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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 15th 2013
   • (edited May 3rd 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSome comments on the implementation of [Atiyah- and Courant Lie n-algebroids](http://ncatlab.org/nlab/show/Courant+algebroid#RelationToAtiyahGroupoids) in [[cohesive homotopy type theory]], and some general thoughts on "0-strict" $\infty$-groupoids ($\infty$-groupoids equipped with a 1-epimorphism out of a 0-truncated object). Let $\mathbf{H}$ be some [[cohesive (infinity,1)-topos]]. The [[homotopy Lie-Rinehart pair]]-perspective on an $\infty$-groupoid $\mathcal{X} \in \mathbf{H}$ which is equipped with a [[1-epimorphism]] $\phi \colon X \to \mathcal{X}$ is to consider the pair consisting of $X$ and of the [[infinity-group]] [[group of bisections|of bisections]] $$ BiSect(\mathcal{X},X) \coloneqq \mathbf{Aut}_{\mathcal{X}}(X) \coloneqq \underset{\mathcal{X}}{\prod} Equiv_{/\mathcal{X}}(X)_ \,. $$ (Traditionally in a (homtopy-)[[Lie-Rinehart algebra]] of course one only remembers the [[L-infinity algebra]] $Lie( \mathbf{Aut}_{\mathcal{X}}(X) )$ obtained from this under [[Lie differentiation]], but here I will stick to the complete Lie integrated picture.) Now, it turns out that famous homotopy Lie-Rinehart algebras out there are constructed (secretly, but one can see that this is what happens) by starting with a map $$ \chi \colon X \to \mathbf{F} $$ to some [[moduli infinity-stack]] $\mathbf{F}$ and then taking the group of bisections to be the automorphism group of this $\chi$ over $\mathbf{F}$. For instance * the [[Atiyah Lie algebroid]] assigned to a circle principal bundle modulated by $\nabla^0 \colon X \to \mathbf{B}U(1)$ is the Lie differentiaton of $(X, \mathbf{Aut}_{\mathbf{B}U(1)}(\nabla^0))$; * the [[Courant Lie 2-algebroid]] assigned to a map $\nabla^1 \colon X \to \mathbf{B}^2 U(1)_{conn^1}$ modulating a "bundle gerbe with connective data but no curving" is the Lie differentiation of $(X, conc \mathbf{Aut}_{\mathbf{B}^2 U(1)_{conn^1}}(\nabla^1))$. (here $conc$ stands for "differential concretification", a technical subtlety related to the right cohesive structure on these objects, which for the purpose of the present discussion one should ignore) Now given such an "integrated homotopy Lie-Rinehart pair" consisting of an object $X$ and an automorphism $\infty$-group of a map $\chi : X \to \mathbf{F}$ over $\mathbf{F}$, can we canonically find for it the corresponding $\infty$-groupoid, hence the $\mathcal{X} \in \mathbf{H}$ such that 1. $X \to \mathcal{X}$ is a 1-epimorphism; 1. $\mathbf{Aut}_{\mathbf{F}}(\chi) \simeq \mathbf{Aut}_{\mathcal{X}}(X)$ ? Well, that's just the [[1-image]] of $\chi$: $$ \mathcal{X} \simeq im_1(\chi) $$ isn't it? Let's look at the first example in the above series: Let $\mathbf{H} = $ [[SmoothGrpd]], let $X \in SmthMfd \hookrightarrow \mathbf{H}$ be a smooth manifold, and let $\nabla^0 \colon X \to \mathbf{B}U(1)$ be the map modulating a circle principal bundle $P \to X$. Then what is $$ X \to \mathcal{X}\coloneqq im_1(\nabla^0) \,. $$ To check this, remember that the 1-image of a map may be computed as the homotopy colimit over its homotopy Cech nerve. Doing so here, we find that $$ im_1(\nabla^0) \simeq ( P \times_{U(1)} P \stackrel{\to}{\to} X) \,. $$ This is the [[Lie integration]] of the [[Atiyah Lie algebroid]] of the circle bundle $P \to X$ modulated by $\nabla^0$.

   Some comments on the implementation of Atiyah- and Courant Lie n-algebroids in cohesive homotopy type theory, and some general thoughts on “0-strict” $\infty$-groupoids ($\infty$-groupoids equipped with a 1-epimorphism out of a 0-truncated object).

   Let $\mathbf{H}$ be some cohesive (infinity,1)-topos.

   The homotopy Lie-Rinehart pair-perspective on an $\infty$-groupoid $\mathcal{X} \in \mathbf{H}$ which is equipped with a 1-epimorphism $\phi \colon X \to \mathcal{X}$ is to consider the pair consisting of $X$ and of the infinity-group of bisections

   $$BiSect(\mathcal{X},X) \coloneqq \mathbf{Aut}_{\mathcal{X}}(X) \coloneqq \underset{\mathcal{X}}{\prod} Equiv_{/\mathcal{X}}(X)_ \,.$$

   (Traditionally in a (homtopy-)Lie-Rinehart algebra of course one only remembers the L-infinity algebra $Lie( \mathbf{Aut}_{\mathcal{X}}(X) )$ obtained from this under Lie differentiation, but here I will stick to the complete Lie integrated picture.)

   Now, it turns out that famous homotopy Lie-Rinehart algebras out there are constructed (secretly, but one can see that this is what happens) by starting with a map

   $$\chi \colon X \to \mathbf{F}$$

   to some moduli infinity-stack $\mathbf{F}$ and then taking the group of bisections to be the automorphism group of this $\chi$ over $\mathbf{F}$.

   For instance

   • the Atiyah Lie algebroid assigned to a circle principal bundle modulated by $\nabla^0 \colon X \to \mathbf{B}U(1)$ is the Lie differentiaton of $(X, \mathbf{Aut}_{\mathbf{B}U(1)}(\nabla^0))$;

   • the Courant Lie 2-algebroid assigned to a map $\nabla^1 \colon X \to \mathbf{B}^2 U(1)_{conn^1}$ modulating a “bundle gerbe with connective data but no curving” is the Lie differentiation of $(X, conc \mathbf{Aut}_{\mathbf{B}^2 U(1)_{conn^1}}(\nabla^1))$.

     (here $conc$ stands for “differential concretification”, a technical subtlety related to the right cohesive structure on these objects, which for the purpose of the present discussion one should ignore)

   Now given such an “integrated homotopy Lie-Rinehart pair” consisting of an object $X$ and an automorphism $\infty$-group of a map $\chi : X \to \mathbf{F}$ over $\mathbf{F}$, can we canonically find for it the corresponding $\infty$-groupoid, hence the $\mathcal{X} \in \mathbf{H}$ such that

   1. $X \to \mathcal{X}$ is a 1-epimorphism;

   2. $\mathbf{Aut}_{\mathbf{F}}(\chi) \simeq \mathbf{Aut}_{\mathcal{X}}(X)$

   ?

   Well, that’s just the 1-image of $\chi$:

   $$\mathcal{X} \simeq im_1(\chi)$$

   isn’t it?

   Let’s look at the first example in the above series:

   Let $\mathbf{H} =$ SmoothGrpd, let $X \in SmthMfd \hookrightarrow \mathbf{H}$ be a smooth manifold, and let $\nabla^0 \colon X \to \mathbf{B}U(1)$ be the map modulating a circle principal bundle $P \to X$.

   Then what is

   $$X \to \mathcal{X}\coloneqq im_1(\nabla^0) \,.$$

   To check this, remember that the 1-image of a map may be computed as the homotopy colimit over its homotopy Cech nerve. Doing so here, we find that

   $$im_1(\nabla^0) \simeq ( P \times_{U(1)} P \stackrel{\to}{\to} X) \,.$$

   This is the Lie integration of the Atiyah Lie algebroid of the circle bundle $P \to X$ modulated by $\nabla^0$.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 19th 2013
   • (edited Feb 20th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have now started to write out a bunch of details along the above lines in a new entry * _[[higher Atiyah groupoid]]_ (Not proof-read yet, as it is getting too late for me now. Will fix typos and furher refine tomorrow...)

   I have now started to write out a bunch of details along the above lines in a new entry

   • higher Atiyah groupoid

   (Not proof-read yet, as it is getting too late for me now. Will fix typos and furher refine tomorrow…)