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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 29th 2014
   • (edited Dec 29th 2014)
   • PermaLink
   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 $X$ an object in a differentially cohesive infinity-topos, say that a framing of $X$ is a trivialization of the infinitesimal disk bundle $p :T_{inf} X \to X$, 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): 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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 30th 2014
   • (edited Dec 30th 2014)
   • PermaLink
   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 $\infty Grpd$ 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 $\infty$-group is canonically framed here.