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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
$X \to \mathcal{X}$ is a 1-epimorphism;
$\mathbf{Aut}_{\mathbf{F}}(\chi) \simeq \mathbf{Aut}_{\mathcal{X}}(X)$
?
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 $\nabla^0$.
I have now started to write out a bunch of details along the above lines in a new entry
(Not proof-read yet, as it is getting too late for me now. Will fix typos and furher refine tomorrow…)
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