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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2014
    • (edited Dec 28th 2014)

    have added to regular epimorphism the statement (here) that a pullback square of regular epis is also a pushout.

    This must be true for effective epimorphisms in an \infty-topos, too. What’s the proof?

    • CommentRowNumber2.
    • CommentAuthorZhen Lin
    • CommentTimeDec 29th 2014

    If I’m not mistaken, the \infty-analogue is false. For instance,

    4 2 2 1\begin{array}{ccc} 4 & \rightarrow & 2 \\ \downarrow & & \downarrow \\ 2 & \rightarrow & 1 \end{array}

    is a (homotopy) pullback square of effective epimorphisms (of sets) in Grpd\infty Grpd but not a homotopy pushout square. (The homotopy pushout is a circle.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2014
    • (edited Dec 29th 2014)

    That’s true of course, stupid me. Thanks.

    I’ll need to consider some further conditions.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 7th 2015
    • (edited Jan 7th 2015)

    So concretely, I am looking at the following situation: XX a VV-manifold (as here), T infXT_{inf}X its infinitesimal disk bundle (as here), UXU \to X a formally étale 1-epi, and the pullback in question being the one that exhibits a local trivialization of the formal disk bundle

    U×𝔻 T infX U X \array{ && U \times \mathbb{D} \\ & \swarrow && \searrow \\ T_{inf}X && && U \\ & \searrow && \swarrow \\ && X }

    So T infXXT_{inf}X \to X as well as U×𝔻UU \times \mathbb{D} \to U are \Re-equivalences (\Re the reduction modality). This means that applying \Re to this diagram it does become a homotopy pushout. Since \Re preserves homotopy pushouts, this means that the homotopy pushout T infXU×𝔻UT_{inf} X \underset{U \times \mathbb{D}}{\coprod} U differs from XX at most in some infinitesimal extension.

    So for achieving the desired XT infXU×𝔻UX \simeq T_{inf} X \underset{U\times \mathbb{D}}{\coprod} U I am reduced to arguing (or else to try to arrange by further assumptions) that also this infinitesimal difference vanishes. This should follow since U×𝔻T infXU \times \mathbb{D}\to T_{inf}X is something like a fiberwise equivalence. Hm…

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 7th 2015
    • (edited Jan 28th 2015)

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 7th 2015
    • (edited Jan 28th 2015)

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 8th 2015
    • (edited Jan 8th 2015)

    In more detail, what I am after is described around def. 4.7 of this note on parameterized WZW terms (pdf). My real goal is to get a useful condition that makes the necessary obstruction of theorem 4.5 also a sufficient obstruction, as in corollary 4.11. Possibly there is an altogether better way to do this than I have there right now.

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