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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*{foliation} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topology}{}\paragraph*{{Topology}}\label{topology} [[!include topology - contents]] \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{higher_lie_theory}{}\paragraph*{{Higher Lie theory}}\label{higher_lie_theory} [[!include infinity-Lie 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{original_definition}{Original definition}\dotfill \pageref*{original_definition} \linebreak \noindent\hyperlink{alternative_definitions}{Alternative definitions}\dotfill \pageref*{alternative_definitions} \linebreak \noindent\hyperlink{InTermsOfLieAlgebroidsAndLieGroupoids}{In terms of Lie algebroids and Lie groupoids}\dotfill \pageref*{InTermsOfLieAlgebroidsAndLieGroupoids} \linebreak \noindent\hyperlink{of_higher_smooth_spaces}{Of higher smooth spaces}\dotfill \pageref*{of_higher_smooth_spaces} \linebreak \noindent\hyperlink{InCohesiveHigherGeometry}{In terms of differential cohesive higher geometry}\dotfill \pageref*{InCohesiveHigherGeometry} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{leaf_space}{Leaf space}\dotfill \pageref*{leaf_space} \linebreak \noindent\hyperlink{classification}{Classification}\dotfill \pageref*{classification} \linebreak \noindent\hyperlink{characteristic_classes}{Characteristic classes}\dotfill \pageref*{characteristic_classes} \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} A \emph{foliation} of a [[manifold]] $X$ is a decomposition into [[submanifolds]]. These submanifolds are called the \emph{leaves} of the foliation and one says that $X$ is \emph{foliated by the leaves}. In order to have a useful notion, leaves are required to behave sufficiently well locally. In particular, if all leaves have the same [[dimension]], then one speaks of a \emph{regular foliation}, which is the case discussed here. If the dimension of leaves is allowed to vary, one speaks instead a \emph{[[singular foliation]]}; see there for more details. For [[smooth manifolds]], smooth foliations are decompositions into [[immersion|immersed]] [[submanifolds]] that locally can be expressed as [[fibers]] of a [[submersion]] (the projection to the [[space of leaves]]). Historically (as they were introduced in \hyperlink{Ehresmann}{Ehresmann} and \hyperlink{Reeb}{Reeb}), one considered foliations for [[smooth manifolds]] $X$ as arising from subbundles of the [[tangent bundle]] $E \hookrightarrow T X$ that are [[integrable distributions]] in the sense that the [[Lie bracket]] of [[vector fields]] that are [[sections]] of $E$ is again a section of $E$: the leaves are the submanifolds whose tangent vectors are sections of $E$. If one thinks of $E$ as encoding a [[differential equation]], then the leaves are the solution spaces to this equation. Expressed in terms of [[higher Lie theory]] such an integrable distribution is a sub-[[Lie algebroid]] of the [[tangent Lie algebroid]] of $X$. Accordingly, under [[Lie integration]] of this structure foliations of $X$ are also equivalently encodes as [[Lie groupoids]] whose space of objects is $X$ and whose [[orbits]] are the leaves of the foliation. Moreover, foliations are classified by [[Cech cohomology]] [[cocycles]] with coefficients in a [[topological groupoid]]/[[Lie groupoid]] called the \emph{[[Haefliger groupoid]]}. These relations make foliation theory of sub-topic of [[Lie groupoid]]-theory. See also at \emph{[[motivation for higher differential geometry]]}. The Haefliger groupoids in fact classifies structures slightly more general than foliations: \emph{[[Haefliger structures]]}. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} There are several equivalent definitions of foliations. \hypertarget{original_definition}{}\subsubsection*{{Original definition}}\label{original_definition} Let $M$ be an $n$-[[dimension|dimensional]] [[topological manifold]]. A decomposition of $M$ as a [[disjoint union]] of [[connected topological space|connected]] subsets $V_\alpha$, called \textbf{leaves}, \begin{displaymath} M = \cup_\alpha V_\alpha \end{displaymath} is called a \textbf{foliation} if there is a [[cover]] of $M$ by a collection of ``special'' [[charts]] of the form $(U, \phi)$, $\phi = (\phi_1,\ldots,\phi_n) : U \to \mathbb{R}^n$ such that for each ``special'' chart and each $\alpha$ there is a number $p\leq n$, called the \emph{dimension of the foliation}, such that the intersection of any given leaf $V_\alpha$ with $U$ is one of the level sets, i.e. the solution of the system $\phi_r(x) = const = const(r,U,\alpha)$ for all $r\gt p$. If the manifold is a [[smooth manifold]], the charts may be required to be smooth too, to obtain the notion of a \emph{smooth foliation} or \emph{foliation in [[differential geometry]]}. In this case, the $p$-dimensional foliations with underlying manifold $X$ are in 1-1 correspondence with [[integrable distributions]] of hyperplanes of dimension $p$ in the [[tangent bundle]] of $X$. \hypertarget{alternative_definitions}{}\subsubsection*{{Alternative definitions}}\label{alternative_definitions} The following equivalent definitions and their relation are discussed for instance in (\hyperlink{MoerdijkMrcun}{IfLg, 1.2}). \begin{defn} \label{}\hypertarget{}{} A \textbf{foliation atlas} of a [[manifold]] $X$ of [[dimension]] $n$ and leaf-[[codimension]] $q$ is an [[atlas]] $\{\phi_i^{-1}: R^n \to X\}_i$ such that the transition functions are globally of the form \begin{displaymath} \phi_{i j} : (x,y) \mapsto (g_{i j}(x,y), h_{i j}(y)) \end{displaymath} with respect to the canonical decomposition $\mathbb{R}^n = \mathbb{R}^{n-q} \times \mathbb{R}^q$. \begin{displaymath} \itexarray{ \mathbb{R}^n &\stackrel{\phi_{i j}}{\to}& \mathbb{R}^n && \\ \downarrow && \downarrow \\ \mathbb{R}^q &\stackrel{h_{i j}}{\to}& \mathbb{R}^q } \,. \end{displaymath} \end{defn} \begin{defn} \label{FoliationAtlasByLocalSubmersions}\hypertarget{FoliationAtlasByLocalSubmersions}{} A \textbf{foliation atlas} of a [[manifold]] $X$ of [[dimension]] $n$ and leaf-[[codimension]] $q$ is an [[open cover]] $\{U_i \to X\}_i$ of $X$ equipped with [[submersions]] $\{ s_i \colon U_i \to \mathbb{R}^q \}$ such that there exists [[diffeomorphisms]] \begin{displaymath} \gamma_{i j} \colon s_j(U_i \cap U_j) \to s_i(U_i \cap U_j) \end{displaymath} satisfying on each $U_i \cap U_j$ the condition \begin{displaymath} s_i = \gamma_{i j} \circ s_j \,. \end{displaymath} \end{defn} \begin{remark} \label{}\hypertarget{}{} Given a foliation atlas as in def. \ref{FoliationAtlasByLocalSubmersions}, the [[diffeomorphisms]] $\{\gamma_{i j}\}_{i,j}$ satisfy the [[Cech cohomology|Cech]] [[cocycle]] condition \begin{displaymath} \gamma_{i j} \circ \gamma_{j k} = \gamma_{i k} \,. \end{displaymath} This is called the \textbf{[[Haefliger cocycle]]} of the foliation atlas. \end{remark} \begin{defn} \label{AsIntegrableDistribution}\hypertarget{AsIntegrableDistribution}{} A smooth foliation of a [[smooth manifold]] $X$ is equivalently an [[integrable distribution]] (or an integrable subbundle) $E \hookrightarrow T X$. \end{defn} \hypertarget{InTermsOfLieAlgebroidsAndLieGroupoids}{}\subsubsection*{{In terms of Lie algebroids and Lie groupoids}}\label{InTermsOfLieAlgebroidsAndLieGroupoids} Definition \ref{AsIntegrableDistribution} above is immediately reformulated equivalently as the following statement in [[higher Lie theory]]. \begin{defn} \label{AsLieAlgebroidsWithInjectiveAnchor}\hypertarget{AsLieAlgebroidsWithInjectiveAnchor}{} For $X$ a [[smooth manifold]], a foliation of $X$ is equivalently a [[Lie algebroid]] over $X$ such that the [[anchor map]] is an [[injection]]. \end{defn} \begin{remark} \label{}\hypertarget{}{} The [[Lie groupoids]] which under [[Lie differentiation]] give rise to Lie algebroids with injective anchors as in def. \ref{AsLieAlgebroidsWithInjectiveAnchor} are precisely those which are Morita-equivalent to [[étale groupoids]] (hence are the \emph{[[foliation groupoids]]}, see there for more details) (\hyperlink{CrainicMoerdijk}{Crainic-Moerdijk 00, theorem 1}). \end{remark} One says: \begin{defn} \label{}\hypertarget{}{} A [[Lie groupoid]] \textbf{integrates} a given foliation, if it [[Lie integration|Lie integrates]] the coresponding [[Lie algebroid]], according to def. \ref{AsLieAlgebroidsWithInjectiveAnchor}. \end{defn} \begin{example} \label{}\hypertarget{}{} For a [[simple foliation]] $\mathcal{D}$ of a manifold $X$, example \ref{SimpleFoliation}, hence one where there is a [[submersion]] \begin{displaymath} p_{\mathcal{D}} \;\colon\; X \to X/\mathcal{D} \end{displaymath} to the leaf space, that map itself is the [[atlas]] of a Lie groupoid $\mathcal{G}$ which integrates the foliation, which is the [[Cech nerve]] \begin{displaymath} \mathcal{G}_\bullet = \left( X \underset{X/\mathcal{D}}{\times} X \stackrel{\to}{\to} X/\mathcal{D} \right) \,. \end{displaymath} \end{example} \begin{example} \label{}\hypertarget{}{} Among all Lie groupoids that integrate a given foliation $\mathcal{F}$ of a manifold $X$, the two special extreme \begin{enumerate}% \item [[holonomy groupoid]] $Hol(X,\mathcal{F})_\bullet$ \item [[monodromy groupoid]] $Monod(X,\mathcal{F})_\bullet$ \end{enumerate} \end{example} \begin{prop} \label{}\hypertarget{}{} Let $\mathcal{G}_\bullet$ be a [[Lie groupoid]] with (for simplicity) [[connected topological space|connected]] source-fibers. Then there are maps \begin{displaymath} hol \;\colon\; Monod(X,\mathcal{F})_\bullet \stackrel{g_{\mathcal{G}}}{\to} \mathcal{G}_\bullet \stackrel{hol_G}{\to} Hol(X,\mathcal{F}) \end{displaymath} which are surjective [[local diffeomorphisms]] and such that the composite is the holonomy morphism (\ldots{}). \end{prop} This is (\hyperlink{CrainicMoerdijk}{Crainic-Moerdijk 00, prop. 1}). \hypertarget{of_higher_smooth_spaces}{}\subsubsection*{{Of higher smooth spaces}}\label{of_higher_smooth_spaces} One can consider the generalization of the notion of foliation of manifolds to foliations of structures in [[higher differential geometry]] such as [[Lie groupoids]] and [[Lie algebroids]]. See at \begin{itemize}% \item [[foliation of a Lie algebroid]] \item [[foliation of a Lie groupoid]] \end{itemize} \hypertarget{InCohesiveHigherGeometry}{}\subsubsection*{{In terms of differential cohesive higher geometry}}\label{InCohesiveHigherGeometry} The following is a suggestion for an axiomatization of foliations in [[higher differential geometry]] in the formalization of [[differential cohesion|differential]] [[cohesion]], followed by some considerations showing how these axioms reproduce traditional theory. \begin{quote}% Under construction. \end{quote} \begin{defn} \label{}\hypertarget{}{} Let $\mathbf{H}$ be a [[cohesive (∞,1)-topos]] equipped with [[differential cohesion]]. As usual, we write $(\int \dashv \flat \dashv \sharp)$ for the [[adjoint triple]] of [[modalities]] that defines the [[cohesion]] ([[shape modality]] $\dashv$ [[flat modality]] $\dashv$ [[sharp modality]]) and we write $(Red \dashv \int_{inf} \dashv \flat_{inf})$ for the adjoint triple of modalities that defines the [[differential cohesion]] ([[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]). \end{defn} Below we are going to axiomatize aspects of the traditional description of foliations by [[Lie groupoids]]/[[foliation groupoids]] as discussed \hyperlink{InTermsOfLieAlgebroidsAndLieGroupoids}{above}, so we start by briefly setting up some terminology on [[groupoid object in an (infinity,1)-category|groupoid objects]] in [[differential cohesion]]. \begin{defn} \label{NotationForGroupoidObjects}\hypertarget{NotationForGroupoidObjects}{} By the [[Giraud-Rezk-Lurie axioms]] we may think of a [[1-epimorphism]] $\mathcal{G}_0 \to \mathcal{G}$ in $\mathbf{H}$ as an [[atlas]] of the cohesive $\infty$-groupoid $\mathcal{G} \in \mathbf{H}$, exhibiting equivalently the corresponding [[groupoid object in an (infinity,1)-category|groupoid object]] which we write \begin{displaymath} \mathcal{G}_\bullet \coloneqq \mathcal{G}_0^{\times^{\bullet+1}_{\mathcal{G}}} \,. \end{displaymath} Hence we use notation where omitting the subscript decorationon a groupoid object $\mathcal{G}_\bullet \in \mathbf{H}^{\Delta^{op}}$ refers to its realization \begin{displaymath} \mathcal{G} \coloneqq {\underset{\rightarrow}{\lim}}_n \mathcal{G}_{n} \;\;\; \in \mathbf{H} \,. \end{displaymath} \end{defn} We have the following ``geometricity'' constraints on groupoid objects. \begin{defn} \label{GeometricAndEtale}\hypertarget{GeometricAndEtale}{} For $f \colon X \to Y$ any morphism in $\mathbf{H}$, write \begin{displaymath} X \stackrel{L(f)}{\to} Y \underset{\int_{inf} Y}{\times} \int_{inf} X \end{displaymath} for the canonical morphism induced by the [[natural transformation|naturality]] of the $\int_{inf}$-[[unit of an adjunction|unit]]. We say that \begin{enumerate}% \item $f$ is a \textbf{[[formally smooth morphism]]} (or \emph{[[submersion]]}) if $L(f)$ is a [[1-epimorphism]]; \item $f$ is a \textbf{[[formally étale morphism]]} (or \emph{[[local diffeomorphism]]}) if $L(f)$ is an [[equivalence in an (infinity,1)-category|equivalence]]. \end{enumerate} Now if $\pi \colon \mathcal{G}_0 \to \mathcal{G}$ is a [[1-epimorphism]], hence an [[atlas]] for the cohesive $\infty$-groupoid $\mathcal{G}$, then we say about the corresponding [[groupoid object in an (infinity,1)-category|groupoid object]] as in def. \ref{NotationForGroupoidObjects}, that \begin{enumerate}% \item $\mathcal{G}_\bullet$ is an \textbf{[[geometric ∞-groupoid]]} if its [[atlas]] $\pi$ is a [[formally smooth morphism]]/[[submersion]]. \item $\mathcal{G}_\bullet$ is an \textbf{[[étale ∞-groupoid]]} if its [[atlas]] $\pi$ is a [[formally étale morphism]]/[[local diffeomorphism]]. \end{enumerate} \end{defn} \begin{defn} \label{FoliationInDifferentialCohesion}\hypertarget{FoliationInDifferentialCohesion}{} For $X \in \mathbf{H}$, a \textbf{[[foliation]]} of $X$ is a morphism $\mathcal{D} \colon X \to X//\mathcal{D}$ in $\mathbf{H}$ which is \begin{enumerate}% \item a [[1-epimorphism]]; \item a [[formally smooth morphism]]. \end{enumerate} Equivalently a foliation of $X$ is a map that exhibits $X$ as an [[atlas]] for a [[geometric ∞-groupoid]], def. \ref{GeometricAndEtale}. Given a foliation $\mathcal{D}$ on $X$ we say that the \textbf{leaf decomposition} of $X$ induced by the foliation is the [[(∞,1)-pullback]] \begin{displaymath} LeafDec(\mathcal{D}) \coloneqq \flat(X//\mathcal{D}) \underset{X//\mathcal{D}}{\times} X \end{displaymath} in \begin{displaymath} \itexarray{ LeafDec(\mathcal{D}) &\stackrel{\iota_{\mathcal{D}}}{\to}& X \\ \downarrow && \downarrow^{\mathrlap{\mathcal{D}}} \\ \flat (X//\mathcal{D}) &\to& X//\mathcal{D} } \,, \end{displaymath} where the bottom map is the [[counit of an adjunction|counit]] of the [[flat modality]]. \end{defn} Now let $\mathbb{G} \in Grp_2(\mathbf{H})$ a [[braided ∞-group]]. Write \begin{displaymath} \Omega^2_{cl} := \Omega^2_{flat}(-,\mathbb{G}) \end{displaymath} for the corresponding coefficient object for [[curvature]] forms of $\mathbb{G}$-[[principal ∞-connections]] (as discussed there). \begin{defn} \label{IsotropicFoliationsInDiffCohe}\hypertarget{IsotropicFoliationsInDiffCohe}{} Given a closed 2-form \begin{displaymath} \omega \;\colon\; X \to \Omega^2_{cl} \end{displaymath} a foliation of $X$ by \textbf{$\omega$-[[isotropic submanifold|isotropic subspaces]]} is a [[foliation]] $\mathcal{D} \colon X \to X//\mathcal{D}$ as in def. \ref{FoliationInDifferentialCohesion} such that the restriction of $\omega$ to the leaf decomposition is equivalent to the 0-form \begin{displaymath} \iota_{\mathcal{D}}^* \omega \simeq 0 \,, \end{displaymath} hence such that the top composite morphism in the diagram \begin{displaymath} \itexarray{ (\flat \mathcal{E}) \underset{\mathcal{E}}{\times} X &\to& X &\stackrel{\omega}{\to}& \Omega^2_{cl} \\ \downarrow && \downarrow \\ \flat (X//\mathcal{D}) &\to& X//\mathcal{D} } \end{displaymath} factors through the point. \end{defn} We now discuss how low-degree examples of this axiomatics interpreted in $\mathbf{H} \coloneqq$ [[SynthDiff∞Grpd]] reproduces the traditional notions of [[foliations]] and [[isotropic submanifolds]] of [[pre-symplectic manifolds]]. In the following we regard [[smooth manifolds]] canonically under the embedding [[SmoothMfd]] $\hookrightarrow$ [[Smooth∞Grpd]] $\stackrel{i_!}{\hookrightarrow}$ [[SynthDiff∞Grpd]] $= \mathbf{H}$ as [[reduced object|reduced]] [[synthetic differential ∞-groupoids]]. \begin{example} \label{TraditionalSubmersions}\hypertarget{TraditionalSubmersions}{} A [[smooth function]] $f \colon X \to Y$ between [[smooth manifolds]] is \begin{enumerate}% \item a [[local diffeomorphism]] in the traditional sense precisely if it is a [[formally étale morphism]] in the sense of def. \ref{GeometricAndEtale}; \item a [[submersion]] in the traditional sense precisely if it is a [[formally smooth morphism]] in the sense of def. \ref{GeometricAndEtale}. \end{enumerate} \end{example} This is discussed at \emph{[[SynthDiff∞Grpd]]}. The idea of the proof is to use the [[∞-cohesive site]] of definition [[CartSp]]${}_{synthdiff}$ and evaluate the [[homotopy pullback]] in def. \ref{GeometricAndEtale} first on all representables of the form $U \times D_1$ where $U$ ranges over [[Cartesian spaces]] and where $D_1$ is the first order ininfitesimal neighbourhood of the origin on $\mathbb{R}^1$ (whose [[smooth algebra]] of fucntions is the [[ring of dual numbers]]). Then the homotopy pullback is represented as an ordinary pullback of sheaves over Cartesian spaces and the naturality diagram in question is the diagram of [[tangent bundles]] \begin{displaymath} \itexarray{ T X &\stackrel{d f}{\to}& T Y \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y } \,. \end{displaymath} With this now the claim is reduced to the traditional characterization of [[submersions]] and [[local diffeomorphisms]]. \begin{example} \label{LieAndEtaleGroupoids}\hypertarget{LieAndEtaleGroupoids}{} Let $\mathcal{G}$ be a [[smooth groupoid]] which has a presentation by a [[simplicial presheaf]] $\mathcal{G}_\bullet$ with values in 2-coskeletak Kan complexes where objects and morphisms are represented by a [[smooth manifold]] each, and consider it equipped with the induced [[atlas]] $\mathcal{G}_0 \to \mathcal{G}$. Then \begin{itemize}% \item if the presentation $\mathcal{G}_\bullet$ is a [[Lie groupoid]] then $\mathcal{G}_0 \to \mathcal{G}$ is a [[geometric ∞-groupoid]] \item if the presentation $\mathcal{G}_\bullet$ is an [[étale groupoid]] then $\mathcal{G}_0 \to \mathcal{G}$ is an [[étale ∞-groupoid]] \end{itemize} in the sense of def. \ref{GeometricAndEtale}. \end{example} This follows by the corresponding discussion at [[SynthDiff∞Grpd]]. The idea of the proof is that one presents the atlas in the projective [[model structure on simplicial presheaves]] by the [[décalage]] [[fibration]] [[resolution]], schematically \begin{displaymath} \itexarray{ && g \\ & \swarrow && \searrow & &&& &&& \mathcal{G}_0 \\ g_1 &&\to&& g_2 \\ \\ &&&& &&&&&& \downarrow \\ \\ g_1 &&\to&& g_2 &&&&&& \mathcal{G} } \,. \end{displaymath} Then the [[homotopy pullback]] $\mathcal{G} \underset{\int_{inf}\mathcal{G}} {\times}\int_{inf} X$ is presented by an ordinary pullback and so example \ref{TraditionalSubmersions} applies degreewise. In degree 0 the above resolution is the target map in the groupoid $\mathcal{G}$ and so by example \ref{TraditionalSubmersions} this is a [[submersion]] or [[local diffeomorphism]], respectively, as claimed. \begin{example} \label{}\hypertarget{}{} Let $X$ be a smooth manifold and let $\mathcal{D}$ be a traditional [[foliation]] on $X$ which is a \emph{[[simple foliation]]}, example \ref{SimpleFoliation}, in that the [[leaf space]] $X/\mathcal{D}$ exists as a smooth manifold and the projection map $X \to X/\mathcal{D}$ is a [[submersion]]. Then by the discussion at [[synthetic differential ∞-groupoid]], this projection map is also a [[formally smooth morphism]] in $\mathbf{H}$ according to def. \ref{GeometricAndEtale}. Moreover, being a quotient projection it is a [[1-epimorphism]] and hence exhibits the corresponding [[foliation groupoid]] \begin{displaymath} \left( X \underset{X/\mathcal{D}}{\times} X \stackrel{\to}{\to} X/\mathcal{D} \right) \end{displaymath} as a [[geometric ∞-groupoid]] in the sense of def. \ref{GeometricAndEtale}. Now $\flat ( X// \mathcal{D})$ is the underlying set of points of the [[leaf space]], regarded as a [[discrete ∞-groupoid]]. So we have the [[pasting]] diagram of pullbacks \begin{displaymath} \itexarray{ L_{l} &\to& \coprod_{l \in X/\mathcal{D}} L_l &\to& X \\ \downarrow && \downarrow && \downarrow^{\mathrlap{\mathcal{D}}} \\ {*} &\stackrel{\vdash l}{\to}& \coprod_{l \in D/\mathcal{X}}{*} &\to& X/\mathcal{D} } \end{displaymath} for every [[leaf]] $L_l$ labeled by the point $l \in X/\mathcal{D}$ in leaf space, which exhibits the leaf decomposition of $X$ under $\mathcal{D}$ according to def. \ref{FoliationInDifferentialCohesion} as the disjoint union of the leaves of $(X,\mathcal{D})$ in the traditional sense, injected into $X$ in the canonical way. \end{example} \begin{example} \label{TraditionalGeneralFoliation}\hypertarget{TraditionalGeneralFoliation}{} Consider now $\mathcal{G}_\bullet$ any [[Lie groupoid]], hence in particular a [[smooth groupoid]] $\mathcal{G} \in \mathbf{H}$ equipped with an [[atlas]] $\mathcal{G}_0 \to \mathcal{G}$, which hence by example \ref{LieAndEtaleGroupoids} exhibits a [[geometric ∞-groupoid]] in the sense of def. \ref{GeometricAndEtale}, hence a foliation $\mathcal{D} \;\colon\; \mathcal{G}_0 \to \mathcal{G}$ in the sense of def. \ref{FoliationInDifferentialCohesion}. Computation of the [[homotopy pullback]] \begin{displaymath} \itexarray{ LeafDec_{\mathcal{D}}(\mathcal{G}_0) &\to& \mathcal{G}_0 \\ \downarrow && \downarrow^{\mathrlap{\mathcal{D}}} \\ \flat( \mathcal{G} ) &\to& \mathcal{G} & = \mathcal{G}_0//\mathcal{D} } \end{displaymath} by the method as in example \ref{LieAndEtaleGroupoids} shows that $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ is the [[smooth groupoid]] presented by the presheaf of groupoids whose \begin{itemize}% \item smoothyl $U$-parameterized collection of objects are smoothly $U$-parameterized collections of morphisms $\{g_0 \to g(u)\}_{u \in U}$ in $\mathcal{G}_\bullet$ with $g_0$ held fixed; \item morphisms are given by precomposing these collections with a fixed (not varying with $U$) morphism in $\mathcal{G}_\bullet$. \end{itemize} This means that if $\mathcal{G}_\bullet$ is an [[étale groupoid]] to start with, then $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ is the disjoint union of all its [[orbit]] leaves (as smooth manifolds), hence that the abstractly defined $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ reproduces the decomposition of $\mathcal{G}_0$ by the [[foliation]] encoded by the [[foliation groupoid]] $\mathcal{G}_\bullet$ as in traditional theory. \end{example} \begin{remark} \label{}\hypertarget{}{} We may suggestively summarize example \ref{TraditionalGeneralFoliation} in words as: ``In cohesive higher geometry, every foliation is a [[simple foliation]].'' Because the quotient map to the leaf space of a general foliation is always a [[submersion]]/[[formally smooth morphism]], just not always onto a manifold, but onto a higher space. \end{remark} \begin{remark} \label{}\hypertarget{}{} If the $\mathcal{G}_\bullet$ in example \ref{TraditionalGeneralFoliation} is not an [[étale groupoid]] to start with but a more general [[Lie groupoid]], then $LeafDec_{\mathcal{D}}(\mathcal{G}_0)$ in general retains information of non-[[discrete group|discrete]] [[isotropy groups]] of $\mathcal{G}_\bullet$. We might decide to rule out this possibility by adding to the axioms in def. \ref{FoliationInDifferentialCohesion} the clause that $X//\mathcal{G}$ (here $\mathcal{G}_0//\mathcal{D}$) be \'e{}tale. However, we might also keep that case and regard it as the first instance of what is certainly a natural phenomenon as we pass to [[higher geometry]], namely that leaves of a foliation no longer need to manifolds but will be (higher) groupoids themselves. \end{remark} Finally, given the above it is clear how [[isotropic submanifold|isotropic]] appear in the cohesive axiomatics. \begin{example} \label{}\hypertarget{}{} For $\mathbb{G} = U(1)$ the smooth [[circle group]], $\Omega^2_{cl}$ is the ordinary sheaf of closed [[differential 2-forms]] under the canonical embedding \begin{displaymath} Sh(CartSp) \hookrightarrow Sh_\infty(CartSp) \simeq Smooth\infty Grpd \stackrel{i_!}{\hookrightarrow} SynthDiff\infty\mathrm{Grpd} \,. \end{displaymath} Then for $X$ a [[smooth manifold]] a morphism $\omega \;\colon\; X \to \Omega^2_{cl}$ is equivalently a [[differential 2-form]]. Then for $\mathcal{D} \;\colon\; X \to X//\mathcal{D}$ a traditional [[foliation]] of $X$ regarded as a foliation in $SynthDiff\infty Grpd$ by example \ref{TraditionalGeneralFoliation}, it follows with the discussion there that $\iota_{\mathcal{D}}^* \omega$ is precisely the collection of restriction of $\omega$ to each of the leaves of the foliation. Therefore this is a foliation by [[isotropic submanifolds]] in the traditional sense precisely if it is an $\omega$-isotropc foliation in the sense of def. \ref{IsotropicFoliationsInDiffCohe}. \end{example} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{example} \label{SimpleFoliation}\hypertarget{SimpleFoliation}{} For $X \to Y$ a [[submersion]] of [[smooth manifolds]], the connected [[fibers]] of the submersion constitute a foliation of $X$ whose [[codimension]] is the [[dimension]] of $Y$. Foliations of this form are called \textbf{[[simple foliations]]}. \end{example} \begin{example} \label{}\hypertarget{}{} Every [[Lie groupoid]] gives a folitation on its space of [[objects]]: the leaves are the [[orbits]]. Conversely, every regular foliation gives rise to its [[holonomy groupoid]]. This is a (not necessarily Hausdorff) Lie groupoid whose orbits are the leaves of the original foliation, and which in some sense is minimal with this condition. \end{example} \begin{example} \label{}\hypertarget{}{} Every [[Poisson manifold]] has a canonical structure of a foliation whose leaves are its maximal [[symplectic manifold|symplectic]] submanifolds, called \emph{[[symplectic leaves]]}. \end{example} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{leaf_space}{}\subsubsection*{{Leaf space}}\label{leaf_space} The set of components of a foliation is typically non-[[Hausdorff space|Hausdorff]], which is one of the motivations of the [[Alain Connes|Connes]]-style [[noncommutative geometry]]. \hypertarget{classification}{}\subsubsection*{{Classification}}\label{classification} Folitation are classified by the [[Haefliger groupoid]]. See at \emph{[[Haefliger theorem]]}. \hypertarget{characteristic_classes}{}\subsubsection*{{Characteristic classes}}\label{characteristic_classes} There is a theory of [[characteristic classes]] for foliations. A most well known example is the Godbillon-Vey characteristic class. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Bott connection]] \item [[holonomy groupoid]], [[monodromy groupoid]] \item [[orbifold]] \item [[Frobenius theorem (differential topology)]] \item [[higher differential geometry applied to plain differential geometry]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} The notion of foliated manifolds was introduced in the 1950s, motivated from [[partial differential equation]] theory, in \begin{itemize}% \item [[Charles Ehresmann]], \ldots{} \item Reeb, \ldots{} \item [[Eli Cartan]], \emph{Sur l'int\'e{}gration des \'e{}quations diff\'e{}rentiels completement int\'e{}grable}, Oeuvres Compl\`e{}tes, Pt. II, Vol. I, 555-561. \end{itemize} A discussion in [[differential geometry]] is in \begin{itemize}% \item Robert Hermann, \emph{On the differential geometry of foliations}, Annals of Mathematics, Second Series, Vol. 72, No. 3 pp. 445-457 (\href{http://www.jstor.org/stable/1970226}{jstor}) \end{itemize} A textbook account with a view to the modern formulation in [[Lie groupoid]] theory is \begin{itemize}% \item [[Ieke Moerdijk]], [[Janez Mr?un]], \emph{[[Introduction to foliations and Lie groupoids]]}, Cambridge Studies in Advanced Mathematics \textbf{91}, 2003. x+173 pp. ISBN: 0-521-83197-0 \end{itemize} Foliations in [[Lie groupoid]] theory are discussed in more detail in \begin{itemize}% \item [[Marius Crainic]], [[Ieke Moerdijk]], \emph{Foliation groupoids and their cyclic homology} (\href{http://arxiv.org/abs/math/0003119}{arXiv:math/0003119}) \end{itemize} The corresponding [[groupoid algebras]] are discussed in chapter 2, section 8 of \begin{itemize}% \item [[Alain Connes]], \emph{[[Noncommutative Geometry]]} \end{itemize} See also \begin{itemize}% \item \href{http://en.wikipedia.org/wiki/Foliation}{wikipedia}, Springer Online Enc. of Math.: \href{http://eom.springer.de/F/f040740.htm}{foliation} \end{itemize} A survey by Fuks in Russian Itogi: \begin{itemize}% \item . . , \emph{}, . . . . ., 1981, \textbf{18}, . 151---213, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=inta&paperid=93&what=fullt&option_lang=rus}{pdf} \end{itemize} Cohomology of formal vector fields and characteristic classes of foliations were originally studied in the papers \begin{itemize}% \item D. B. Fuks, \emph{Cohomology of infinite-dimensional Lie algebras and characteristic classes of foliations} (book, Rus. and Eng. versions) \item I. M. Gelfand, B. L. Fegin, D. B. Fuks, \emph{Cohomology of the Lie algebra of formal vector fields with coefficients in its dual space and variations of characteristic classes of foliations, Funkcional. Anal. i Priloen. 8 (1974), no. 2, 13--29 (Russian original \href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=faa&paperid=2326&option_lang=eng}{mathnet.ru}, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=2326&volume=8&year=1974&issue=2&fpage=13&what=fullt&option_lang=eng}{pdf})} \item Claude Godbillon, \emph{Cohomologies d'alg\`e{}bres de Lie de champs de vecteurs formels}, S\'e{}minaire Bourbaki, 25\`e{}me ann\'e{}e (1972/1973), Exp. No. 421, pp. 69--87. Lecture Notes in Math. \textbf{383}, Springer 1974. \item . . , . . , \emph{ }, . . . ., 1970, \textbf{34}, . 2, . 322---337, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=2418&what=fullt&option_lang=rus}{pdf} \end{itemize} In a series of works of Connes and Moscovici, the local index formulas in the context of transverse geometry of foliations has been studied in connection to a new cyclic homology of a Hopf algebra arising in this context: \begin{itemize}% \item [[A. Connes]], H. Moscovici, \emph{Modular Hecke algebras and their Hopf symmetry}, Mosc. Math. J., 4:1 (2004), 67--109; \href{http://arxiv.org/abs/math/0301089}{math.QA/0301089}, \href{http://www.ams.org/distribution/mmj/vol4-1-2004/connes_moscovici_1.pdf}{ams}; \emph{Hopf algebras, cyclic cohomology and the transverse index theory}, \href{http://arxiv.org/abs/math/9806109}{math.DG/9806109}, Comm. Math. Phys. \textbf{198}, n.1, 1998 \href{http://www.ams.org/mathscinet-getitem?mr=1657389}{MR99m:58186} \href{http://dx.doi.org/10.1007/s002200050477}{doi}; \emph{Rankin-Cohen brackets and the Hopf algebra of transverse geometry}, Mosc. Math. J., 4:1 (2004), 111--130; \emph{Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry}, in: Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math. \textbf{38}, p. 217--255. Enseignement Math., Geneva, 2001 \href{http://www.ams.org/mathscinet-getitem?mr=1929328}{MR2003k:58042} \item A. Connes, \emph{Cyclic cohomology and the transverse fundamental class of a foliation}, in: Geom. methods in operator algebras (Kyoto, 1983), Pitman Res. Notes in Math. \textbf{123}, p. 52--144, 1986 \href{http://www.ams.org/mathscinet-getitem?mr=866491}{MR88k:58149} \end{itemize} More general issues of index theory in noncommutative geometry applied to foliations is in \begin{itemize}% \item Yu. A. Kordyukov, \emph{Noncommutative geometry of foliations}, J. K-Theory, 2:2, Special issue in memory of Y. P. Solovyev, Part 1 (2008), 219--327 \href{http://www.ams.org/mathscinet-getitem?mr=2456103}{MR2009m:58018}; \emph{Index theory and non-commutative geometry on foliated manifolds}, Russian Math. Surveys, 64:2 (2009), 273--391 (original: . . , , 64:2(386) (2009), 73--202); \emph{ - }, , 12:3 (2000), 81--105 \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=1108&what=fullt&option_lang=rus}{pdf} \item W. P. Thurston, \emph{Existence of codimension-one foliations}, Ann. of Math. (2) \textbf{104} (1976), no. 2, 249--268 (\href{http://dx.doi.org/10.1007/BF02566730}{doi}); \emph{Foliations and groups of diffeomorphisms}, Bull. Amer. Math. Soc. 80 (1974), 304--307 (\href{http://www.ams.org/bull/1974-80-02/S0002-9904-1974-13475-0/S0002-9904-1974-13475-0.pdf}{pdf}); \emph{The theory of foliations of codimension greater than one}, Comment. Math. Helv. 49 (1974), 214--231 (\href{http://resolver.sub.uni-goettingen.de/purl?GDZPPN00206135X}{link}) \end{itemize} [[!redirects foliations]] [[!redirects leaf space]] [[!redirects leaf spaces]] [[!redirects space of leaves]] [[!redirects spaces of leaves]] [[!redirects foliation theory]] [[!redirects regular foliation]] [[!redirects regular foliations]] \end{document}