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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*{smooth infinity-groupoid} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes} [[!include cohesive infinity-toposes - contents]] \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{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{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{cohesion}{Cohesion}\dotfill \pageref*{cohesion} \linebreak \noindent\hyperlink{RelativeCohesion}{Relative cohesion}\dotfill \pageref*{RelativeCohesion} \linebreak \noindent\hyperlink{continuous_cohesion}{Continuous cohesion}\dotfill \pageref*{continuous_cohesion} \linebreak \noindent\hyperlink{infinitesimal_cohesion}{Infinitesimal cohesion}\dotfill \pageref*{infinitesimal_cohesion} \linebreak \noindent\hyperlink{truncations}{Truncations}\dotfill \pageref*{truncations} \linebreak \noindent\hyperlink{InfSheavesOnCartSp}{Structures in the cohesive $(\infty,1)$-topos $Smooth \infty Grpd$}\dotfill \pageref*{InfSheavesOnCartSp} \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{smooth $\infty$-groupoid} is an [[∞-groupoid]] equipped with [[cohesive (∞,1)-topos|cohesion]] in the form of [[smooth structure]]. Examples include [[smooth manifolds]], [[Lie groups]], [[Lie groupoids]] and generally [[Lie infinity-groupoids]], but also for instance [[moduli spaces]] of [[differential forms]], [[moduli stacks]] of [[principal connections]] and generally of [[cocycles]] in [[differential cohomology]]. The [[(∞,1)-topos]] $Smooth \infty Grpd$ of all smooth $\infty$-groupoids is a [[cohesive (∞,1)-topos]]. It realizes a [[higher geometry]] version of [[differential geometry]]. Many properties of smooth $\infty$-groupoids are inherited from the underlying [[Euclidean-topological ∞-groupoid]]s. See [[ETop∞Grpd]] for more. There is a refinement of smooth $\infty$-groupoids to [[synthetic differential ∞-groupoid]]s. See [[SynthDiff∞Grpd]] for more on that. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{DifferentiablyGoodOpenCover}\hypertarget{DifferentiablyGoodOpenCover}{} For $X$ a [[smooth manifold]], say an [[open cover]] $\{U_i \to X\}$ is a \textbf{[[differentiably good open cover]]} if each non-empty finite intersection of the $U_i$ is [[diffeomorphism|diffeomorphic]] to a [[Cartesian space]]. \end{defn} \begin{prop} \label{}\hypertarget{}{} Every [[paracompact manifold|paracompact]] [[smooth manifold]] admits a differentiably good open cover. \end{prop} \begin{proof} This is a folk theorem. A detailed proof is at [[good open cover]]. \end{proof} \begin{defn} \label{}\hypertarget{}{} Let [[SmoothMfd]] be the [[large site]] of [[paracompact manifold|paracompact]] [[smooth manifold]]s with [[smooth function]]s between them and equipped with the [[coverage]] of \hyperlink{DifferentiablyGoodOpenCover}{differentiably good open covers}. \end{defn} \begin{defn} \label{}\hypertarget{}{} This does indeed define a coverage. The [[Grothendieck topology]] that is generated from it is the standard [[open cover]] topology. \end{defn} \begin{proof} For $\{U_i \to X\}$ any open cover of a paracompact manifold also $\coprod_i U_i$ is paracompact. Hence we may find a differentiably good open cover $\{K_j \to \coprod_i U_i\}$. This is then a refinement of the original open cover of $X$. \end{proof} \begin{defn} \label{}\hypertarget{}{} Let [[CartSp]]${}_{smooth}$ be the [[site]] of [[Cartesian space]]s with [[smooth function]]s between them and equipped with the [[coverage]] of \hyperlink{DifferentiablyGoodOpenCover}{differentiably good open covers}. \end{defn} \begin{defn} \label{}\hypertarget{}{} The [[(∞,1)-topos]] of \textbf{smooth $\infty$-groupoids} is the [[(∞,1)-category of (∞,1)-sheaves]] on [[CartSp]]${}_{smooth}$: \begin{displaymath} Smooth \infty Grpd := Sh_{(\infty,1)}(CartSp_{smooth}) \,. \end{displaymath} \end{defn} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{cohesion}{}\subsubsection*{{Cohesion}}\label{cohesion} \begin{prop} \label{}\hypertarget{}{} $Smooth \infty Grpd$ is a [[cohesive (∞,1)-topos]]. \end{prop} \begin{proof} The [[site]] [[CartSp]]${}_{smooth}$ is (as discussed there) an [[∞-cohesive site]]. See there for the implication. \end{proof} \begin{defn} \label{}\hypertarget{}{} Let [[SmoothMfd]] be the [[large site]] of [[paracompact topological space|paracompact]] [[smooth manifold]]s with [[smooth function]]s between them and equipped with the [[coverage]] whose [[covering]] families are \emph{differentiably good open covers} : [[open cover]]s $\{U_i \to U\}$ where each non-empty open intersection is [[diffeomorphic]] to a [[Cartesian space]]. \end{defn} \begin{prop} \label{}\hypertarget{}{} This does indeed define a [[coverage]] and the [[Grothendieck topology]] generated by it is the standard [[open cover]] topology. \end{prop} \begin{proof} This is discussed in detail at [[good open cover]]. \end{proof} \begin{prop} \label{SmoothManifoldEmbeds}\hypertarget{SmoothManifoldEmbeds}{} The [[(∞,1)-topos]] $Smooth \infty Grpd$ is equivalent to the [[hypercompletion]] $\hat Sh_{(\infty,1)}(SmoothMfd)$ of the [[(∞,1)-category of (∞,1)-sheaves]] on the large site [[SmoothMfd]] \begin{displaymath} Smooth \infty Grpd \simeq \hat Sh_{(\infty,1)}(SmoothMfd) \,. \end{displaymath} \end{prop} \begin{proof} By the above we have that [[CartSp]]${}_{smooth}$ is a [[dense sub-site]] of [[SmoothMfd]]. With this the claim follows as in the analogous discussion at [[ETop∞Grpd]]. \end{proof} \begin{prop} \label{EmbeddingOfSmoothManifolds}\hypertarget{EmbeddingOfSmoothManifolds}{} The canonical embedding of [[smooth manifold]]s as [[0-truncated]] objects in $Smooth\infty Grpd$ is a [[full and faithful (∞,1)-functor]] \begin{displaymath} SmoothMfd \hookrightarrow Smooth \infty Grpd ,. \end{displaymath} \end{prop} \hypertarget{RelativeCohesion}{}\subsubsection*{{Relative cohesion}}\label{RelativeCohesion} We discuss the relation of $Smooth\infty Grpd$ to other [[cohesive (∞,1)-topos]]es. \hypertarget{continuous_cohesion}{}\paragraph*{{Continuous cohesion}}\label{continuous_cohesion} The [[cohesive (∞,1)-topos]] [[ETop∞Grpd]] of [[Euclidean-topological ∞-groupoid]]s has as [[site]] of definition [[CartSp]]${}_{top}$. There is a canonical [[forgetful functor]] \begin{displaymath} i : CartSp_{smooth} \to CartSp_{top} \end{displaymath} \begin{prop} \label{RelativeTopologicalCohesion}\hypertarget{RelativeTopologicalCohesion}{} The functor $i$ extends to an [[essential (∞,1)-geometric morphism]] \begin{displaymath} (i_! \dashv i^* \dashv i_*) : Smooth\infty Grpd \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}} ETop\infty Grpd \end{displaymath} such that the [[(∞,1)-Yoneda embedding]] is factored through the induced inclusion [[SmoothMfd]] $\stackrel{i}{\hookrightarrow}$ [[Mfd]] as \begin{displaymath} \itexarray{ SmoothMfd &\hookrightarrow& Smooth\infty Grpd \\ \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{i_!}} \\ Mfd &\hookrightarrow& ETop\infty Grpd } \, \end{displaymath} \end{prop} \begin{proof} Using the observation that $i$ preserves [[covering]]s and [[pullback]]s along morphism in covering families, the proof follows precisely the steps of the proof of . (Both of these are special cases of a general statement about morphisms of [[(∞,1)-site]]s, which should eventually be stated in full generality somewhere). \end{proof} \begin{prop} \label{UnderlyingETopologicalInftyGroupoids}\hypertarget{UnderlyingETopologicalInftyGroupoids}{} The [[essential geometric morphism|essential]] [[global section]] [[(∞,1)-geometric morphism]] of $Smooth \infty Grpd$ factors through that of [[ETop∞Grpd]] \begin{displaymath} (\Pi_{Smooth} \dashv Disc_{Smooth} \dashv \Gamma_{Smooth}) : Smooth \infty Grpd \stackrel{\overset{i_!}{\to}}{\stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}}} ETop\infty Grpd \stackrel{\overset{\Pi_{ETop}}{\to}}{\stackrel{\overset{Disc_{ETop}}{\leftarrow}}{\underset{\Gamma_{ETop}}{\to}}} \infty Grpd \end{displaymath} \end{prop} \begin{proof} This follows from the essential uniqueness of the [[global section]] [[(∞,1)-geometric morphism]] and of [[adjoint (∞,1)-functor]]s. \end{proof} \begin{prop} \label{}\hypertarget{}{} The functor $i_!$ here is the [[forgetful functor]] that forgets [[smooth structure]] and only remembers [[Euclidean topology]]-structure. \end{prop} \hypertarget{infinitesimal_cohesion}{}\paragraph*{{Infinitesimal cohesion}}\label{infinitesimal_cohesion} Observe that [[CartSp]]${}_{smooth}$ is (the [[syntactic category]] of) a [[Lawvere theory]]: the [[algebraic theory]] of [[smooth algebra]]s ($C^\infty$-rings). Write $SmoothAlg := Alg(C)$ for the category of its [[algebras over a Lawvere theory|algebras]]. Let $InfPoint \hookrightarrow SmoothAlg^{op}$ be the [[full subcategory]] on the [[infinitesimally thickened point]]s. \begin{defn} \label{}\hypertarget{}{} Let [[CartSp]]${}_{synthdiff} \hookrightarrow SmoothAlg^{op}$ be the [[full subcategory]] on the objects of the form $U \times D$ with $D \in CartSp_{smooth} \hookrightarrow SmoothAlg^{op}$ and $D \in InfPoint \hookrightarrow SmoothAlg^{op}$. Write \begin{displaymath} i : CartSp_{smooth} \hookrightarrow CartSp_{synthdiff} \end{displaymath} for the canonical inclusion. \end{defn} \begin{prop} \label{}\hypertarget{}{} The inclusion exhibits an of $Smooth \infty Grpd$ \begin{displaymath} (i_! \dashv i^* \dashv i_* \dashv i^!) : Smooth \infty Grpd \hookrightarrow SynthDiff\infty Grpd \,, \end{displaymath} where [[SynthDiff∞Grpd]] is the [[cohesive (∞,1)-topos]] of [[synthetic differential ∞-groupoid]]s: the [[(∞,1)-category of (∞,1)-sheaves]] over $CartSp_{synthdiff}$. \end{prop} \begin{proof} This follows as a special case of after observing that $CartSp_{synthdiff}$ is an infinitesimal neighbourhood site of $CartSp_{smooth}$ in the sense defined there. \end{proof} In [[SynthDiff∞Grpd]] we have [[∞-Lie algebra]]s and [[∞-Lie algebroid]]s as actual [[infinitesimal object]]s. See there for more details. \hypertarget{truncations}{}\subsubsection*{{Truncations}}\label{truncations} The [[(n,1)-topos|(1,1)-topos]] on the [[0-truncated]] smooth $\infty$-groupoids is \begin{displaymath} Sh(CartSp) \simeq Smooth \infty Grpd_{\leq 0} \hookrightarrow Smooth\infty Grpd \,, \end{displaymath} the [[sheaf topos]] on [[SmthMfd]]/[[CartSp]] discussed at \emph{[[smooth space]]}. The [[concrete objects]] in there \begin{displaymath} Smooth\infty Grpd_{\leq 0}^{conc} \hookrightarrow Smooth \infty Grpd \end{displaymath} are precisely the [[diffeological spaces]]. \hypertarget{InfSheavesOnCartSp}{}\subsection*{{Structures in the cohesive $(\infty,1)$-topos $Smooth \infty Grpd$}}\label{InfSheavesOnCartSp} We discuss the general abstract realized in $Smooth \infty Grpd$. This section is at \begin{itemize}% \item [[smooth infinity-groupoid -- structures]] \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[cohesive (∞,1)-topos]] \begin{itemize}% \item [[discrete ∞-groupoid]] \item [[Euclidean topological ∞-groupoid]] \item \textbf{smooth $\infty$-groupoid} \begin{itemize}% \item [[smooth 2-group]] \item [[Kan-fibrant simplicial manifold]] \item [[smooth spectrum]] \end{itemize} \end{itemize} \end{itemize} [[!include geometries of physics -- table]] \begin{itemize}% \item [[complex analytic ∞-groupoid]] \end{itemize} Smooth $\infty$-groupoids and related cohesive structures play a central role in the discussion at \begin{itemize}% \item [[geometry of physics]]. \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} For standard references on [[differential geometry]] and [[Lie groupoid]]s see there. The $(\infty,1)$-topos $Smooth \infty Grpd$ is discussed in section 3.3 of \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} A discussion of smooth $\infty$-groupoids as $(\infty,1)$-sheaves on $CartSp$ and the presentaton of the $\infty$-Chern-Weil homomorphism on these is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{Cech cocycles for differential characteristic classes -- An $\infty$-Lie theoretic construction} (). \end{itemize} For references on [[Chern-Weil theory in Smooth∞Grpd]] and [[connection on a smooth principal ∞-bundle]], see there. The results on differentiable [[Lie group cohomology]] used above are in \begin{itemize}% \item P. Blanc, \emph{Cohomologie diff\'e{}rentiable et changement de groupes} Ast\'e{}risque, vol. 124-125 (1985), pp. 113-130. \end{itemize} and \begin{itemize}% \item [[Jean-Luc Brylinski]], \emph{Differentiable Cohomology of Gauge Groups} (\href{http://arxiv.org/abs/math/0011069}{arXiv}) \end{itemize} which parallels \begin{itemize}% \item [[Graeme Segal]], \emph{Cohomology of topological groups} , Symposia Mathematica, Vol IV (1970) (1986?) p. 377 \end{itemize} A review is in section 4 of \begin{itemize}% \item [[Chris Schommer-Pries]], \emph{A finite-dimensional String 2-group} (\href{http://arxiv.org/abs/0911.2483}{arXiv:0911.2483}) \end{itemize} Classification of topological [[principal 2-bundle]]s is discussed in \begin{itemize}% \item [[John Baez]], [[Danny Stevenson]], \emph{The classifying space of a topological 2-group} Algebraic Topology Abel Symposia, 2009, Volume 4, 1-31 (\href{http://arxiv.org/abs/0801.3843}{arXiv:0801.3843}) \end{itemize} and the generalization to classification of smooth [[principal 2-bundle]]s is in \begin{itemize}% \item [[Thomas Nikolaus]], [[Konrad Waldorf]], \emph{Four Equivalent Versions of Non-Abelian Gerbes} (\href{http://arxiv.org/abs/1103.4815}{arXiv:1103.4815}) \end{itemize} Further discussion of the [[shape modality]] on smooth $\infty$-groupoids is in \begin{itemize}% \item [[David Carchedi]], \emph{On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces} (\href{http://arxiv.org/abs/1504.02394}{arXiv:1504.02394}) \end{itemize} [[!redirects Smooth∞Grpd]] [[!redirects ∞-Lie groupoid cohomology]] [[!redirects infinity-Lie groupoid cohomology]] [[!redirects smooth ∞-groupoid]] [[!redirects smooth ∞-groupoids]] [[!redirects smooth infinity-groupoid]] [[!redirects smooth infinity-groupoids]] [[!redirects Lie infinity-groupoid]] [[!redirects Lie infinity-groupoids]] [[!redirects smooth ∞-group]] [[!redirects smooth ∞-groups]] [[!redirects smooth infinity-group]] [[!redirects smooth infinity-groups]] [[!redirects smooth 2-groupoid]] [[!redirects smooth 2-groupoids]] [[!redirects higher differentiable stack]] [[!redirects higher differentiable stacks]] [[!redirects smooth homotopy type]] [[!redirects smooth homotopy types]] [[!redirects Smooth∞Groupoid]] [[!redirects Smooth∞Groupoids]] \end{document}