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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2014
    • (edited Dec 29th 2014)

    For XX an object in a differentially cohesive infinity-topos, say that a framing of XX is a trivialization of the infinitesimal disk bundle p:T infXXp :T_{inf} X \to X, which is the homotopy pullback

    T infX X p X ʃ infX \array{ T_{inf} X &\longrightarrow& X \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\longrightarrow& ʃ_{inf} X }

    of the unit at XX of the infinitesimal shape modality ʃ infʃ_{inf} along itself.

    Now it is a simple standard fact that Lie groups GG are canonically framed by left translation of their tangent space at the neutral element over the group.

    In which generality does the analogous statement remain true for \infty-group objects in differentially cohesive \infty-toposes?

    I suppose one is to proceed by the homotopy-Mayer-Vietoris sequence in the \infty-topos (see the other discussion): if GG is an \infty-group then (using also that ʃ infʃ_{inf} preserves \infty-limits and \infty-colimits and hence \infty-group structure) the defining homotopy pullback diagram for T infGT_{inf}G is equivalently the following pasting of homotopy pullbacks

    T infG 𝔻 e * e G×G ()() 1 G ʃ infG \array{ T_{inf} G &\longrightarrow& \mathbb{D}_e &\longrightarrow& \ast \\ \downarrow && \downarrow && \downarrow^{\mathrlap{e}} \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G &\longrightarrow& ʃ_{inf} G }

    where hence 𝔻 eG\mathbb{D}_e \to G is the infinitesimal neighbourhood of the neutral element in the \infty-group GG.

    Now I want to conclude that hence the left pullback square gives the desired trivialization statement T infGG×𝔻 eT_{inf} G \simeq G \times \mathbb{D}_e.

    What’s a good way to conclude this. If I have a 1-site of definition then I know that GG is presented by a presheaf of simplicial groups and then I may conclude by falling back to the 1-categorical statement.

    But there is probably a better and more general argument.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2014
    • (edited Dec 30th 2014)

    As Marc highlights here of course the statement directly follows from it being true in Grpd\infty Grpd by arguing pointwise over a site of definition.

    But is there also an internal argument?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 30th 2014

    I have added statement and proof of how every differentially cohesive \infty-group is canonically framed here.