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I've been trying to understand how to differentiate Lie $\infty$-groupoids to get a Lie $\infty$-algebroid. First of all, I will state the definitions that I'm assuming.

A Lie $\infty$-groupoid is a Kan fibrant simplicial smooth manifold, i.e., a simplicial smooth manifold $\mathscr{G}: \Delta^{op} \longrightarrow \textbf{Mfld}$ such that the canonical restriction $\textbf{Mfld} (\Delta^n, \mathscr{G}) \longrightarrow \textbf{Mfld} (\Lambda^n_k, \mathscr{G})$ is a surjective submersion.

A Lie $\infty$-algebroid is a commutative semifree DG $\mathscr{C}^{\infty} (\mathscr{G}_0)$-algebra of finite rank with $\mathscr{C}^{\infty} (\mathscr{G}_0)$ in degree 0 (see the definition here http://ncatlab.org/nlab/show/Lie+infinity-algebroid). However I'm ok, if anyone find the Lie $\infty$-algebroid as a simplicial vector bundle or as a graded vector bundle satisfying the analogous axioms (under the Dold-Kan correspondence and picking global sections).

There is a discussion (answer) here http://nforum.ncatlab.org/discussion/4630/differentiation-of-lie-infinity-groupoids/, however I was unable to understand it correctly and it deals with more general spaces (infinity stack of cartesian spaces) that need not to be Kan fibrant. In this discussion, as I understand itś used infinitesimal disks to get a presheaf of infinitesimally thickened points and then I don't know how to proceed.

So, I would like to know how to get the Lie differentiation in simple simplicial terms (using just simplicial manifolds) without anything fancier (like smooth cohesion or synthetic stuff). Furthermore I would like to know if this definition of Lie differentiation is compatible with Lie integration as in http://arxiv.org/abs/math/0603563 .

So the question is: given a Lie $\infty$-groupoid $\mathscr{G}$, what's $\text{Lie} (\mathscr{G})$ as a Lie $\infty$-algebroid?

Thanks in advance.

EDIT I have found these other articles http://arxiv.org/abs/math/0612349 and http://arxiv.org/pdf/1403.7185.pdf . The first one, describe how to compute the object that will represent the Lie $\infty$-algebroid, while the second one proves the existence of a semi-strict Lie 2-algebra law for the object obtained (via the same procedure as the previous paper) applied to any semi-strict Lie 2-groups.

I will describe what I've understood until now. Let $D_n$ denotes the infinitesimal line $\mathbb{R}^{0|n}$ or a point together with the stalk $\mathbb{R} [x_1, …, x_n] /(x_ix_j - x_jx_i)$ (so I will use synthetic methods). Furthermore I will assume manifolds and super manifolds to mean the same thing. Let $$Y: \textbf{Mfld}^{\Delta^{op}} \longrightarrow \widehat{\textbf{SSM}}$$ where $\textbf{SSM}$ is the category composed by surjective submersions and commutative squares as morphisms. This $Y$ is a kind of Yoneda embedding given by $$Y(K)(M \rightarrow N) = \textbf{Mfld}^{\Delta^{op}} (N(M \times_N M), K)$$, where $N (M \times_N M)$ is the nerve of the groupoid given on 1-simplices by $M \times_N M$ and by $M$ on $0$-simplices.

Let $\mathbf{SSM}_n$ be the full subcategory of $\mathbf{SSM}$ given by the submersions $$D_n \times M \xrightarrow{pr_2} M$$. Let $J_n : \textbf{Mfld}^{\Delta^{op}} \longrightarrow \widehat{\textbf{SSM}_n}$ be defined as the restriction of $Y$. In the first article, this functor is called the $n$-jet (actually the definition is a little different but equivalent) and the author proves that the presheaf $J_n (K)$ is represented by a simplicial manifold that is a subobject of $\textbf{Mfld}((D_n)^{\bullet}, K_{\bullet})$. In the second article, it's given an explicit $L_{\infty}$ structure in $J_1 (BG)$ for any semi-strict Lie 2-group. Anyway, I'm facing some difficulties in understanding the physicist notations in both these articles.

In the first article, it's said that this construction gives the Lie algebraic of a Lie groupoid, so this construction will probably give the infinity version. In the forum discussion cited above it's said that the construction of the Lie algebra in the Lie-Rinehart algebra is given by picking the Lie algebra of the $\infty$-group of bisections. Therefore if the construction on the nforum and the two articles are the same, understanding the the semi-strict Lie 2-group as described in the second article is enough.

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    $\begingroup$ This thesis arxiv.org/abs/1512.04209 contains some elaboration on the first article you are confused with. $\endgroup$ – Ma Ming Oct 12 '16 at 8:50
  • $\begingroup$ @MaMing Thank you! I will try reading it carefully later when I have more time. But, for now I would like to ask 2 things, do you know a way of explicitly computing the $Q$ on $Hom (P, X)$ or better the brackets of the $L_{\infty}$-algebra for, say a 2-groupoid (that is not a 2-group).?Furthermore, what's the relation between the stack-theoretic version $T[-1] X = T\Omega X$ and $Hom (P, X)$? I suppose that they should coincide by picking a chart of the given stack (and taking the Cech cover) and using Koszul duality (for that Maurer-Cartan functor) and they should respect Morita equivalence. $\endgroup$ – user40276 Oct 12 '16 at 9:39

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