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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 set} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry} [[!include synthetic differential geometry - contents]] \hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive toposes}}\label{cohesive_toposes} [[!include cohesive infinity-toposes - contents]] \hypertarget{smooth_spaces}{}\section*{{Smooth spaces}}\label{smooth_spaces} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak \noindent\hyperlink{Cohesion}{Cohesion}\dotfill \pageref*{Cohesion} \linebreak \noindent\hyperlink{topos_points_and_stalks}{Topos points and stalks}\dotfill \pageref*{topos_points_and_stalks} \linebreak \noindent\hyperlink{distribution_theory}{Distribution theory}\dotfill \pageref*{distribution_theory} \linebreak \noindent\hyperlink{VariantsAndGeneralizations}{Variants and generalizations}\dotfill \pageref*{VariantsAndGeneralizations} \linebreak \noindent\hyperlink{synthetic_differential_geometry}{Synthetic differential geometry}\dotfill \pageref*{synthetic_differential_geometry} \linebreak \noindent\hyperlink{higher_smooth_geometry}{Higher smooth geometry}\dotfill \pageref*{higher_smooth_geometry} \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} The concept of a \emph{smooth set} or \emph{smooth space}, in the sense discussed here, is a generalization of that of [[smooth manifolds]] beyond that of [[diffeological spaces]]: A smooth set is a [[generalized smooth space]] that may be \emph{probed} by smooth [[Cartesian spaces]]. For expository details see at \emph{[[geometry of physics -- smooth sets]]}. Alternatively, the smooth test spaces may be taken to be more generally all [[smooth manifolds]]. But since [[manifolds]] themselves are built from gluing together smooth [[open balls]] $D^n_{int} \subset \mathbb{R}^n$ or equivalently [[Cartesian spaces]] $\mathbb{R}^n$, one may just as well consider Cartesian spaces test spaces. Finally, since $D^n$ is diffeomorphic to $\mathbb{R}^n$, one can just as well take just the cartesian smooth spaces $\mathbb{R}^n$ as test objects. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} The category of \textbf{smooth spaces} is the [[sheaf topos]] \begin{displaymath} SmoothSp := Sh(Diff) \end{displaymath} of [[sheaves]] on the [[site]] [[Diff]] of smooth manifolds equipped with its standard [[coverage]] ([[Grothendieck topology]]) given by open covers of manifolds. Since $Diff$ is [[equivalence of categories|equivalent]] to the category of [[manifolds]] \emph{embedded} into $\mathbb{R}^\infty$, $Diff$ is an [[essentially small category]], so there are no size issues involved in this definition. But since manifolds themselves are defined in terms of gluing conditons, the [[Grothendieck topos]] $SmoothSp$ depends on much less than all of $Diff$. Let \begin{displaymath} Ball := \{ (D^n_{int} \to D^m_{int}) \in Diff | n,m \in \mathbb{N}\} \end{displaymath} and \begin{displaymath} CartSp := \{ (\mathbb{R}^n \to \mathbb{R}^m) \in Diff | n,m \in \mathbb{N}\} \end{displaymath} be the full [[subcategory|subcategories]] $Ball$ and [[CartSp]] of $Diff$ on open balls and on cartesian spaces, respectively. Then the corresponding [[sheaf]] [[topos]]es are still those of smooth spaces: \begin{displaymath} \begin{aligned} SmoothSp &\simeq Sh(Ball) \\ & \simeq Sh(CartSp) \end{aligned} \,. \end{displaymath} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item The category of ordinary [[manifolds]] is a full subcategory of smooth spaces: \begin{displaymath} Diff \hookrightarrow SmoothSp \,. \end{displaymath} When one regards smooth spaces concretely as sheaves on $Diff$, then this inclusion is of course just the [[Yoneda embedding]]. \item The full [[subcategory]] \begin{displaymath} DiffSp \subset SmoothSp \end{displaymath} on [[concrete sheaf|concrete sheaves]] is called the category of [[diffeological spaces]]. \begin{itemize}% \item The standard class of examples of smooth spaces that motivate their use even in cases where one starts out being intersted just in [[smooth manifolds]] are \textbf{mapping spaces}: for $X$ and $\Sigma$ two smooth spaces (possibly just ordinary smooth manifolds), by the [[closed monoidal structure on presheaves]] the \textbf{mapping space} $[\Sigma,X]$, i.e. the space of smooth maps $\Sigma \to X$ exists again naturally as a smooth. By the general formula it is given as a [[sheaf]] by the assignment \begin{displaymath} [\Sigma,X] : U \mapsto SmoothSp(\Sigma \times U, X) \,. \end{displaymath} If $X$ and $\Sigma$ are ordinary manifolds, then the [[hom-set]] on the right sits inside that of the underlying sets $SmoothSp(\Sigma \times U , X) \subset Set(|\Sigma| \times |U|, |X| )$ so that $[\Sigma,X]$ is a [[diffeological space]]. The above formula says that a $U$-parameterized family of maps $\Sigma \to X$ is smooth as a map into the smooth space $[\Sigma,X ]$ precisely if the corresponding map of sets $U \times \Sigma \to X$ is an ordinary morphism of smooth manifolds. \end{itemize} \item The canonical examples of smooth spaces that are not diffeological spaces are the sheaves of (closed) differential forms: \begin{displaymath} K^n : U \mapsto \Omega^n_{closed}(U) \,. \end{displaymath} \item The category \begin{displaymath} SimpSmoothSp := SmoothSp^{\Delta^{op}} \end{displaymath} equivalently that of sheaves on $Diff$ with values in [[simplicial sets]] \begin{displaymath} \cdots \simeq Sh(Diff, SSet) \end{displaymath} of [[simplicial objects]] in smooth spaces naturally carries the structure of a [[homotopical category]] (for instance the [[model structure on simplicial sheaves]] or that of a Brown [[category of fibrant objects]] (if one restricts to locally Kan simplicial sheaves)) and as such is a [[presentable (∞,1)-category|presentation]] for the [[(∞,1)-topos]] of [[smooth ∞-stacks]]. \end{itemize} \hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties} \hypertarget{Cohesion}{}\subsubsection*{{Cohesion}}\label{Cohesion} \begin{prop} \label{SmoothSetsFormACohesiveTopos}\hypertarget{SmoothSetsFormACohesiveTopos}{} \textbf{([[smooth sets]] form a [[cohesive topos]])} The [[category]] $SmoothSet$ of [[smooth sets]] is a [[cohesive topos]] \begin{equation} SmoothSet \itexarray{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\longleftarrow} } Set \label{SheafToposAdjointQuadruple}\end{equation} \end{prop} \begin{proof} First of all (by \href{geometry+of+physics+--+smooth+sets#CategoryOfSmoothSets}{this Prop}) smooth sets indeed form a [[sheaf topos]], over the [[site]] [[CartSp]] of [[Cartesian spaces]] $\mathbb{R}^n$ with [[smooth functions]] between them, and equipped with the [[coverage]] of differentiably-[[good open covers]] (\href{geometry+of+physics+--+smooth+sets#DifferentiallyGoodOpenCover}{this def.}) \begin{displaymath} SmoothSet \simeq Sh(CartSp) \,. \end{displaymath} Hence, by Prop. \ref{CategoriesOfSheavesOnCohesiveSiteIsCohesive}, it is now sufficient to see that [[CartSp]] is a [[cohesive site]] (Def. \ref{OneCohesiveSite}). It clearly has [[finite products]]: The [[terminal object]] is the [[point]], given by the 0-[[dimension|dimensional]] [[Cartesian space]] \begin{displaymath} \ast = \mathbb{R}^0 \end{displaymath} and the [[Cartesian product]] of two [[Cartesian spaces]] is the Cartesian space whose [[dimension]] is the [[sum]] of the two separate dimensions: \begin{displaymath} \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \;\simeq\; \mathbb{R}^{ n_1 + n_2 } \,. \end{displaymath} This establishes the first clause in Def. \ref{OneCohesiveSite}. For the second clause, consider a differentiably-[[good open cover]] $\{U_i \overset{}{\to} \mathbb{R}^n\}$ (\href{geometry+of+physics+--+smooth+sets#DifferentiallyGoodOpenCover}{this def.}). This being a [[good cover]] implies that its [[Cech groupoid]] is, as an [[internal groupoid]] (via \href{geometry+of+physics+--+categories+and+toposes#PresheavesOfGroupoidsAsInternalGroupoidsInPresheaves}{this remark}), of the form \begin{equation} C(\{U_i\}_i) \;\simeq\; \left( \itexarray{ \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} y(U_i) } \right) \,. \label{CechGroupoidForCartSp}\end{equation} where we used the defining property of [[good open covers]] to identify $y(U_i) \times_X y(U_j) \simeq y( U_i \cap_X U_j )$. The [[colimit]] of \eqref{CechGroupoidForCartSp}, regarded just as a [[presheaf]] of [[reflexive graph|reflexive]] [[directed graph|directed]] [[graphs]] (hence ignoring [[composition]] for the moment), is readily seen to be the [[graph]] of the [[colimit]] of the components (the [[universal property]] follows immediately from that of the component colimits): \begin{equation} \begin{aligned} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \itexarray{ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \right) \\ & \simeq \left( \itexarray{ \underset{i,j}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i) } \right) \\ & \simeq \left( \itexarray{ \underset{i,j}{\coprod} \ast \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \ast } \right) \end{aligned} \,. \label{ColimitOfCechGroupoidOverCartSp}\end{equation} Here we first used that [[colimits commute with colimits]], hence in particular with [[coproducts]] (\href{geometry+of+physics+--+categories+and+toposes#LimitsCommuteWithLimits}{this prop.}) and then that the colimit of a [[representable presheaf]] is [[generalized the|the]] [[singleton]] set (\href{geometry+of+physics+--+categories+and+toposes#ColimitOfRepresentableIsSingleton}{this Lemma}). This colimiting [[graph]] carries a unique [[composition]] structure making it a [[groupoid]], since there is at most one morphism between any two objects, and every object carries a morphism from itself to itself. This implies that this groupoid is actually the colimiting groupoid of the Cech groupoid: hence the groupoid obtained from replacing each representable summand in the Cech groupoid by a point. Precisely this operation on [[Cech groupoids]] of [[good open covers]] of [[topological spaces]] is what \emph{[[Borsuk's nerve theorem]]} is about, a classical result in [[topology]]/[[homotopy theory]]. This theorem implies directly that the set of [[connected components]] of the groupoid \eqref{ColimitOfCechGroupoidOverCartSp} is in [[bijection]] with the set of [[connected components]] of the [[Cartesian space]] $\mathbb{R}^n$, regarded as a [[topological space]]. But this is evidently a [[connected topological space]], which finally shows that, indeed \begin{displaymath} \pi_0 \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; \ast \,. \end{displaymath} The second item of the second clause in Def. \ref{OneCohesiveSite} follows similarly, but more easily: The [[limit]] of the [[Cech groupoid]] is readily seen to be, as before, the unique groupoid structure on the limiting underlying graph of presheaves. Since $CartSp$ has a [[terminal object]] $\ast = \mathbb{R}^0$, which is hence an [[initial object]] in the [[opposite category]] $CartSp^{op}$, limits over $CartSp^{op}$ yield simply the evaluation on that object: \begin{equation} \begin{aligned} \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \itexarray{ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \phantom{A} \right) \\ & \simeq \left( \itexarray{ \underset{i,j}{\coprod} Hom_{CartSp}\left( \ast, U_i \underset{\mathbb{R}^n}{\cap} U_j \right) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} Hom_{CartSp}( \ast, U_i ) } \right) \end{aligned} \,. \label{ColimitOfCechGroupoidOverCartSp}\end{equation} Here we used that [[colimits]] (here [[coproducts]]) of [[presheaves]] are computed objectwise, and then the definition of the [[Yoneda embedding]] $y$. But the [[equivalence relation]] induced by this graph on its set of objects $\underset{i}{\coprod} Hom_{CartSp}( \ast, U_i )$ precisely identifies pairs of points, one in $U_i$ the other in $U_j$, that are actually the same point of the $\mathbb{R}^n$ being covered. Hence the set of [[equivalence classes]] is the set of points of $\mathbb{R}^n$, which is just what remained to be shown: \begin{displaymath} \pi_0 \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; Hom_{CartSp}(\ast, \mathbb{R}^n) \,. \end{displaymath} \end{proof} \hypertarget{topos_points_and_stalks}{}\subsubsection*{{Topos points and stalks}}\label{topos_points_and_stalks} \begin{lemma} \label{}\hypertarget{}{} For every $n \in N$ there is a [[point of a topos|topos point]] \begin{displaymath} D^n : Set \stackrel{\stackrel{(D^n)^*}{\leftarrow}} {\stackrel{D^n_*}{\to}} SmoothSp \end{displaymath} where the [[inverse image]] morphism -- the [[stalk]] -- is given on $A \in SmoothSp$ by \begin{displaymath} (D^n)^* A := \colim_{\mathbb{R}^n \supset U \ni 0} A(U) \,, \end{displaymath} where the colimit is over all open neighbourhoods of the origin in $\mathbb{R}^n$. \end{lemma} \begin{lemma} \label{}\hypertarget{}{} SmoothSp has [[point of a topos|enough points]]: they are given by the $D^n$ for $n \in \mathbb{N}$. \end{lemma} \hypertarget{distribution_theory}{}\subsubsection*{{Distribution theory}}\label{distribution_theory} Since a space of [[smooth functions]] on a [[smooth manifold]] is canonically a smooth set, it is natural to consider the \emph{smooth} [[linear functionals]] on such [[mapping spaces]]. These turn out to be equivalent to the [[continuous linear functionals]], hence to [[distributional densities]]. See at \emph{[[distributions are the smooth linear functionals]]} for details. \hypertarget{VariantsAndGeneralizations}{}\subsection*{{Variants and generalizations}}\label{VariantsAndGeneralizations} \hypertarget{synthetic_differential_geometry}{}\subsubsection*{{Synthetic differential geometry}}\label{synthetic_differential_geometry} The [[site]] [[CartSp]]${}_{smooth}$ may be replaced by the site [[CartSp]]${}_{th}$ (see there) whose objects are products of smooth Cartesian spaces with [[infinitesimally thickened points]]. The corresponding [[sheaf topos]] $Sh(CartSp_{th})$ is called the \emph{[[Cahiers topos]]}. It contains smooth spaces with possibly infinitesimal extension and is a model for [[synthetic differential geometry]] (a ``[[smooth topos]]''), which $Sh(CartSp)$ is not. The two toposes are related by an [[adjoint quadruple]] of functors that witness the fact that the objects of $Sh(CartSp_{th})$ are possiby infinitesimal extensions of objects in $Sh(CartSp)$. For more discussion of this see [[synthetic differential ∞-groupoid]] \hypertarget{higher_smooth_geometry}{}\subsubsection*{{Higher smooth geometry}}\label{higher_smooth_geometry} The topos of smooth spaces has an evident generalization from [[geometry]] to [[higher geometry]], hence from [[differential geometry]] to [[higher differential geometry]]: to an [[(∞,1)-topos]] of \emph{[[smooth ∞-groupoids]]}. See there for more details. $\,$ \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} [[!include geometries of physics -- table]] $\,$ \hypertarget{references}{}\subsection*{{References}}\label{references} Lecture notes are at \begin{itemize}% \item \emph{[[geometry of physics -- smooth sets]]} \end{itemize} Aspects of the category of smooth sets is discussed, with an eye towards its generalization to [[smooth ∞-groupoids]] and their [[homotopy]] [[localization]] in \begin{itemize}% \item [[Daniel Dugger]], section 3.4, from page 29 on in: \emph{Sheaves and Homotopy Theory} (\href{http://www.uoregon.edu/~ddugger/cech.html}{web}, \href{http://ncatlab.org/nlab/files/cech.pdf}{pdf}) \end{itemize} The [[point of a topos|topos points]] of $Sh(Diff)$ are discussed there in example 4.1.2 on p. 36. (they are mentioned before on p. 31). As a [[cohesive topos]], smooth sets are discussed in \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} [[!redirects smooth space]] [[!redirects smooth spaces]] [[!redirects smooth sets]] [[!redirects SmoothSet]] \end{document}