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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*{point of a topos} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory} [[!include topos theory - contents]] \hypertarget{the_points_of_a_topos}{}\section*{{The points of a topos}}\label{the_points_of_a_topos} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{in_presheaf_toposes}{In presheaf toposes}\dotfill \pageref*{in_presheaf_toposes} \linebreak \noindent\hyperlink{in_localic_sheaf_toposes}{In localic sheaf toposes}\dotfill \pageref*{in_localic_sheaf_toposes} \linebreak \noindent\hyperlink{in_sheaf_toposes}{In sheaf toposes}\dotfill \pageref*{in_sheaf_toposes} \linebreak \noindent\hyperlink{in_classifying_toposes}{In classifying toposes}\dotfill \pageref*{in_classifying_toposes} \linebreak \noindent\hyperlink{of_toposes_with_enough_points}{Of toposes with enough points}\dotfill \pageref*{of_toposes_with_enough_points} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{Rings}{Rings and algebraic theories}\dotfill \pageref*{Rings} \linebreak \noindent\hyperlink{of_a_local_topos}{Of a local topos}\dotfill \pageref*{of_a_local_topos} \linebreak \noindent\hyperlink{over_cohesive_sites}{Over $\infty$-cohesive sites}\dotfill \pageref*{over_cohesive_sites} \linebreak \noindent\hyperlink{toposes_with_enough_points}{Toposes with enough points}\dotfill \pageref*{toposes_with_enough_points} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{}\hypertarget{}{} A \textbf{point $x$ of a [[topos]] $E$} is a [[geometric morphism]] \begin{displaymath} x : Set \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} E \end{displaymath} from the base topos [[Set]] to $E$. For $A \in E$ an object, its [[inverse image]] $x^* A \in Set$ under such a point is called the [[stalk]] of $A$ at $x$. If $x$ is given by an [[essential geometric morphism]] we say that it is an \textbf{essential point} of $E$. \end{defn} \begin{remark} \label{}\hypertarget{}{} Since [[Set]] is the [[terminal object]] in the category [[Topos|GrothendieckTopos]] of [[Grothendieck topos]]es, for $E$ a [[sheaf topos]] this is a [[global element]] of the topos. Since $Set = Sh(*)$ is the [[category of sheaves]] on the one-point [[locale]], the notion of point of a topos is indeed the direct analog of a point of a [[locale]] (under [[localic reflection]]). \end{remark} \begin{defn} \label{EnoughPoints}\hypertarget{EnoughPoints}{} A [[topos]] is said to have \textbf{enough points} if isomorphy can be tested [[stalk]]wise, i.e. if the [[inverse image]] functors from all of its points are jointly [[conservative functor|conservative]]. More explicity: $E$ has enough points if for any morphism $f : A \to B$, we have that if for every point $p$ of $E$, the morphism of [[stalks]] $p^* f : p^* A \to p^* B$ is an [[isomorphism]], then $f$ itself is an isomorphism. \end{defn} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{in_presheaf_toposes}{}\subsubsection*{{In presheaf toposes}}\label{in_presheaf_toposes} \begin{prop} \label{}\hypertarget{}{} For $C$ a [[small category]], the points of the [[presheaf topos]] $[C^{op}, Set]$ are the [[flat functor]]s $C \to Set$: there is an [[equivalence of categories]] \begin{displaymath} Topos(Set, [C^{op}, Set]) \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} FlatFunc(C,Set) \,. \end{displaymath} \end{prop} This appears for instance as (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk, theorem VII 2}). \hypertarget{in_localic_sheaf_toposes}{}\subsubsection*{{In localic sheaf toposes}}\label{in_localic_sheaf_toposes} For the special case that $E = Sh(X)$ is the [[category of sheaves]] on a [[category of open subsets]] $Op(X)$ of a [[sober topological space|sober]] [[topological space]] $X$ the notion of topos points comes from the ordinary notion of points of $X$. For notice that \begin{itemize}% \item $Set = Sh(*)$ is simply the [[topos]] of [[sheaf|sheaves]] on a one-[[point]] [[topological space|space]]. \item [[geometric morphisms]] $f : Sh(Y) \to Sh(X)$ between [[category of sheaves|sheaf topoi]] are in a bijection with continuous functions of topological spaces $f : Y \to X$ (denoted by the same letter, by convenient abuse of notation); for this to hold $X$ needs to be sober. \end{itemize} It follows that for $E = Sh(X)$ points of $E$ in the sense of points of topoi are in bijection with the ordinary points of $X$. The action of the [[direct image]] $x_* : Set \to Sh(X)$ and the [[inverse image]] $x^* : Sh(X) \to Set$ of a point $x : Set \to Sh(X)$ of a sheaf topos have special interpretation and relevance: \begin{itemize}% \item The [[direct image]] of a set $S$ under the point $x : {*} \to X$ is, by definition of [[direct image]] the sheaf \begin{displaymath} x_*(S) : (U \subset X) \mapsto S(x^{-1}(U)) = \left\{ \itexarray{ S & \text{ if } x \in U \\ {*} & \text{otherwise} } \right. \end{displaymath} This is the [[skyscraper sheaf]] $skysc_x(S)$ with value $S$ supported at $x$. (In the first line on the right in the above we identify the set $S$ with the unique sheaf on the point it defines. Notice that $S(\emptyset) = pt$). \item The [[inverse image]] of a sheaf $A$ under the point $x : {*} \to X$ is by definition of [[inverse image]] (see the [[Kan extension]] formula discussed there), the set \begin{displaymath} \begin{aligned} x^*(A) & = colim_{{*} \to x^{-1}(V)} A(V) \\ &= colim_{V\subset X| x \in V} F(V) \end{aligned} \,. \end{displaymath} This is the [[stalk]] of $A$ at the point $x$, \begin{displaymath} x^*(-) = stalk_x(-) \,. \end{displaymath} \end{itemize} By definition of [[geometric morphisms]], taking the stalk at $x$ is [[left adjoint]] to forming the skyscraper sheaf at $x$: for all $S \in Set$ and $A \in Sh(X)$ we have \begin{displaymath} Hom_{Set}(stalk_x(A), S) \simeq Hom_{Sh(X)}(A, skysc_x(S)) \,. \end{displaymath} Note that the observation that the points of $Sh(X)$ are in bijection with the points of $X$ actually factors over an intermediate concept, namely that of [[point of a locale|points of a locale]]. Firstly, any topological space gives rise to a locale; if the space is sober, its points are in bijection with the locale-theoretic points of the induced locale. Secondly, for any locale ([[topological locale|spatial]] or not), its locale-theoretic points correspond to the points of its induced sheaf topos. \hypertarget{in_sheaf_toposes}{}\subsubsection*{{In sheaf toposes}}\label{in_sheaf_toposes} The following characterization of points in [[sheaf toposes]] a special case of the general statements at [[morphism of sites]]. \begin{prop} \label{}\hypertarget{}{} For $C$ a [[site]], there is an [[equivalence of categories]] \begin{displaymath} Topos(Set, Sh(C)) \simeq Site(C,Set) \,. \end{displaymath} (Morphisms of sites $C\to Set$ are precisely the [[cover-preserving functor|continuous]] [[flat functors]].) \end{prop} This appears for instance as (\hyperlink{MacLaneMoerdijk}{MacLaneMoerdijk, corollary VII.5.4}). \begin{prop} \label{}\hypertarget{}{} If $E$ is a [[Grothendieck topos]] with enough points (def. \ref{EnoughPoints}), there is a \emph{[[small set]]} of points of $E$ which are jointly conservative, and therefore a [[geometric morphism]] $Set/X \to E$, for some set $X$, which is [[surjective geometric morphism|surjective]]. \end{prop} This appears as (\hyperlink{Johnstone}{Johnstone, lemma 2.2.11, 2.2.12}). (In general, of course, a topos can have a proper class of non-isomorphic points.) \begin{prop} \label{}\hypertarget{}{} A Grothendieck topos has enough points (def. \ref{EnoughPoints}) precisely when it underlies a bounded [[ionad]]. \end{prop} \hypertarget{in_classifying_toposes}{}\subsubsection*{{In classifying toposes}}\label{in_classifying_toposes} From the above it follows that if $E$ is the [[classifying topos]] of a [[geometric theory]] $T$, then a point of $E$ is the same as a [[model]] of $T$ in [[Set]]. \hypertarget{of_toposes_with_enough_points}{}\subsubsection*{{Of toposes with enough points}}\label{of_toposes_with_enough_points} \begin{prop} \label{}\hypertarget{}{} If a [[sheaf topos]] $E$ has \emph{enough points} (def. \ref{EnoughPoints}) then \begin{itemize}% \item there exists a [[topological space]] $X$ whose [[cohomology]] and [[homotopy theory]] is the [[cohomology|intrinsic cohomology]] and [[homotopy groups in an (infinity,1)-topos|intrinsic homotopy theory]] of the topos; \item such that $E$ is the category of [[equivariant cohomology|equivariant]] objects in the [[sheaf topos]] $Sh(X)$ with respect to some groupoid action on $X$. \end{itemize} \end{prop} This is due to (\hyperlink{Butz}{Butz}) and (\hyperlink{Moerdijk}{Moerdijk}). \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item For $X$ any [[topological space]], the [[category of sheaves|topos of sheaves]] on (the [[category of open subsets]] of) $X$ has enough points (def. \ref{EnoughPoints}): a morphism of sheaves is a mono-/epi-/isomorphism precisely if it is so on every [[stalk]]. \item Points of [[over-toposes]] are discussed at . \end{itemize} \hypertarget{Rings}{}\subsubsection*{{Rings and algebraic theories}}\label{Rings} The category of points of the presheaf topos over $Ring_{fp}^{op}$, the dual of the category of finitely presented [[rings]], is the category of all rings (without a size or presentation restriction). In fact this holds for any [[algebraic theory]], not only for the theory of commutative rings. See also at \emph{[[Gabriel-Ulmer duality]]}, \emph{[[flat functors]]}. \hypertarget{of_a_local_topos}{}\subsubsection*{{Of a local topos}}\label{of_a_local_topos} A [[local topos]] $(\Delta \dashv \Gamma \dashv coDisc) : E \to Set$ has a canonical point, $(\Gamma \dashv coDisc) : Set \to E$. Morover, this point is an [[initial object]] in the category of all points of $E$ (see \emph{\href{local+geometric+morphism#EquivalentCharacterizations}{Equivalent characterizations}} at \emph{[[local topos]]}.) \hypertarget{over_cohesive_sites}{}\subsubsection*{{Over $\infty$-cohesive sites}}\label{over_cohesive_sites} \begin{itemize}% \item Let [[Diff]] be a [[small category]] version of the category of smooth manifolds (for instance take it to be the category of manifolds embedded in $\mathbb{R}^\infty$). Then the sheaf topos $Sh(Diff)$ has precisely one point $p_n$ per natural number $n \in \mathbb{N}$ , corresponding to the $n$-ball: the [[stalk]] of a sheaf on $Diff$ at that point is the colimit over the result of evaluating the sheaf on all $n$-dimensional smooth balls. This is discussed for instance in (\hyperlink{Dugger}{Dugger, p. 36}) in the context of the [[model structure on simplicial presheaves]]. \end{itemize} \hypertarget{toposes_with_enough_points}{}\subsubsection*{{Toposes with enough points}}\label{toposes_with_enough_points} The following classes of topos have enough points (def. \ref{EnoughPoints}). \begin{itemize}% \item every [[presheaf topos]]; \item every [[coherent topos]] (due to the [[Deligne completeness theorem]]); \item every [[Galois topos]] (see (\hyperlink{Zoghaib}{Zoghaib})). \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[topological locale]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} Textbook references are section 7.5 of \begin{itemize}% \item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]} \end{itemize} as well as section C2.2 of \begin{itemize}% \item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]} \end{itemize} In \begin{itemize}% \item Carsten Butz, \emph{Logical and cohomological aspects of the space of points of a topos} (\href{http://www.academia.edu/9139781/Logical_and_Cohomological_Aspects_of_the_Space_of_Points_of_a_Topos}{web}) \end{itemize} is a discussion of how for every topos with enough points there is a [[topological space]] whose [[cohomology]] and [[homotopy theory]] is related to the [[cohomology|intrinsic cohomology]] and [[homotopy groups in an (infinity,1)-topos|intrinsic homtopy theory]] of the topos. More on this is in \begin{itemize}% \item [[Ieke Moerdijk]], \emph{Classifying toposes for toposes with enough points} , Milan Journal of Mathematics Volume 66, Number 1, 377-389 \end{itemize} See also \begin{itemize}% \item Sam Zoghaib, \emph{A few points in topos theory} (\href{http://www.normalesup.org/~zoghaib/math/points-tt.pdf}{pdf}) \end{itemize} Points of the sheaf topos over the category of [[manifold]]s are discussed in \begin{itemize}% \item [[Dan Dugger]], \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} [[!redirects points of a topos]] [[!redirects points of toposes]] [[!redirects points of topoi]] [[!redirects point of topos]] [[!redirects enough points]] [[!redirects topos with enough points]] [[!redirects topos point]] [[!redirects topos points]] \end{document}