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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 29th 2014
   • (edited Dec 29th 2014)
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   Author: Urs
   Format: MarkdownItexFor $X$ an object in a [[differential cohesion|differentially cohesive infinity-topos]], say that a _framing_ of $X$ is a trivialization of the [infinitesimal disk bundle](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#GLnTangentBundles) $p :T_{inf} X \to X$, which is the homotopy pullback $$ \array{ T_{inf} X &\longrightarrow& X \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\longrightarrow& ʃ_{inf} X } $$ of the unit at $X$ of the [[infinitesimal shape modality]] $ʃ_{inf}$ along itself. Now it is a simple standard fact that Lie groups $G$ 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](http://nforum.mathforge.org/discussion/6403/generality-of-mayervietoris-in-an-infinitytopos-/?Focus=51404#Comment_51404)): if $G$ is an $\infty$-group then (using also that $ʃ_{inf}$ preserves $\infty$-limits and $\infty$-colimits and hence $\infty$-group structure) the defining homotopy pullback diagram for $T_{inf}G$ is equivalently the following pasting of homotopy pullbacks $$ \array{ T_{inf} G &\longrightarrow& \mathbb{D}_e &\longrightarrow& \ast \\ \downarrow && \downarrow && \downarrow^{\mathrlap{e}} \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G &\longrightarrow& ʃ_{inf} G } $$ where hence $\mathbb{D}_e \to G$ is the infinitesimal neighbourhood of the neutral element in the $\infty$-group $G$. Now I want to conclude that hence the left pullback square gives the desired trivialization statement $T_{inf} G \simeq G \times \mathbb{D}_e$. What's a good way to conclude this. If I have a 1-site of definition then I know that $G$ 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.

   For an object in a differentially cohesive infinity-topos, say that a framing of is a trivialization of the infinitesimal disk bundle , which is the homotopy pullback

   of the unit at of the infinitesimal shape modality along itself.

   Now it is a simple standard fact that Lie groups 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 -group objects in differentially cohesive -toposes?

   I suppose one is to proceed by the homotopy-Mayer-Vietoris sequence in the -topos (see the other discussion): if is an -group then (using also that preserves -limits and -colimits and hence -group structure) the defining homotopy pullback diagram for is equivalently the following pasting of homotopy pullbacks

   where hence is the infinitesimal neighbourhood of the neutral element in the -group .

   Now I want to conclude that hence the left pullback square gives the desired trivialization statement .

   What’s a good way to conclude this. If I have a 1-site of definition then I know that 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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 30th 2014
   • (edited Dec 30th 2014)
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   Author: Urs
   Format: MarkdownItexAs Marc highlights [here](http://nforum.mathforge.org/discussion/6403/generality-of-mayervietoris-in-an-infinitytopos-/?Focus=51406#Comment_51406) of course the statement directly follows from it being true in $\infty Grpd$ by arguing pointwise over a site of definition. But is there also an internal argument?

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

   But is there also an internal argument?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 30th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added statement and proof of how every differentially cohesive $\infty$-group is canonically framed [here](http://ncatlab.org/nlab/show/differential+cohesive+%28infinity%2C1%29-topos#DifferentialCohesiveInfinityGroupIsCanonicallyFramed).

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