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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*{locally connected topos} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory} [[!include topos 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{characterization_over_locally_connected_sites}{Characterization over locally connected sites}\dotfill \pageref*{characterization_over_locally_connected_sites} \linebreak \noindent\hyperlink{equivalent_conditions}{Equivalent conditions}\dotfill \pageref*{equivalent_conditions} \linebreak \noindent\hyperlink{Connected}{Locally connected and connected}\dotfill \pageref*{Connected} \linebreak \noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A [[topos]] may be thought of as a generalized [[topological space]]. Accordingly, the notions of \begin{itemize}% \item [[locally connected space]] \item locally 2-[[connected]] space \item etc. \ldots{} \item [[locally contractible space]] \end{itemize} have analogs for [[topos]]es and [[(∞,1)-topos]]es \begin{itemize}% \item \textbf{locally connected topos} \item [[locally n-connected (n,1)-topos]] \item etc. \ldots{} \end{itemize} \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{ConnectedObject}\hypertarget{ConnectedObject}{} An object $A$ in a [[topos]] $\mathcal{E}$ is called a \textbf{[[connected object]]} if the [[hom-functor]] $\mathcal{E}(A, -)$ preserves finite [[coproduct]]s. Equivalently, an object $A$ is connected if it is nonempty (non[[initial object|initial]]) and cannot be expressed as a coproduct of two nonempty [[subobject]]s. \end{defn} \begin{defn} \label{}\hypertarget{}{} A [[Grothendieck topos]] $\mathcal{E}$ is called a \textbf{locally connected topos} if every object $A \in \mathcal{E}$ is a [[coproduct]] of connected objects $\{A_i\}_{i \in I}$, $A = \coprod_{i \in I} A_i$. \end{defn} It follows that the index set $I$ is unique up to isomorphism, and we write \begin{displaymath} \pi_0(A) = I \,. \end{displaymath} This construction defines a functor $\Pi_0 : \mathcal{E} \to Set : A \mapsto \pi_0(A)$ which is [[left adjoint]] to the [[constant sheaf]] functor, the [[left adjoint]] part of the [[global section]] [[geometric morphism]]. Thus, for a locally connected topos we have \begin{displaymath} (\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{Const}{\leftarrow}}{\underset{\Gamma}{\to}}} Set \,. \end{displaymath} This is the \textbf{connected component functor}. It generalises the functor, also denoted $\pi_0$ or $\Pi_0$, which to a [[topological space]] assigns the set of [[connected]] components of that space. See the \hyperlink{Examples}{examples} below. The following proposition asserts that the existence of $\Pi_0$ already characterizes locally connected toposes. \begin{prop} \label{LocalConnectednessByEssentialGeometricMorphism}\hypertarget{LocalConnectednessByEssentialGeometricMorphism}{} A [[Grothendieck topos]] $\mathcal{E}$ is locally connected precisely if the [[global section]] [[geometric morphism]] $\Gamma : \mathcal{E} \to Set$ is an [[essential geometric morphism]] $(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set$. \end{prop} A proof appears as (\hyperlink{Johnstone}{Johnstone, lemma C.3.3.6}). \begin{proof} Suppose that $(\Pi_0 \dashv L Const \dashv \Gamma) : \mathcal{E} \to Set$ exists. First notice that an object $A$ is connected in the \hyperlink{ConnectedObject}{above sense} precisely if $\Pi_0(A) = *$. Because for all $S \in Set$ the connectivity condition demands that \begin{displaymath} \mathcal{E}(A, \coprod_S L Const *) \simeq \coprod_S \mathcal{E}(A,*) \simeq \coprod_S * \simeq S \end{displaymath} but by the $(\Pi_0 \dashv L Const)$-hom-equivalence the first term is \begin{displaymath} \cdots \simeq \mathcal{E}(A, L Const \coprod_S *) \simeq Set(\Pi_0(A), S) \end{displaymath} and the last set is isomorphic to $S$ precisely for $\Pi_0(A)$ is the singleton set. So we need to show that given the extra left adjoint $\Pi_0$, every object of $\mathcal{E}$ is a coproduct of objects for which $\Pi_0(-)$ is the point. For that purpose consider for every object $A \in \mathcal{E}$ the [[pullback]] diagram \begin{displaymath} \itexarray{ i_A^* {\lim_\to}_{\Pi_0(A)} * &\to& {\lim_\to}_{\Pi_0(A)} * \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ A &\stackrel{i_A}{\to}& L Const \Pi_0 (A) } \,, \end{displaymath} where the bottom morphism is the $(\Pi \dashv L Const)$-[[unit of an adjunction|unit]] and the right [[isomorphism]] is the identification of any set as the [[colimit]] (here: [[coproduct]]) of the functor over the set itself that is constant on the point. Since pullbacks of isomorphism are isomorphisms, also the left morphism is an iso. By [[universal colimits]] this left morphism is equivalently \begin{displaymath} {\lim_\to}_{s \in \Pi_0(A)} (i_A^* *_s) \stackrel{\simeq}{\to} A \end{displaymath} and hence expresses $A$ as a coproduct of objects $i_A^* *_s$, each of which is a [[pullback]] \begin{displaymath} \itexarray{ i_A^* *_s &\to& L Const * \\ \downarrow && \downarrow^{\mathrlap{s}} \\ A &\stackrel{i_A}{\to}& L Const \Pi_0 A } \,, \end{displaymath} where the right morphism includes the element $s$ into the set $\Pi_0 A$. By applying $\Pi_0$ to this diagram and pasting on the $(\Pi_0 \dashv L Const)$-[[unit of an adjunction|counit]] we get \begin{displaymath} \itexarray{ \Pi_0(i_A^* *_s) &\to& \Pi_0 L Const * &\to& * \\ \downarrow && \downarrow^{} && \downarrow \\ \Pi_0(A) &\stackrel{\Pi_0(i_A)}{\to}& \Pi_0 L Const \Pi_0 A &\to& \Pi_0 A } \end{displaymath} and by the [[zig-zag identity]] the bottom morphism is the identity. This says that in \begin{displaymath} \Pi_0( {\lim_{\to}}_{\Pi_0 A} i_A^* *_s \stackrel{\simeq }{\to} A) \simeq ({\lim_\to}_{\Pi_0 A} \Pi_0(i_A^* *_s) \stackrel{\simeq}{\to} \Pi_0(A)) \end{displaymath} all the component maps out of the coproduct factor through the point. This means that this can only be an isomorphism if all these component maps are point inclusions, hence if $\Pi_0(i_A^* *_s) \simeq *$ for all $s \in \Pi_0 A$. \end{proof} However, this doesn't mean that essential geometric morphisms are the ``relative'' analog of locally connected toposes; in general one needs to impose an additional condition, which is automatic in the case of the global sections morphism, to obtain the notion of a [[locally connected geometric morphism]]. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{characterization_over_locally_connected_sites}{}\subsubsection*{{Characterization over locally connected sites}}\label{characterization_over_locally_connected_sites} See at \emph{[[locally connected site]]}. \hypertarget{equivalent_conditions}{}\subsubsection*{{Equivalent conditions}}\label{equivalent_conditions} \begin{defn} \label{}\hypertarget{}{} For $C$ and $C$ [[cartesian closed categories]], a [[functor]] $F : C \to D$ that preserves [[product]]s is called a \textbf{[[cartesian closed functor]]} if the canonical [[natural transformation]] \begin{displaymath} F(B^A) \to (F(B))^{F(A)} \end{displaymath} (which is the [[adjunct]] of $F(A) \times F(B^A) \simeq F(A \times B^A) \to F(B)$) is an [[isomorphism]]. \end{defn} \begin{prop} \label{}\hypertarget{}{} The [[constant sheaf]]-functor $\Delta : \mathcal{S} \to \mathcal{E}$ is a [[cartesian closed functor]] precisely if $\mathcal{E}$ is a locally connected topos. \end{prop} \hypertarget{Connected}{}\subsubsection*{{Locally connected and connected}}\label{Connected} A [[topos]] $E$ is called a [[connected topos]] if the [[left adjoint]] $L Const : Set \to E$ is a [[full and faithful functor]]. \begin{prop} \label{}\hypertarget{}{} If $\Gamma \colon E\to Set$ is a locally connected topos, then it is also a [[connected topos]] --- in that $L Const$ is full and faithful --- if and only if the [[left adjoint]] $\Pi_0$ of $L Const$ preserves the [[terminal object]]. \end{prop} This is (\hyperlink{Johnstone}{Johnstone, C3.3.3}). Notice that for a connected and locally connected topos, the adjunction \begin{displaymath} Set \stackrel{\overset{\Pi_0}{\leftarrow}}{\hookrightarrow} E \end{displaymath} exhibits [[Set]] as a [[reflective subcategory]] of $E$. We may think then of [[Set]] as being the [[localization]] of $E$ at those morphisms that induce isomorphisms of connected components. \hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples} \begin{example} \label{LocallyConnectedTopologicalSpace}\hypertarget{LocallyConnectedTopologicalSpace}{} For $X$ a [[topological space]], the [[category of sheaves]] $Sh(X) \coloneqq Sh(Op(X))$ is a locally connected topos precisely if $X$ is a [[locally connected space]]. The functor $\Pi_0$ sends a sheaf $F \in Sh(X)$ to the set of connected components of the corresponding [[etale space]]. \end{example} \begin{example} \label{SmoothSpaces}\hypertarget{SmoothSpaces}{} For $C =$ [[CartSp]] the [[site]] of [[Cartesian spaces]] with its [[good open cover]] [[coverage]], the topos $Sh(CartSp)$ of [[smooth spaces]] is locally connected. An arbitrary $X \in Sh(CartSp)$ is sent to the [[colimit]] $\lim_\to X \in Set$. If $X$ is a [[diffeological space]] or even a [[smooth manifold]], then this is the set of connected components of the underlying topological space. \end{example} \begin{example} \label{LCC}\hypertarget{LCC}{} Every locally connected geometric morphism is a [[locally cartesian closed functor]]. \end{example} \begin{example} \label{}\hypertarget{}{} Suppose that $C$ is a [[site]] such that constant [[presheaves]] on $C$ are [[sheaves]]. Then the left adjoint $\Pi_0$ exists and is given by the [[colimit]] functor: if we write $L : PSh(C) \to Sh(C)$ for [[sheafification]], then for any sheaf $X$, we have \begin{displaymath} Hom_{Sh(C)}(X, L Const S) \simeq Hom_{PSh(C)}(X, L Const S) \simeq Hom_{PSh(C)}(X, Const S) \simeq Hom_{Set}(\lim_\to X, S) \,. \end{displaymath} In particular, this is the case if every covering [[sieve]] in $C$ is connected, i.e. $C$ is a [[locally connected site]]. If $C$ furthermore has a terminal object $1$, then the global sections functor $\Gamma\colon Sh(C)\to Set$ (the right adjoint of $L Const$) is simply given by evaluation at $1$, and so the unit $S \to \Gamma L Const S \cong L Const S(1)$ is an isomorphism. Thus in this case $Sh(C)$ is additionally [[connected topos|connected]]. This situation also applies to $C=CartSp$ in example \ref{SmoothSpaces} above. \end{example} \begin{example} \label{}\hypertarget{}{} If $C$ is a category with all [[finite limits]] and if the unique functor $\pi \colon C \to \ast$ to the [[terminal category]] is a [[cover-preserving functor]] (for $\ast$ equipped with the trivial topology/[[coverage]]) then $Sh(C)$ is locally connected. (In particular, this holds for presheaf toposes). This is because the inclusion of the terminal object $i \colon \ast \to C$ provides a [[right adjoint]] to $\pi$, so that there is an [[adjoint quadruple]] of functors on [[presheaf categories]] \begin{displaymath} (\pi_! \simeq Lan_\pi) \dashv (\pi^\ast \simeq i_! \simeq Lan_i) \dashv (\pi_\ast \simeq i^\ast ) \dashv (\pi^! \simeq i_* \simeq Ran_i) \;\colon\; PSh(C) \leftrightarrow PSh(\ast) \simeq Sh(C) \simeq Set \,, \end{displaymath} where $Lan_{(-)}$ and $Ran_{(-)}$ denote let and right [[Kan extension]], respectively. Now if $C \to \ast$ indeed preserves covers and using that $C \to \ast$ trivially preserves finite limits and hence is a [[flat functor]], then by the discussion at \emph{[[morphism of sites]]} the first three functors here descend to [[sheaves]] and hence exhibit $Sh(C)$ as being locally connected. But \textbf{beware} that the assumptions here are stronger than they may seem: that $C \to \ast$ preserves covers is not automatic, but is a strong condition. It is violated as soon as $C$ contains an empty object with [[empty set|empty]] cover, such as is the case in most categories of [[spaces]], notably in [[categories of open subsets]] $Op(X)$ of a [[topological space]] $X$, as in example \ref{LocallyConnectedTopologicalSpace}. \end{example} \hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries} \begin{itemize}% \item [[essential geometric morphism]] \item \textbf{locally connected topos} / [[locally ∞-connected (∞,1)-topos]] \item [[connected topos]] / [[∞-connected (∞,1)-topos]] \item [[totally connected topos]] / [[totally connected (∞,1)-topos]] \item [[local topos]] / [[local (∞,1)-topos]]. \item [[cohesive topos]] / [[cohesive (∞,1)-topos]] \end{itemize} \hypertarget{References}{}\subsection*{{References}}\label{References} Section C1.5 and C3.3 of \begin{itemize}% \item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]} . \end{itemize} A variant is in \begin{itemize}% \item [[Marta Bunge]], Funk, \emph{Quasi locally connected toposes} (\href{http://www.tac.mta.ca/tac/volumes/18/8/18-08.pdf}{pdf}) \end{itemize} Discussion of characterizations of [[sites]] of definition of locally connected toposes is in \begin{itemize}% \item [[Olivia Caramello]], \emph{Site characterizations for geometric invariants of toposes}, Theory and Applications of Categories, Vol. 26, 2012, No. 25, pp 710-728. (\href{http://www.tac.mta.ca/tac/volumes/26/25/26-25abs.html}{TAC}) \end{itemize} [[!redirects locally connected toposes]] [[!redirects locally connected topoi]] \end{document}