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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Lie integration} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory} [[!include infinity-Lie theory - contents]] \hypertarget{integration_theory}{}\paragraph*{{Integration theory}}\label{integration_theory} [[!include integration theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{HigherPathsInAnInfinityLieAlgebroid}{Higher dimensional paths in an $\infty$-Lie algebroid}\dotfill \pageref*{HigherPathsInAnInfinityLieAlgebroid} \linebreak \noindent\hyperlink{IntToBareGrpd}{Integration to a discrete $\infty$-groupoid}\dotfill \pageref*{IntToBareGrpd} \linebreak \noindent\hyperlink{SmoothIntegration}{Integration to a smooth $\infty$-groupoid}\dotfill \pageref*{SmoothIntegration} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{PropertiesHomotopyGroups}{Homotopy groups}\dotfill \pageref*{PropertiesHomotopyGroups} \linebreak \noindent\hyperlink{QuillenAdjunction}{Quillen adjunction}\dotfill \pageref*{QuillenAdjunction} \linebreak \noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak \noindent\hyperlink{LieAlgebrasToLieGroups}{Interating Lie algebras to Lie groups}\dotfill \pageref*{LieAlgebrasToLieGroups} \linebreak \noindent\hyperlink{IntegrationToLineNGroup}{Integrating to line/circle Lie $n$-groups}\dotfill \pageref*{IntegrationToLineNGroup} \linebreak \noindent\hyperlink{integrating_the_string_lie_2algebra_to_the_string_lie_2group}{Integrating the string Lie 2-algebra to the string Lie 2-group}\dotfill \pageref*{integrating_the_string_lie_2algebra_to_the_string_lie_2group} \linebreak \noindent\hyperlink{integrating_lie_algebroids_to_stacky_lie_groupoids}{Integrating Lie algebroids to (stacky) Lie groupoids}\dotfill \pageref*{integrating_lie_algebroids_to_stacky_lie_groupoids} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} \emph{Lie integration} is a process that assigns to a [[Lie algebra]] $\mathfrak{g}$ -- or more generally to an [[L-∞-algebra|∞-Lie algebra]] or [[∞-Lie algebroid]] -- a [[Lie group]] -- or more generally [[∞-Lie groupoid]] -- that is [[infinitesimal space|infinitesimally]] modeled by $\mathfrak{g}$. It is essentially the reverse operation to [[Lie differentiation]], except that there are in general several objects Lie integrating a given Lie algebraic datum, due to the fact that the infinitesimal data does not uniquely determine global topological properties. Classically, Lie integration of [[Lie algebras]] is part of [[Lie's three theorems]], which in particular finds an unique (up to [[isomorphism]]) [[simply connected]] Lie group integrating a given finite-dimensional Lie algebra. One may observe that the [[simply connected]] [[Lie group]] integrating a (finite-dimensional) Lie algebra is equivalently realized as the collection of [[equivalence classes]] of [[Lie algebra valued 1-forms]] on the interval where two such are identified if they are interpolated by a \emph{flat} Lie-algebra valued 1-form on the [[disk]]. (\hyperlink{DuistermaatKolk00}{Duistermaat-Kolk 00, section 1.14}, see also the example \hyperlink{LieAlgebrasToLieGroups}{below}). This \emph{path method} of Lie integration stands out as having natural generalizations to [[higher Lie theory]] (\hyperlink{Severa01}{\v{S}evera 01}). In its evident generalization from [[Lie algebra valued differential forms]] to [[Lie algebroid valued differential forms]] this provides a means for Lie integration of [[Lie algebroids]] (e.g. \hyperlink{Crainic}{Crainic-Fernandes 01}). In another direction, one may observe that [[L-∞ algebras]] are [[formal dual|formally dually]] incarnated by their [[Chevalley-Eilenberg algebra|Chevalley-Eilenberg]] [[dg-algebras]], and that under this identification the evident generalization of the path method to [[L-∞ algebra valued differential forms]] is essentially the [[Sullivan construction]], known from [[rational homotopy theory]], applied to these dg-algebras (\hyperlink{Hinich97}{Hinich 97}, \hyperlink{Getzler04}{Getzler 04}). Or rather, the bare such construction gives the [[geometrically discrete ∞-groupoid|geometrically discrete]] [[∞-group]] underlying what should be the Lie integration to a [[smooth ∞-group]]. This is naturally obtained, as in the classical case, by suitably smoothly parameterizing the [[∞-Lie algebroid valued differential forms]] (\hyperlink{Henriques}{Henriques 08}, \hyperlink{Roytenberg09}{Roytenberg 09}, \hyperlink{FSS12}{FSS 12}). Both these directions may be combined via the evident concept of [[∞-Lie algebroid valued differential forms]] to yield a Lie integration of [[∞-Lie algebroids]] to [[smooth ∞-groupoids]]. (Moreover, the same formula directly generalizes from $L_\infty$-algebroids to [[A-infinity categories]] to yield the \emph{[[dg-nerve]]} construction.) While the construction exists and behaves as expected in examples, there is to date no good general theory providing higher analogs of, say, [[Lie's three theorems]]. But people are working on it. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Throughout, let $\mathfrak{a}$ be an [[∞-Lie algebroid]] (for instance a [[Lie algebra]], or a [[Lie algebroid]] or an [[L-∞-algebra]]). Write \begin{displaymath} CE(\mathfrak{a}) \in dgAlg \end{displaymath} for its [[Chevalley-Eilenberg algebra]], a [[dg-algebra]]. Notice that, by the discussion at \emph{[[L-∞ algebra]]} and at \emph{[[∞-Lie algebroid]]}, the Chevalley-Eilenberg dg-algebras $CE(\mathfrak{a})$ is the [[formal dual]] of $\mathfrak{a}$, in that the [[functor]] \begin{displaymath} CE \colon \infty LieAlgd \hookrightarrow dgAlg^{op} \end{displaymath} is a [[fully faithful functor]]. Indeed, the following definition of Lie integration (being just a smooth refinement of the [[Sullivan construction]]) makes sense just as well for any dg-algebra, not necessarily in the [[essential image]] of this embedding. But only for dg-algebras in the essential image of this embedding do the examples come out as expected for [[higher Lie theory]]. An [[∞-Lie algebra]] is equivalently a [[pointed object|pointed]] [[∞-Lie algebroid]] whose base space is the point. We write $\mathfrak{b}\mathfrak{g} \in \infty Lie Algd$ for objects of this form (``[[delooping]]'') \begin{displaymath} \itexarray{ \infty LieAlgd &\stackrel{CE}{\hookrightarrow}& dgAlg^{op} \\ \uparrow^{\mathrlap{b}} & \nearrow_{\mathrlap{CE}} \\ \infty LieAlg \,. } \end{displaymath} Notice that this induces some degree shifts that may be a little ambiguous in situations like the [[line Lie n-algebra]]: as an [[L-∞ algebra]] this is $b^{n-1}\mathbb{R}$, the corresponding [[∞-Lie algebroid]] is $b^n \mathbb{R}$. \hypertarget{HigherPathsInAnInfinityLieAlgebroid}{}\subsubsection*{{Higher dimensional paths in an $\infty$-Lie algebroid}}\label{HigherPathsInAnInfinityLieAlgebroid} \begin{example} \label{}\hypertarget{}{} For $X$ a [[smooth manifold]] (possibly [[nLab:manifold with boundary|with boundary]] and [[manifold with corners|with corners]]) then its [[tangent Lie algebroid]] $T X$ is the one whose [[Chevalley-Eilenberg algebra]] is the [[de Rham complex]] \begin{displaymath} CE(T X) = (\Omega^\bullet(X), d_{dR}) \,. \end{displaymath} \end{example} \begin{defn} \label{kPath}\hypertarget{kPath}{} For $k \in \mathbb{N}$ write $\Delta^k$ for the standard $k$-[[simplex]] regarded as a [[smooth manifold]] ([[nLab:manifold with boundary|with boundary]] and [[manifold with corners|with corners]]). A \textbf{$k$-path} in the $\infty$-Lie algebroid $\mathfrak{a}$ is a morphism of $\infty$-Lie algebroids of the form \begin{displaymath} \Sigma \; \coloneqq \; T \Delta^k \longrightarrow \mathfrak{a} \end{displaymath} from the [[tangent Lie algebroid]] $T \Delta^k$ of the standard smooth $k$-simplex to $\mathfrak{a}$. Dually this is equivalently a [[homomorphism]] of [[dg-algebras]] \begin{displaymath} \Omega^\bullet(\Delta^k) \longleftarrow CE(\mathfrak{a}) \;\colon\; \Sigma^* \end{displaymath} from the [[Chevalley-Eilenberg algebra]] of $\mathfrak{a}$ to the [[de Rham complex]] of $\Delta^d$. \end{defn} See also at \emph{[[differential forms on simplices]]}. \begin{remark} \label{}\hypertarget{}{} A $k$-path in $\mathfrak{a}$, def. \ref{kPath}, is equivalently \begin{enumerate}% \item a \emph{flat} $\mathfrak{a}$-[[infinity-Lie algebroid valued differential form|valued differential form]] on $\Delta^k$; \item a [[Maurer-Cartan element]] in $\Omega^\bullet(\Delta^k)\otimes \mathfrak{a}$. \end{enumerate} \end{remark} The Lie integration of $\mathfrak{a}$ is essentially the [[simplicial object]] whose $k$-cells are the $d$-paths in $\mathfrak{a}$. However, in order for this to be well-behaved, it is possible and useful to restrict to $d$-paths that are sufficiently well-behaved towards the [[boundary]] of the simplex: \begin{defn} \label{}\hypertarget{}{} Regard the smooth simplex $\Delta^k$ as embedded into the [[Cartesian space]] $\mathbb{R}^{k+1}$ in the standard way, and equip $\Delta^k$ with the [[metric space]] structure induced this way. A smooth [[differential form]] $\omega$ on $\Delta^k$ is said to have \textbf{sitting instants} along the boundary if, for every $(r \lt k)$-face $F$ of $\Delta^k$ there is an [[open neighbourhood]] $U_F$ of $F$ in $\Delta^k$ such that $\omega$ restricted to $U$ is constant in the directions perpendicular to the $r$-face on its value restricted to that face. More generally, for any $U \in$ [[CartSp]] a smooth differential form $\omega$ on $U \times\Delta^k$ is said to have sitting instants if there is $0 \lt \epsilon \in \mathbb{R}$ such that for all points $u : * \to U$ the pullback along $(u, \mathrm{Id}) : \Delta^k \to U \times \Delta^k$ is a form with sitting instants on $\epsilon$-[[open neighbourhood|neighbourhood]]s of faces. Smooth forms with sitting instants clearly form a sub-dg-algebra of all smooth forms. We write $\Omega^\bullet_{si}(U \times \Delta^k)$ for this sub-dg-algebra. We write $\Omega_{si,vert}^\bullet(U \times \Delta^k)$ for the further sub-dg-algebra of [[vertical differential form]]s with respect to the projection $p : U \times \Delta^k \to U$, hence the [[coequalizer]] \begin{displaymath} \Omega^\bullet(U) \stackrel{\stackrel{p^*}{\longrightarrow}}{\underset{0}{\longrightarrow}} \Omega^\bullet_{si}(U \times \Delta^k) \to \Omega^\bullet_{si, vert}(U \times \Delta^k) \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} The dimension of the normal direction to a face depends on the dimension of the face: there is one perpendicular direction to a codimension-1 face, and $k$ perpendicular directions to a vertex. \end{remark} \begin{example} \label{}\hypertarget{}{} \begin{itemize}% \item A smooth 0-form (a [[smooth function]]) has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it is constant. A smooth function $f : U \times \Delta^1 \to \mathbb{R}$ is in $\Omega^0_{\mathrm{vert}}(U \times \Delta^1)$ if there is $0 \lt \epsilon \in \mathbb{R}$ such that for each $u \in U$ the function $f(u,-) : \Delta^1 \simeq [0,1] \to \mathbb{R}$ is constant on $[0,\epsilon) \coprod (1-\epsilon,1)$. \item A smooth 1-form has sitting instants on $\Delta^1$ if in a neighbourhood of the endpoints it vanishes. \item Let $X$ be a [[smooth manifold]], $\omega \in \Omega^\bullet(X)$ be a smooth differential form. Let \begin{displaymath} \phi \colon \Delta^k \to X \end{displaymath} be a [[smooth function]] that has [[sitting instants]] as a function: towards any $j$-face of $\Delta^k$ it eventually becomes perpendicularly constant. Then the [[pullback of differential forms|pullback]] form $\phi^* \omega \in \Omega^\bullet(\Delta^k)$ is a form with sitting instants. \end{itemize} \end{example} \begin{remark} \label{}\hypertarget{}{} The condition of sitting instants serves to make smooth differential forms not be affected by the boundaries and corners of $\Delta^k$. Notably for $\omega_j \in \Omega^\bullet(\Delta^{k-1})$ a collection of forms with sitting instants on the $(k-1)$-cells of a [[horn]] $\Lambda^k_i$ that coincide on adjacent boundaries, and for \begin{displaymath} p \colon \Delta^k \to \Lambda^{k-1}_i \end{displaymath} a standard piecewise smooth [[retract]], the [[pullback of differential forms|pullbacks]] \begin{displaymath} p^* \omega_i \end{displaymath} glue to a single smooth differential form (with sitting instants) on $\Delta^k$. \end{remark} \begin{remark} \label{}\hypertarget{}{} That $\omega \in \Omega^\bullet(\Delta^k)$ having sitting instants does not imply that there is a neighbourhood of the boundary of $\Delta^k$ on which $\omega$ is entirely constant. It is important for the following constructions that in the vicinity of the boundary $\omega$ is allowed to vary parallel to the boundary, just not perpendicular to it. \end{remark} \hypertarget{IntToBareGrpd}{}\subsubsection*{{Integration to a discrete $\infty$-groupoid}}\label{IntToBareGrpd} Here we discuss the [[discrete ∞-groupoid]]s underlying the [[smooth ∞-groupoid]]s to which an [[∞-Lie algebroid]] integrates. For $\mathfrak{a}$ an $\infty$-Lie algebroid, the $d$-paths in $\mathfrak{a}$ naturally form a [[simplicial set]] as $d$ varies: \begin{displaymath} \begin{aligned} \exp(\mathfrak{a})_{bare} & \coloneqq \left( \cdots Hom_{\infty LieAlgd}(T \Delta^2, \mathfrak{a}) \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Hom_{\infty LieAlgd}(T \Delta^1, \mathfrak{a}) \stackrel{\longrightarrow}{\longrightarrow} Hom_{\infty LieAlgd}(T \Delta^0, \mathfrak{g}) \right) \\ & = ( \cdots Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^2)) \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^1)) \stackrel{\longrightarrow}{\longrightarrow} Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(\Delta^0)) ) \end{aligned} \,. \end{displaymath} (We are indicating only the face maps, not the degeneracy maps, just for notational simplicity). If here instead of smooth differential forms one uses [[polynomial differential forms]] then this is precisely the [[Sullivan construction]] of [[rational homotopy theory]] applied to $CE(\mathfrak{a})$. We next realize [[smooth structure]] on this and hence realize this as an object in [[higher Lie theory]]. \begin{remark} \label{SpuriousHomotopyGroups}\hypertarget{SpuriousHomotopyGroups}{} \textbf{(spurious homotopy groups)} For $\mathfrak{a}$ a [[infinity-Lie algebroid|Lie n-algebroid]] (an $n$-[[truncated]] $\infty$-Lie algebroid) this construction will not yield in general an $n$-[[truncated]] [[∞-groupoid]] $\exp(\mathfrak{a})$. To see this, consider the example (discussed in detail \hyperlink{LieAlgebrasToLieGroups}{below}) that $\mathfrak{a} = \mathfrak{g}$ is an ordinary [[Lie algebra]]. Then $\exp(\mathfrak{g})_n$ is canonically identified with the set of smooth based maps $\Delta^n \to G$ into the simply connected [[Lie group]] that integrates $\mathfrak{g}$ in ordinary [[Lie theory]]. This means that the simplicial [[homotopy group]]s of $\exp(\mathfrak{g})$ are the topological homotopy groups of $G$, which in general (say for $G$ the [[orthogonal group]] or [[unitary group]]) will be non-trivial in arbitrarily higher degree, even though $\mathfrak{g}$ is just a Lie 1-algebra. This phenomenon is well familiar from [[rational homotopy theory]], where a classical theorem asserts that the rational homotopy groups of $\exp(\mathfrak{g})$ are generated from the generators in a [[minimal Sullivan model]] resolution of $\mathfrak{g}$. \end{remark} For the purposes of [[higher Lie theory]] therefore instead one wants to [[truncated|truncate]] $\exp(\mathfrak{g})$ to its $(n+1)$-[[coskeleton]] \begin{displaymath} \mathbf{cosk}_{n+1}\exp(\mathfrak{a})_{bare} \,. \end{displaymath} This divides out [[n-morphisms]] by $(n+1)$-morphisms and forgets all higher higher nontrivial morphisms, hence all higher homotopy groups. $\,$ \hypertarget{SmoothIntegration}{}\subsubsection*{{Integration to a smooth $\infty$-groupoid}}\label{SmoothIntegration} We now discuss Lie integration of $\infty$-Lie algebroids to [[smooth ∞-groupoid]]s, [[presentable (∞,1)-category|presented]] by the [[model structure on simplicial presheaves]] $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ over the [[site]] [[CartSp]]${}_{smooth}$. For the following definition recall the [[presentable (∞,1)-category|presentation]] of [[smooth ∞-groupoids]] by the [[model structure on simplicial presheaves]] over the [[site]] [[CartSp]]${}_{smooth}$. \begin{defn} \label{}\hypertarget{}{} For $\mathfrak{a}$ an [[L-∞ algebra]] of [[finite type]] with [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ define the [[simplicial presheaf]] \begin{displaymath} \exp(\mathfrak{a}) \;\colon\; CartSp_{smooth}^{op} \to sSet \end{displaymath} by \begin{displaymath} \exp(\mathfrak{a}) \;\colon\; (U,[k]) \mapsto Hom_{dgAlg}(CE(\mathfrak{a}), \Omega^\bullet(U \times \Delta^k)_{si,vert}) \,, \end{displaymath} for all $U \in$ [[CartSp]] and $[k] \in \Delta$. \end{defn} \begin{remark} \label{}\hypertarget{}{} Compared to the integration to [[discrete ∞-groupoids]] \hyperlink{IntToBareGrpd}{above} this definition knows about $U$-parametrized \emph{smooth families} of $k$-paths in $\mathfrak{g}$. The underlying [[discrete ∞-groupoid]] is recovered as that of the $\mathbb{R}^0 = *$-parameterized family: \begin{displaymath} \exp(\mathfrak{a}) \colon \mathbb{R}^0 \mapsto \exp(\mathfrak{a})_{disc} \,. \end{displaymath} \end{remark} \begin{prop} \label{}\hypertarget{}{} The objects $\exp(\mathfrak{g})$ are indeed [[Kan complexes]] over each $U \in$ [[CartSp]]. \end{prop} \begin{proof} Observe that the standard [[continuous function|continuous]] [[horn]] [[retracts]] $f : \Delta^k \to \Lambda^k_i$ are [[smooth function|smooth]] away from the preimages of the $(r \lt k)$-faces of $\Lambda[k]^i$. For $\omega \in \Omega^\bullet_{si,vert}(U \times \Lambda[k]^i)$ a differential form with sitting instants on $\epsilon$-neighbourhoods, let therefore $K \subset \partial \Delta^k$ be the set of points of distance $\leq \epsilon$ from any subface. Then we have a smooth function \begin{displaymath} f : \Delta^k \setminus K \to \Lambda^k_i \setminus K \,. \end{displaymath} The [[pullback of differential forms|pullback]] $f^* \omega \in \Omega^\bullet(\Delta^k \setminus K)$ may be extended constantly back to a form with sitting instants on all of $\Delta^k$. The resulting assignment \begin{displaymath} (CE(\mathfrak{g}) \stackrel{A}{\longrightarrow} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i)) \mapsto (CE(\mathfrak{g}) \stackrel{A}{\to} \Omega^\bullet_{si,vert}(U \times \Lambda^k_i) \stackrel{f^*}{\to} \Omega^\bullet_{si,vert}(U \times \Delta^n)) \end{displaymath} provides fillers for all [[horns]] over all $U \in$ [[CartSp]]. \end{proof} \begin{defn} \label{}\hypertarget{}{} Write $\mathbf{cosk}_{n+1} \exp(a)$ for the simplicial presheaf obtained by postcomposing $\exp(\mathfrak{a}) : CartSp^{op} \to sSet$ with the $(n+1)$-[[coskeleton]] [[functor]] $\mathbf{cosk}_{n+1} : sSet \stackrel{tr_n}{\longrightarrow} sSet_{\leq n+1} \stackrel{cosk_{n+1}}{\to} sSet$. \end{defn} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{PropertiesHomotopyGroups}{}\subsubsection*{{Homotopy groups}}\label{PropertiesHomotopyGroups} \ldots{}(\hyperlink{Henriques}{Henriques, theorem 6.4})\ldots{} remark \ref{SpuriousHomotopyGroups}\ldots{} \hypertarget{QuillenAdjunction}{}\subsubsection*{{Quillen adjunction}}\label{QuillenAdjunction} The \hyperlink{SmoothIntegration}{above} construction of Lie integraton to [[smooth ∞-groupoids]] clearly applies to all [[differential graded-commutative algebras]], not necessarily just those which are [[Chevalley-Eilenberg algebras]] of [[L-∞ algebras]]. (but up to [[weak equivalence]], there is no difference). With this generalization, the higher Lie integration extends to a [[Quillen adjunction]] (Prop. \ref{LieIntegrationIsRightQuillenFunctor} below). In order to state this conveniently, we first make more explicit the [[functor]] assigning smooth families of [[smooth differential forms on simplices]] (Def. \ref{SmoothDifferentialFormsOnSimplicesWithSittingInstants} below). $\,$ \begin{defn} \label{SmoothDifferentialFormsOnSimplicesWithSittingInstants}\hypertarget{SmoothDifferentialFormsOnSimplicesWithSittingInstants}{} \textbf{(smooth families of [[smooth differential forms on simplices]] with [[sitting instants]])} For $k \in \mathbb{N}$, write $\Delta^k_{mfd}$ for the [[k-simplex]] canonically regarded as a [[smooth manifold with boundaries and corners]]. For $n \in \mathbb{N}$, regard $\mathbb{R}^n \times \Delta^k_{mfd} \overset{p_1}{to} \mathbb{R}^n$ as the [[trivial bundle|trivial]] [[fiber bundle]] over the [[Cartesian space]] $\mathbb{R}^n$ with [[fiber]] that smooth [[k-simplex]]. Write \begin{displaymath} \Omega^\bullet_{vert}(\mathbb{R}^n \times \Delta^k_{mfd}) \hookrightarrow \Omega^\bullet(\mathbb{R}^n \times \Delta^k_{mfd}) \end{displaymath} for the [[subobject|sub-]][[dgc-algebra]] of the [[de Rham algebra]] on the [[vertical differential forms]] with respect to this bundle structure. Moreover, write \begin{displaymath} \Omega^\bullet_{vert, si}\left(\mathbb{R}^n \times \Delta^k_{mfd}\right) \hookrightarrow \Omega^\bullet_{vert}\left(\mathbb{R}^n \times \Delta^k_{mfd}\right) \end{displaymath} for the further [[subobject|sub-]][[dgc-algebra]] on those [[vertical differential forms]] which have [[sitting instants]] towards the [[boundary]] of the [[k-simplex]]. Via [[pullback of differential forms]] this construction provides a [[functor]] \begin{displaymath} \Omega^\bullet_{vert,si} \;\colon\; CartSp \times \Delta \longrightarrow dgcAlg_{\mathbb{R}, conn}^{op} \end{displaymath} from the [[product category]] of the [[category]] [[CartSp]] of [[Cartesian spaces]] and [[smooth functions]] between them, with the [[simplex category]], to the [[opposite category|opposite]] of the [[category]] of connective [[dgc-algebras]] over the [[real numbers]]. \end{defn} (\hyperlink{FSS12}{FSS 12, Def. 4.2.1}, see \hyperlink{BraunackMayer18}{Braunack-Mayer 18, Def. 3.1.3}) \begin{prop} \label{LieIntegrationIsRightQuillenFunctor}\hypertarget{LieIntegrationIsRightQuillenFunctor}{} \textbf{(Lie integration is [[right Quillen functor]] to [[smooth ∞-groupoids]])} There is a [[Quillen adjunction]] \begin{displaymath} dgcAlg^{op}_{\mathbb{R}, \geq 0, proj} \; \underoverset {\underset{ Spec }{\longrightarrow}} {\overset{ \mathcal{O} }{\longleftarrow}} {\phantom{A}\phantom{{}_{Qu}}\bot_{Qu}\phantom{A}} \; [CartSp^{op},sSet_{Qu}]_{proj,loc} \end{displaymath} between \begin{enumerate}% \item the [[projective local model structure on simplicial presheaves]] over [[CartSp]], regarded as a [[site]] via the [[good open cover]] [[coverage]] (i.e. [[locally presentable (∞,1)-category|presenting]] [[smooth ∞-groupoids]]); \item the [[opposite model category|opposite]] [[projective model structure on connective dgc-algebras]] over the [[real numbers]] \end{enumerate} given by [[nerve and realization]] with respect to the [[functor]] of [[smooth differential forms on simplices]] $CartSp \times \Delta \overset{\Omega^\bullet_{vert,si}}{\longrightarrow} dgcAlg_{\mathbb{R}, conn}^{op}$ from Def. \ref{SmoothDifferentialFormsOnSimplicesWithSittingInstants}: \begin{enumerate}% \item the [[right adjoint]] $Spec$ sends a [[dgc-algebra]] $A \in dgcAlg_{\mathbb{R},\geq 0}$ to the [[simplicial presheaf]] which in degree $k$ is the set of [[dg-algebra]]-[[homomorphism]] form $A$ into the [[dgc-algebras]] of [[smooth differential forms on simplices]] $\Omega^\bullet_{si,vert}(-)$ (Def. \ref{SmoothDifferentialFormsOnSimplicesWithSittingInstants}): \begin{displaymath} Spec(A) \;\colon\; \mathbb{R}^n \times \Delta[k] \;\mapsto\; Hom_{dgcAlg_{\mathbb{R}}} \left( A , \Omega^\bullet_{si, vert}(\mathbb{R}^n \times \Delta^k_{mfd}) \right) \end{displaymath} \item the [[left adjoint]] $\mathcal{O}$ is the [[Yoneda extension]] of the [[functor]] $\Omega^\bullet_{vert,si} \;\colon\; CartSp \times \Delta \to dgcAlg_{\mathbb{R},conn}^{op}$ assigning [[dgc-algebras]] of [[smooth differential forms on simplices]] from Def. \ref{SmoothDifferentialFormsOnSimplicesWithSittingInstants}, hence which acts on a [[simplicial presheaf]] $\mathbf{X} \in [CartSp^{op}, sSet] \simeq [\CartSp^{op} \times \Delta^{op}, Set]$, expanded via the [[co-Yoneda lemma]] as a [[coend]] of [[representable presheaves|representables]], as \begin{displaymath} \mathcal{O} \;\colon\; \mathbf{X} \simeq \int^{n,k} y(\mathbb{R}^n \times \Delta[k]) \times \mathbf{X}(\mathbb{R}^n)_k \;\mapsto\; \int_{n,k} \underset{\mathbf{X}(\mathbb{R}^n)_k}{\prod} \Omega^\bullet_{si,vert}\left(\mathbb{R}^n \times \Delta^k_{mfd}\right) \end{displaymath} \end{enumerate} \end{prop} (\hyperlink{BraunackMayer18}{Braunack-Mayer 18, theorem 3.1.10}) \hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2} See also at \emph{[[smooth ∞-groupoid -- structures]]} the section \emph{\href{smooth+infinity-groupoid+--+structures#StrucLieAlg}{Exponentiated ∞-Lie algebras}}. \hypertarget{LieAlgebrasToLieGroups}{}\subsubsection*{{Interating Lie algebras to Lie groups}}\label{LieAlgebrasToLieGroups} Let $\mathfrak{g} \in L_\infty$ be an ordinary (finite dimensional) [[Lie algebra]]. Standard [[Lie theory]] (see [[Lie's three theorems]]) provides a [[simply connected]] [[Lie group]] $G$ integrating $\mathfrak{g}$. With $G$ regarded as a [[smooth ∞-groupoid|smooth ∞-group]] write $\mathbf{B}G \in$ [[Smooth∞Grpd]] for its [[delooping]]. The standard presentation of this on $[CartSp_{smooth}^{op}, sSet]$ is by the [[simplicial presheaf]] \begin{displaymath} \mathbf{B}G_c \colon U \mapsto N(C^\infty(U,G) \stackrel{\longrightarrow}{\longrightarrow} *) \,. \end{displaymath} See at \emph{\href{smooth+infinity-groupoid+--+structures#LieGroups}{smooth infinity-groupoid -- structures -- Lie groups}} for more details. \begin{prop} \label{ResultOfIntegrationOfOrdinaryLieAlgebra}\hypertarget{ResultOfIntegrationOfOrdinaryLieAlgebra}{} The operation of [[parallel transport]] $P \exp(\int -) : \Omega^1([0,1], \mathfrak{g}) \to G$ yields a weak equivalence (in $[CartSp^{op}, sSet]_{proj}$) \begin{displaymath} P \exp(\int (-) ) \;\colon\; \mathbf{cosk}_3 \exp(\mathfrak{g}) \;\simeq\; \mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G_c \,. \end{displaymath} \end{prop} This follows from the [[Steenrod-Wockel approximation theorem]] and the following observation. \begin{lemma} \label{}\hypertarget{}{} For $X$ a [[simply connected]] [[smooth manifold]] and $x_0 \in X$ a basepoint, there is a canonical [[bijection]] \begin{displaymath} \Omega^1_{flat}(X,\mathfrak{g}) \simeq C^\infty_*(X,G) \end{displaymath} between the set of [[Lie-algebra valued 1-form]]s on $X$ whose [[curvature]] 2-form vanishes, and the set of [[smooth function]]s $X\to G$ that take $x_0$ to the neutral element $e \in G$. \end{lemma} \begin{proof} The bijection is given as follows. For $A \in \Omega^1_{flat}(X,\mathfrak{g})$ a flat 1-form, the corresponding function $f_A : X \to G$ sends $x \in X$ to the [[parallel transport]] along any path $x_0 \to x$ from the base point to $x$ \begin{displaymath} f_A : x \mapsto tra_A(x_0 \to x) \,. \end{displaymath} Because of the assumption that the [[curvature]] 2-form of $A$ vanishes and the assumption that $X$ is [[simply connected]], this assignment is independent of the choice of path. Conversely, for every such function $f : X \to G$ we recover $A$ as the pullback of the [[Maurer-Cartan form]] on $G$ \begin{displaymath} A = f^* \theta \,. \end{displaymath} \end{proof} From this we obtain \begin{proof} The $\infty$-groupoid $\mathbf{cosk}_2 \exp(\mathfrak{g})$ is equivalent to the [[groupoid]] with a single object (no non-trivial 1-form on the point) whose morphisms are equivalence classes of smooth based paths $\Delta^1 \to G$ (with sitting instants), where two of these are taken to be equivalent if there is a smooth [[homotopy]] $D^2 \to G$ (with sitting instant) between them. Since $G$ is [[simply connected]], these equivalence classes are labeled by the endpoints of these paths, hence are canonically identified with $G$. \end{proof} \begin{remark} \label{}\hypertarget{}{} We do not need to fall back to classical [[Lie theory]] to obtain $G$ in the above argument. A detailed discussion of how to find $G$ with its group structure and smooth structure from $d$-paths in $\mathfrak{g}$ is in (\hyperlink{Crainic}{Crainic}). \end{remark} \hypertarget{IntegrationToLineNGroup}{}\subsubsection*{{Integrating to line/circle Lie $n$-groups}}\label{IntegrationToLineNGroup} \begin{defn} \label{}\hypertarget{}{} For $n \in \mathbb{N}, n \geq 1$ write $b^{n-1} \mathbb{R}$ for the [[L-∞-algebra]] whose [[Chevalley-Eilenberg algebra]] is given by a single generator in degree $n$ and vanishing differential. We may call this the \textbf{[[line Lie n-algebra]]}. Write $\mathbf{B}^{n} \mathbb{R}$ for the . \end{defn} \begin{prop} \label{}\hypertarget{}{} The [[discrete ∞-groupoid]] underlying $\exp(b^{n-1} \mathbb{R})$ is given by the [[Kan complex]] that in degree $k$ has the set of closed differential $n$-forms (with sitting instants) on the $k$-[[simplex]] \begin{displaymath} \exp(b^{n-1} \mathbb{R})_{disc} : [k] \mapsto \Omega^n_{si,cl}(\Delta^k) \end{displaymath} \end{prop} \begin{prop} \label{LieIntegrationOfLinenLieAlgebra}\hypertarget{LieIntegrationOfLinenLieAlgebra}{} The $\infty$-Lie integration of $b^{n-1} \mathbb{R}$ is the [[circle n-group]] $\mathbf{B}^{n} \mathbb{R}$. Moreover, with $\mathbf{B}^n \mathbb{R}_{chn} \in [CartSp_{smooth}^{op}, sSet]$ the standard presentation given under the [[Dold-Kan correspondence]] by the chain complex of sheaves concentrated in degree $n$ on $C^\infty(-, \mathbb{R})$ the equivalence is induced by the [[fiber integration]] of differential $n$-forms over the $n$-[[simplex]]: \begin{displaymath} \int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \stackrel{\simeq}{\longrightarrow} \mathbf{B}^{n} \mathbb{R}_{chn} \,. \end{displaymath} \end{prop} \begin{proof} First we observe that the map \begin{displaymath} \int_{\Delta^\bullet} : (\omega \in \Omega^n_{si,vert,cl}(U \times \Delta^k)) \mapsto \int_{\Delta^k} \omega \in C^\infty(U, \mathbb{R}) \end{displaymath} is a morphism of [[simplicial presheaves]] $\exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ on [[CartSp]]${}_{smooth}$. Since it goes between presheaves of abelian [[simplicial group]]s by the [[Dold-Kan correspondence]] it is sufficient to check that we have a morphism of [[chain complex]]es of presheaves on the corresponding [[Moore complex|normalized chain complex]]es. The only nontrivial degree to check is degree $n$. Let $\lambda \in \Omega_{si,vert,cl}^n(\Delta^{n+1})$. The differential of the [[Moore complex|normalized chains complex]] sends this to the signed sum of its restrictions to the $n$-faces of the $(n+1)$-simplex. Followed by the integral over $\Delta^n$ this is the piecewise integral of $\lambda$ over the boundary of the $n$-simplex. Since $\lambda$ has sitting instants, there is $0 \lt \epsilon \in \mathbb{R}$ such that there are no contributions to this integral in an $\epsilon$-neighbourhood of the $(n-1)$-faces. Accordingly the integral is equivalently that over the smooth surface inscribed into the $(n+1)$-simplex, as indicated in the following diagram Since $\lambda$ is a closed form on the $n$-simplex, this surface integral vanishes, by the [[Stokes theorem]]. Hence $\int_{\Delta^\bullet}$ is indeed a chain map. It remains to show that $\int_{\Delta^\bullet} : \exp(b^{n-1} \mathbb{R}) \to \mathbf{B}^{n}\mathbb{R}_{chn}$ is an [[isomorphism]] on all the [[simplicial homotopy]]s group over each $U \in CartSp$. This amounts to the statement that \begin{enumerate}% \item a smooth family of closed $n$-forms with sitting instants on the boundary of $\Delta^{n+1}$ may be extended to a smooth family of closed forms with sitting instants on $\Delta^{n+1}$ precisely if their smooth family of integrals over the boundary vanishes; \item Any smooth family of closed $n \lt k$-forms with sitting instants on the boundary of $\Delta^{k+1}$ may be extended to a smooth family of closed $n$-forms with sitting instants on $\Delta^{k+1}$. \end{enumerate} To demonstrate this, we want to work with forms on the $(k+1)$-[[ball]] instead of the $(k+1)$-[[simplex]]. To achieve this, choose again $0 \lt \epsilon \in \mathbb{R}$ and construct the [[diffeomorphism|diffeomorphic]] image of $S^k \times [1,1-\epsilon]$ inside the $(k+1)$-simplex as indicated in the above diagram: outside an $\epsilon$-neighbourhood of the corners the image is a rectangular $\epsilon$-thickening of the faces of the simplex. Inside the $\epsilon$-neighbourhoods of the corners it bends smoothly. By the [[Steenrod-Wockel approximation theorem]] the diffeomorphism from this $\epsilon$-thickening of the smoothed boundary of the simplex to $S^k \times [1-\epsilon,1]$ extends to a smooth function from the $(k+1)$-simplex to the $(k+1)$-ball. By choosing $\epsilon$ smaller than each of the sitting instants of the given $n$-form on $\partial \Delta^{k+1}$, we have that this $n$-form vanishes on the $\epsilon$-neighbourhoods of the corners and is hence entirely determined by its restriction to the smoothed simplex, identified with the $(k+1)$-ball. It is now sufficient to show: a smooth family of smooth $n$-forms $\omega \in \Omega^n_{vert,cl}(U \times S^k)$ extends to a smooth family of closed $n$-forms $\hat \omega \in \Omega^n_{vert,cl}(U \times B^{k+1})$ that is radially constant in a neighbourhood of the boundary for all $n \lt k$ and for $k = n$ precisely if its smooth family of integrals vanishes, $\int_{S^k} \omega = 0 \in C^\infty(U, \mathbb{R})$. Notice that over the point this is a direct consequence of the [[de Rham theorem]] ([[kernel of integration is the exact differential forms]]): an $n$-form $\omega$ on $S^k$ is exact precisely if $n \lt k$ or if $n = k$ and its integral vanishes. In that case there is an $(n-1)$-form $A$ with $\omega = d A$. Choosing any smoothing function $f : [0,1] \to [0,1]$ (smooth, surjective, non-decreasing and constant in a neighbourhood of the boundary) we obtain an $n$-form $f \wedge A$ on $(0,1] \times S^k$, vertically constant in a neighbourhood of the ends of the interval, equal to $A$ at the top and vanishing at the bottom. Pushed forward along the canonical $(0,1] \times S^k \to D^{k+1}$ this defines a form on the $(k+1)$-ball, that we denote by the same symbol $f \wedge A$. Then the form $\hat \omega := d (f \wedge A)$ solves the problem. To complete the proof we have to show that this simple argument does extend to smooth families of forms, i.e., that we can choose the $(n-1)$-form $A$ in a way depending smoothly on the the $n$-form $\omega$. One way of achieving this is using [[Hodge theory]]. Fix a [[Riemannian metric]] on $S^n$, and let $\Delta$ be the corresponding [[Laplace operator]], and $\pi$ the projection on the space of [[harmonic forms]]. Then the for [[compact topological space|compact]] [[Riemannian manifold]]s states that the operator $\pi$, seen as an operator from the [[de Rham complex]] to itself, is a cochain map [[homotopy|homotopic]] to the identity, via an explicit homotopy $P := d^* G$ expressed in terms of the adjoint $d^*$ of the de Rham differential and of the [[Green operator]] $G$ of $\Delta$. Since the $k$-form $\omega$ is exact its projection on [[harmonic form]]s vanishes. Therefore \begin{displaymath} \begin{aligned} \omega & = (Id-\pi)\omega \\ & = d (P\omega)+P (d\omega) \\ & = d (P\omega). \end{aligned} \end{displaymath} Hence $A := P\omega$ is a solution of the differential equation $d A=\omega$ depending smoothly on $\omega$. \end{proof} \hypertarget{integrating_the_string_lie_2algebra_to_the_string_lie_2group}{}\subsubsection*{{Integrating the string Lie 2-algebra to the string Lie 2-group}}\label{integrating_the_string_lie_2algebra_to_the_string_lie_2group} Let $\mathfrak{string} = \mathfrak{g}_\mu$ be the [[string Lie 2-algebra]]. Then $\mathbf{cosk}_3 \exp(\mathfrak{g}_\mu)$ is equivalent to the [[2-groupoid]] $\mathbf{B}String$ \begin{itemize}% \item with a single object; \item whose morphisms are based paths in $G$; \item whose 2-morphisms are equivalence class of pairs $(\Sigma,c)$, where \begin{itemize}% \item $\Sigma : D^2_* \to G$ is a smooth based map (where we use a [[homeomorphism]] $D^2 \simeq \Delta^2$ which away from the corners is smooth, so that forms with sitting instants there do not see any non-smoothness, and the basepoint of $D^2_*$ is the 0-vertex of $\Delta^2$) \item and $c \in U(1)$, and where two such are equivalent if the maps coincides at their boundary and if for any 3-ball $\phi : D^3 \to G$ filling them the labels $c_1, c_2 \in U(1)$ differ by the integral $\int_{D^3} \phi^* \mu(\theta) \;\; mod \;\; \mathbb{Z}$,, \end{itemize} \end{itemize} where $\theta$ is the [[Maurer-Cartan form]], $\mu(\theta) = \langle \theta\wedge [\theta \wedge \theta]\rangle$ the 3-form obtained by plugging it into the cocycle. This is the [[string Lie 2-group]]. It's construction in terms of integration by paths is due to (\hyperlink{Henriques}{Henriques}) \hypertarget{integrating_lie_algebroids_to_stacky_lie_groupoids}{}\subsubsection*{{Integrating Lie algebroids to (stacky) Lie groupoids}}\label{integrating_lie_algebroids_to_stacky_lie_groupoids} Unlike finite dimensional Lie algebras, not every Lie algebroid may be integrated to a Lie groupoid. There exists topological obstruction coming from $\pi_2(L)$ of the leaf $L\subset M$ of a Lie algebroid $A\to M$. The integrability criteria is completely classified through the behaviour of certain monodromy groups and that is the achievement of \hyperlink{Crainic}{Crainic-Fernandes 01}. These monodromy groups, providing obstructions of integration, may be seen as the image of a transgression map of the long exact sequence of [[Lie algebroid homotopy groups]] induced by the natural [[Lie algebroid fibration]] $A|_L \to L$. (see \hyperlink{BZ}{Brahic-Zhu 10} ). Let us now describe the construction of the universal groupoid for a Lie algebroid $A$. This is a major step contained in \hyperlink{Crainic}{Crainic-Fernandes 01} and was earlier discovered in the setting of Poisson manifolds as the phase space of [[Poisson sigma model]] in \hyperlink{CattaneoFelder01}{Cattaneo-Felder 00}. Given a Lie algebroid $A\to M$, an $A$-path (a Lie algebroid path), is a Lie algebroid morphism from $T\Delta^1 \to A$, that is, it is a path $a$ in $A$ such that $a=\rho(\gamma)$ where $\gamma=\pi(a)$ is the base path of $a$ (see \href{https://ncatlab.org/nlab/show/Lie+groupoid#example}{example} ). An (end-fixing) $A$-homotopy (a Lie algebroid homotopy) is a Lie algebroid morphism from $T\square \to A$ satisfying certain boundary conditions. Let us be more precise: a vector bundle morphism $T\square \to A$ can be denoted by $a dt + b ds$ (see \href{https://ncatlab.org/nlab/show/Lie+groupoid#example_2}{example} ). Then the boundary condition is that $b(s,0)=0$ and $b(s,1)=0$. The fact that $adt+bds$ is a Lie algebroid morphism is equivalent to the following PDE: \begin{displaymath} \partial_t b- \partial_s a= \nabla_{\rho \alpha} \beta - \nabla_{\rho \beta} \alpha +[\alpha, \beta] \end{displaymath} where $\nabla$ is a TM connection on $A$ and $\alpha$ and $\beta$ are certain time dependent sections of $A$ extending $a$ and $b$ respectively. Notice that there is also a way writing down the right hand side independent of choice of sections. See Section 1 of \hyperlink{Crainic}{Crainic-Fernandes 01}. Then the universal groupoid associated to $A$ is the space of $A$-paths ($P_a A$) dividing by $A$-homotopies. It is naturally a topological groupoid. But only when the obstruction vanishes, it becomes a Lie groupoid and is the source-simply connected Lie groupoid integrating $A$. Fortunately $A$-homotopies form finite codimensional foliation $F$, even though the quotient might not be always representible. Thus, $Mon_F(P_aA)$ represents a differentiable stack which becomes a stacky Lie groupoid over $M$. It turns out that there is a one-to-one correspondence between 'etale stacky Lie groupoid and Lie algebroid. This correspondence provides a positive answer to [[Lie's third theorem]] for Lie algebroids. \hyperlink{TZ}{Tseng-Zhu 04}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Kan-fibrant simplicial manifold]] \item [[parallel transport]], [[higher parallel transport]] \item [[holonomy]], [[higher holonomy]] \begin{itemize}% \item [[nonabelian Stokes theorem]] \end{itemize} \item [[dg-nerve]] \item [[function algebras on infinity-stacks]] \end{itemize} [[!include infinitesimal and local - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} The ``path method'' of integrating Lie algebras to simply connected Lie groups appears in \begin{itemize}% \item [[Hans Duistermaat]], J. A. C. Kolk, section 1.14 of \emph{Lie groups}, 2000 \end{itemize} The idea of identifying the [[Sullivan construction]] applied to [[Chevalley-Eilenberg algebras]] as Lie integration to [[discrete ∞-groupoid]]s appears in \begin{itemize}% \item [[Vladimir Hinich]], \emph{Descent of Deligne groupoids}, Internat. Math. Res. Notices, 5:223--239, 1997 (\href{http://dx.doi.org/10.1155/S1073792897000160}{doi}, \href{http://math.haifa.ac.il/hinich/WEB/mypapers/ddg.pdf}{preprint ddg.pdf}) \end{itemize} and for general [[∞-Lie algebra]]s in \begin{itemize}% \item [[Ezra Getzler]], \emph{Lie theory for nilpotent $L_\infty$ algebras}, Annals of Mathematics, 170 (2009), 271--301 (\href{http://arxiv.org/abs/math/0404003}{math.AT/0404003}) \end{itemize} (whose main point is the discussion of a gauge condition applicable for nilpotent $L_\infty$-algebras that cuts down the result of the Sullivan construction to a much smaller but equivalent model) . This was refined from integration to bare $\infty$-groupoids to an integration to [[internal ∞-groupoid]]s in [[Banach manifold]]s in \begin{itemize}% \item [[André Henriques]], \emph{Integrating $L_\infty$ algebras}, Compos. Math. \textbf{144} (2008), no. 4, 1017--1045 (\href{http://dx.doi.org/10.1112/S0010437X07003405}{doi},\href{http://arxiv.org/abs/math.AT/0603563}{math.AT/0603563}) \end{itemize} (whose origin possibly preceeds that of Getzler's article). For general [[∞-Lie algebroids]] the general idea of the integration process by ``$d$-paths'' had been indicated in \begin{itemize}% \item [[Pavol Severa]], \emph{[[Some title containing the words ``homotopy'' and ``symplectic'', e.g. this one]]}, based on a talk at \emph{\href{http://www.lpthe.jussieu.fr/~dito/poissongeometry/Poisson2000/index.html}{Poisson 2000}}, \href{https://www.cirm-math.fr/}{CIRM Marseille Luminy}, June 2000 (\href{http://arxiv.org/abs/math/0105080}{arXiv:0105080}) \item Pavol \v{S}evera, Michal \v{S}ira, \emph{Integration of differential graded manifolds}, \href{http://arxiv.org/abs/1506.04898}{arxiv/1506.04898} \end{itemize} Lie integration of [[dg-modules]] to [[smooth spectrum|smooth]] [[parameterized spectra]] ([[twisted cohomology|twisted]] [[differential cohomology theories]]); \begin{itemize}% \item [[Vincent Braunack-Mayer]], section 3.1 of \emph{[[schreiber:thesis Braunack-Mayer|Rational parameterized stable homotopy theory]]}, Zurich 2018 (\href{https://ncatlab.org/schreiber/show/thesis+Braunack-Mayer#pdf}{pdf}) \end{itemize} Discussion of Lie integration of [[Lie algebroids]] by the path method is due to \begin{itemize}% \item [[Marius Crainic]], [[Rui Fernandes]], \emph{Integrability of Lie brackets}, Ann. of Math. (2), Vol. 157 (2003), no. 2, 575--620 (\href{http://arxiv.org/abs/math/0105033}{arXiv:math.DG/0105033}) \end{itemize} following (\hyperlink{DuistermaatKolk00}{Duistermaat-Kolk 00, section 1.14}) and following the discussion of the special case of Lie integration of [[Poisson Lie algebroids]] to [[symplectic groupoids]] in \begin{itemize}% \item [[Alberto Cattaneo]], [[Giovanni Felder]], \emph{Poisson sigma models and symplectic groupoids}, in \emph{Quantization of Singular Symplectic Quotients}, (ed. [[Klaas Landsman]], M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198 (Birkh\"a{}user, 2001), 61--93. (\href{http://arxiv.org/abs/math/0003023}{arXiv:math/0003023}) \end{itemize} upgraded to the stacky version by \begin{itemize}% \item Tseng Hsiang-Hua, [[Chenchang Zhu]], \emph{Integrating Lie algebroids via stacks}, Compositio Mathematica, Volume 142 (2006), Issue 01, pp 251-270, \href{http://arxiv.org/abs/math/0405003}{arXiv:math/0405003}. \end{itemize} A general proof that equivalent $L_\infty$-algebras integrate to equivalent Lie $\infty$-groupoids is in \begin{itemize}% \item [[Christopher Rogers]], [[Chenchang Zhu]], \emph{On the homotopy theory for Lie ∞-groupoids, with an application to integrating L∞-algebras} (\href{https://arxiv.org/abs/1609.01394}{arXiv:1609.01394}) \end{itemize} Obstruction interpreted as transgression of a Lie algebroid fibration by \begin{itemize}% \item Olivier Brahic, [[Chenchang Zhu]], \emph{Lie algebroid fibrations}, Adv. Math. 226 (2011), no. 4, 3105--3135, \href{http://arxiv.org/abs/1001.4904}{arXiv:1001.4904}. \end{itemize} A description of Lie integration with values in [[smooth ∞-groupoids]] regarded as [[simplicial presheaves]] on [[CartSp]] (and further the Lie integration of [[infinity-Lie algebra cohomology|L-infinity cocycles]]) is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Cech Cocycles for Differential characteristic Classes]]}, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (\href{http://arxiv.org/abs/1011.4735}{arXiv:1011.4735}) \end{itemize} Essentially the same integration prescription is considered in \begin{itemize}% \item [[Dmitry Roytenberg]], \emph{Differential graded manifolds and associated stacks: an overview} (\href{https://sites.google.com/site/dmitryroytenberg/unpublished/dg-stacks-overview.pdf}{pdf}) \end{itemize} The Lie integration- of [[Lie infinity-algebroid representation|Lie algebroid representations]] $\mathfrak{a} \to end(V)$ to morphisms of [[∞-categories]] $A \to Ch_\bullet^\circ$ / [[higher parallel transport]] is discussed in \begin{itemize}% \item [[Camilo Arias Abad]], [[Florian Schätz]], \emph{The $A_\infty$ de Rham theorem and integration of representations up to homotopy} (\href{http://arxiv.org/abs/1011.4693}{arXiv:1011.4693}) \end{itemize} Application to the problem of Lie integrating ordinary but infinite-dimensional Lie algebras is in \begin{itemize}% \item [[Christoph Wockel]], [[Chenchang Zhu]], \emph{Integrating central extensions of Lie algebras via Lie 2-groups} (\href{http://arxiv.org/abs/1204.5583}{arXiv:1204.5583}) \end{itemize} A generalization of [[Lie integration]] to conjectural Leibniz groups has been conjectured by [[J-L. Loday]]. A local version via local Lie [[rack]]s has been proposed in \begin{itemize}% \item Simon Covez, \emph{The local integration of Leibniz algebras}, \href{http://arxiv.org/abs/1011.4112}{arXiv:1011.4112}; \emph{On the conjectural cohomology for groups}, \href{http://arxiv.org/abs/1202.2269}{arXiv:1202.2269}; \emph{L'int\'e{}gration locale des alg\`e{}bres de Leibniz}, Thesis (2010), \href{http://tel.archives-ouvertes.fr/docs/00/49/54/69/PDF/THESE_Simon_Covez.pdf}{pdf} \end{itemize} [[!redirects Lie integrations]] [[!redirects higher Lie integration]] [[!redirects higher Lie integrations]] category: Lie theory \end{document}