pullback to along , which is the trivial flat form bundle . Hence the above gives a section of that hence a flat form
That worries me a bit. It would be good, but somehow it seems too good to be true. Maybe it is an indication that I am missing something and my intended formalization of #1 still does not work.
]]>Right, the action is that by gauge transformations. If is a Lie 1-group, then
takes to
In the dcct pdf this is prop. 5.4.44, see also example 3.9.37.
(And the statement about the -equivariance of is now also on the Lab here).
]]>Or sections of the adjoint bundle P×_G g, such as appears in the left of the algebroid Atiyah sequence? But you said that we shouldn’t be thinking of the adjoint action (though I may be thinking of the wrong bit, I can’t read back over the thread to unravel it now)
]]>Right, I should have said more about it, sorry. So let’s write for the -principal -bundle which is modulated by that map . Then one name of that thing without name is the associated -bundle
Moreover, a more evocative notaton for is , the sheaf of -algebra valued differential forms, so this is
and that makes most manifest how this is a fiberwise varying collection of such differential form coefficients. One might consider the étalification of this associated bundle to a concrete bundle, and that would be then something like a -valued cotangent bundle of .
]]>Ah, I see. I thought it might be something that already had a name.
]]>Whatever name you want to give to the pullback given by the bottom right square.
]]>@Urs, what’s missing in the middle of the diagram in #28?
]]>No problem, I learned some things. (-:
]]>Yeah, sorry, I was misled. For the absolute MC form I needed a group object, so I thought I should look for one in the slice, too. My bad, sorry for the wild goose chase.
]]>Re: #27, that’s basically just the definition of , right? I actually thought of that, but you really seemed to want the adjoint action on so I assumed it wasn’t right. (-:
]]>And so to come back to #1, the answer is that given a -principal -bundle on (classified by some ) with a trivialization of its pullback to some correspondence space , then this induces the fiberwise flat -valued differential form that is denoted in this pasting diagram of homotopy pullbacks:
Or rather, this is half the answer for #1. Next I need to sum up these fiberwise flat forms to a single non-flat form on the total space…
]]>Ah, I think I have the answer. It’s super-easy. But my insistence above on having a group object in the slice was misled.
Instead of using the adjoint action on , we should use the right action. That’s just
from there we want to map to
Well, that’s easy, there is actually an essentially unique map
Now while that does certainly induce a map on the homotopy fibers, one tends to worry that this will be the trivial map. But, no, it is the MC form. This follows from the pasting law (and it will be trivial to you, but let me spell it out anyway):
the morphism on homotopy fibers which is induced from the previous diagram is that from the pullback diagram
to the pullback diagram
via the universal map in
Here the bottom square and the total rectangle are homotopy pullbacks by assumption. Hence so is the top square. Hence
is indeed the MC form.
]]>Maybe we don’t actually want the fiber to be . Maybe the variant is actually “right”.
In the motivating example of #1 there is in fact an extra condition: the MC forms in arXiv:1305.4870 are produced patchwise in the fibers in (4.13); but then above (4.15) and above (4.17) further cohomological conditions are used to produce, in (4.18), (4.23) (4.24), an actual fiberwise MC form.
]]>Is there a way to say “” synthetically?
Asked this way, the answer would be: sure, this is again just the MC form , i.e. the image of g under .
Something tautological or other needs to be done here to make all this fall into place…
]]>the conjugation action on goes to the inhomogeneous transformation
Ah, right. So that means that we don’t want the “obvious” adjoint action on , we want this sort of “twisted” one. So we do need some other fibration over whose fiber is and which admits a map from . Is there a way to say “” synthetically?
]]>Doesn’t that require the action to be free as well as transitive?
That’s true, right. In fact the part acting non-freely is just again, and this is what gives . So I was wrong in the last equivalence in #17. Instead of the quotient of the diagonal action being one copy of it’s more something like the product of that with .
I don’t think that can happen unless BG is discrete.
Right, that’s also true. In components the reason is that the conjugation action on goes to the inhomogeneous transformation on differential forms, which prevents the 0-form from being the invariant section.
So I need to think more about all this…
]]>Wait a minute, I am confused. is the free loop space . But the free loop space of any object always has a section, assigning to each the constant path. So if there is a map over , then also has a section; but I don’t think that can happen unless is discrete.
]]>…that -action on flat forms is transitive and so in that case
Doesn’t that require the action to be free as well as transitive?
]]>What are the two actions of on ?
So generally for a group homomorphism then
exhibits the induced -action on .
Here the group homomorphism in question is now (since I am assuming to be braided). The action in question is hence via that homomorphism, or rather its diagonal.
Do you mean by this “for a 0-truncated G in the standard models”?
Yes
Do you know whether the map exists in the standard models when is not 0-truncated?
I don’t even know that, unfortunately.
]]>In case is 0-truncated and in the standard models
Do you mean by this “for a 0-truncated in the standard models”? Do you know whether the map exists in the standard models when is not 0-truncated?
]]>What are the two actions of on ?
]]>Quick remark before I need to rush off:
that diagram in #16 gives a map from to what deserves to be written , hence an adjoint-equivariant map from to .
In case is 0-truncated and in the standard models, then that -action on flat forms is transitive and so in that case . In general I suppose this equivalence won’t hold. But maybe that extra quotient construction then is the next closest thing that exists in general.
]]>Yes, I am after just that canonical action, but I didn’t see how to produce a map
This map exists in the models at least for ordinary groups, but I didn’t see how to producte it abstractly. So instead I tried to force such kind of map. But my attempt via the pushout was no good, as you pointed out.
I am thinking of another way to do it, but not sure how to make that work:
Assume that is at least braided, i.e. that it is . Then consider the map of cospan diagrams
Forming pullbacks gives a map over , where “somewhere” knows something about a -action on .
If we could go from there to
we’d be done, as the limit over that last diagram is .
Hm.
]]>Here is a naive question: since is a fiber of a map to , it automatically comes with a “tautological” action by . You seem to be saying that this action is not (or not obviously) the adjoint action, since you’re trying to exhibit as a fiber of a different map to and calling that the adjoint action. What is this tautological action then? I would have thought that the adjoint action is the only available action of on .
]]>