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am splitting off simplicial principal bundle from simplicial group
Readability concern: The first place where $\bar{W}G$ appears should have a link to an entry where $\bar{W}G$ is defined (I do not know which entry has it). I know it is somewhat standard, but not everybody is educated enough.
right, that’s a remnant from the material being copied from simplicial group. I’ll fix it. Thanks.
okay, I added some remarks about $\bar W G$ to simplicial principal bundle. But the entry is still pretty stubby.
It’s of course not the same, in general. There is a condition missing in the entry.
The point is that for simplicial bundles, which are meant (explicitly or implicity) to model principal $\infty$-bnundles, the 1-categorical definition of principal action is not the intended one.
Instead one wants a free action that is “weakly principal” in that the shear map it induces is a weak homotopy equivalence.
I am too tired now to deal with the entry. But if it doesn’t say that, it needs fixing.
Instead one wants a free action that is “weakly principal” in that the shear map it induces is a weak homotopy equivalence.
But being a free action is a cofibrancy condition that presumably one does not want in a weak definition.
I can envision at least two different definitions:
The strict definition says that a principal G-bundle for a simplicial group G is a G-equivariant simplicial map E→B, where the G-action on B is trivial and the induced map E/G→B is an isomorphism.
The weak definition says that a principal G-bundle for a simplicial group G is a G-equivariant simplicial map E→B, where the G-action on B is trivial and the induced map E//G→B is a weak equivalence, where // denotes the homotopy quotient.
One can prove that the ∞-categories of strict and weak principal G-bundles are equivalent.
Which definition do we want here?
Does the weak definition imply $E\times_B E$ is equivalent to $E\times G$?
Re #9: Yes (with a homotopy fiber product): E ⨯^h_B E = E ⨯^h_{E//G} E = E ⨯^h (pt ⨯^h_{pt//G} pt) = E ⨯ G.
[ this is referring to arXiv:1207.0249, web ]
Since the classical model structure on simplicial sets is right proper, a pullback diagram is a homotopy pullback already when one of the two maps is a fibration, with no further condition on the objects. This is the second item of this Prop..
Ahh, thank you! So do I understand it correctly that the Kan condition on a simplicial set $X$ in the section 4 (The universal simplicial $G$-principal bundle) is redundant?
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