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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 17th 2018

    Fixed some hickups in the very first sentence.

    diff, v12, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 17th 2018

    One must be careful with transitive action about the empty set, but all is fine here? I guess any group acting on the empty set is a free action.

    What’s the HoTT version of freeness? Something like IsSet( *:BGV(*))IsSet(\sum_{\ast: BG} V(\ast)).

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeApr 17th 2018

    I would probably state it as: for any xx, the type of gg such that gx=xg x = x is contractible (or, equivalently, a proposition, since it’s always inhabited by ee).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2021

    added definition in terms of the shear map being a monomorphism.

    diff, v19, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 1st 2021

    Typed up in the Sandbox the observation/proof that infinity-quotients of infinity-free actions in an infinity-topos are 0-truncated. But need to go offline right now. Will port and polish tomorrow.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2021

    added (here) statement and detailed proof that, in an \infty-topos, \infty-quotients of \infty-free \infty-actions (i.e. all higher shear maps are (1)(-1)-truncated) are plain quotients of the underlying 0-truncated action.

    diff, v22, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2021
    • (edited Nov 2nd 2021)

    [ thought I could weaken the assumptions of the prop, but maybe I can’t ]

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2021

    What I was after in #7 does work, after all, one has to appeal to the groupoidal Segal conditions:

    I have stepped back and added a Definition-subsection

    with the declaration that an \infty-action is free if its shear map “in degree 1” is (-1)-truncated, and a proof that this implies that also all “higher shear maps” are (-1)-truncated.

    WIth this, the Prop. from #6 above (here) now says that as soon as the shear map X×GX×XX \times G \to X \times X of an \infty-action is (1)(-1)-truncated, the homotopy quotient of the \infty-action is the plain quotient of its 0-truncated version.

    diff, v25, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2021

    I should add a commentary to the entry on the following point, which can be confusing:

    The \infty-action on a principal \infty-bundle is always fiber-wise free, but not globally so unless it’s base space is 0-truncated. Partially-conversely, every \infty-action is fiberwise free (in fact principal) over its homotopy quotient. (all internal to some \infty-topos).