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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 20th 2010
• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeJul 20th 2010

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 20th 2010

right, that’s a remnant from the material being copied from simplicial group. I’ll fix it. Thanks.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 20th 2010

okay, I added some remarks about $\bar W G$ to simplicial principal bundle. But the entry is still pretty stubby.

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeSep 28th 2020

Corrected a serious mistake in the definition: previously, the action of G_n on E_n was not required to be transitive!

• CommentRowNumber6.
• CommentAuthorDmitri Pavlov
• CommentTimeSep 28th 2020

Removed transitivity, since it is introduced later.

Is a “principal action” really the same thing as a “free action”?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 28th 2020

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.

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeSep 28th 2020

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?

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeSep 28th 2020

Does the weak definition imply $E\times_B E$ is equivalent to $E\times G$?

• CommentRowNumber10.
• CommentAuthorDmitri Pavlov
• CommentTimeSep 28th 2020

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.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeSep 29th 2020

I have fixed the definition to that of weakly-principal bundles (here). Please be invited to add further variants.

• CommentRowNumber12.
• CommentAuthorGuest
• CommentTimeAug 1st 2021
According to the point of view of the articles //ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications (https), higher principal bundles (discrete case included) are modeled by homotopy pullbacks which are, in turn, computed as ordinary pullbacks on fibrantly replaced diagrams. If I am not wrong, this necessarily means that we have to obtain (in simplicial sets) twisted Cartesian products with Kan fibrant base, because all objects in fibrantly repalced diagrams are Kan fibrant. But on the other hand, there is no such condition imposed on base of principal twisted Cartesian product in works of May and others. Hence the question is - is the Kan fibrant condition on base needed? If so, why? If not, how to understand the concept of simplicial principal bundles established in the classical literature?
• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeAug 1st 2021
• (edited Aug 1st 2021)

[ 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..

• CommentRowNumber14.
• CommentAuthorGuest
• CommentTimeAug 3rd 2021

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?