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

    Readability concern: The first place where W¯G\bar{W}G appears should have a link to an entry where W¯G\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 W¯G\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!

    diff, v9, current

    • 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”?

    diff, v9, current

    • 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× BEE\times_B E is equivalent to E×GE\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.

    diff, v10, current

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