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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*{classifying topos} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory} [[!include topos theory - contents]] \hypertarget{yoneda_lemma}{}\paragraph*{{Yoneda lemma}}\label{yoneda_lemma} [[!include Yoneda lemma - 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{background_on_the_theory_of_theories}{Background on the theory of theories}\dotfill \pageref*{background_on_the_theory_of_theories} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{GeometricMorphismsAndMorphismsOfSites}{Geometric morphisms equivalent to morphisms of sites}\dotfill \pageref*{GeometricMorphismsAndMorphismsOfSites} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{ForObjects}{For objects}\dotfill \pageref*{ForObjects} \linebreak \noindent\hyperlink{ForPointedObjects}{For pointed objects}\dotfill \pageref*{ForPointedObjects} \linebreak \noindent\hyperlink{for_groups}{For groups}\dotfill \pageref*{for_groups} \linebreak \noindent\hyperlink{for_rings}{For rings}\dotfill \pageref*{for_rings} \linebreak \noindent\hyperlink{ForLinearOrders}{For (inhabited) linear orders}\dotfill \pageref*{ForLinearOrders} \linebreak \noindent\hyperlink{ForIntervals}{For intervals}\dotfill \pageref*{ForIntervals} \linebreak \noindent\hyperlink{for_abstract_circles}{For abstract circles}\dotfill \pageref*{for_abstract_circles} \linebreak \noindent\hyperlink{for_local_rings}{For local rings}\dotfill \pageref*{for_local_rings} \linebreak \noindent\hyperlink{local_rings}{Local rings}\dotfill \pageref*{local_rings} \linebreak \noindent\hyperlink{StrictLocalRings}{Strict local rings}\dotfill \pageref*{StrictLocalRings} \linebreak \noindent\hyperlink{PrincipalBund}{For principal bundles}\dotfill \pageref*{PrincipalBund} \linebreak \noindent\hyperlink{BareGTor}{Over bare groups}\dotfill \pageref*{BareGTor} \linebreak \noindent\hyperlink{BareGTorGeomTheo}{In terms of geometric theories}\dotfill \pageref*{BareGTorGeomTheo} \linebreak \noindent\hyperlink{TopGTor}{Over topological groups}\dotfill \pageref*{TopGTor} \linebreak \noindent\hyperlink{UniversalBundle}{The universal $G$-bundle topos}\dotfill \pageref*{UniversalBundle} \linebreak \noindent\hyperlink{ForLocalicGroupoids}{For general localic groupoids}\dotfill \pageref*{ForLocalicGroupoids} \linebreak \noindent\hyperlink{for_flat_functors}{For flat functors}\dotfill \pageref*{for_flat_functors} \linebreak \noindent\hyperlink{CoverPreservingFLatFunctors}{For geometric theories / cover-preserving flat functors on a site}\dotfill \pageref*{CoverPreservingFLatFunctors} \linebreak \noindent\hyperlink{LocalAlgebras}{For local algebras}\dotfill \pageref*{LocalAlgebras} \linebreak \noindent\hyperlink{as_a_generalization_of_the_notion_of_classifying_space_in_topology}{As a generalization of the notion of classifying space in topology}\dotfill \pageref*{as_a_generalization_of_the_notion_of_classifying_space_in_topology} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{ReferencesRelationToForcing}{Relation to forcing}\dotfill \pageref*{ReferencesRelationToForcing} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{classifying topos} for a given type of mathematical [[structure]] $T$ --- for example the structures: ``[[group]]'', ``[[torsor]]'', ``[[ring]]'', ``[[category]]'' etc. --- is a ([[Grothendieck topos|Grothendieck]]) [[topos]] $S[T]$ such that [[geometric morphisms]] $f: E \to S[T]$ are the same as structures of this sort in the topos $E$, i.e. groups [[internalization|internal to]] $E$, torsors internal to $E$, etc. In other words, a classifying topos is a [[representing object]] for the functor which sends a topos $E$ to the category of structures of the desired sort in $E$. In particular for $E$ a [[sheaf topos]] on a [[topological space]] $X$ and $G$ a (bare, i.e. discrete) [[group]], a $G$-torsor in $E$ is a $G$-[[principal bundle]] over $X$. There is a classifying topos denoted $B G$, such that the [[groupoid]] $G Bund(X)$ of $G$-[[principal bundle]]s over $X$ is equivalent to geometric morphims $Sh(X) \to B G$: \begin{displaymath} G Bund(X) \simeq Topos(Sh(X), B G) \,. \end{displaymath} This is evidently analogous to the notion of [[classifying space]] in [[topology]], which for the discrete group $G$ is a [[topological space]] $\mathcal{B} G$ such that \begin{displaymath} \pi_0 G Bund(X) \simeq \pi_0 Top(X, \mathcal{B}G) \,. \end{displaymath} Hence one can think of classifying topoi as a grand generalization of the notion of [[classifying space]] in topology. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} In a tautological way, every [[topos]] $F$ is the classifying topos for something, namely for the categories of [[geometric morphisms]] $E \to F$ into it. The concept of [[geometric theory]] allows one to usefully interpret these categories as \emph{categories of certain structures in $E$} : as decribed in \emph{\href{geometric+theory#InTermsOfSheafTopoi}{Geometric theories -- In terms of sheaf topoi}}, every [[sheaf topos]] $F$ is a completion $S[T]$ of the [[syntactic category]] $C_T$ of \emph{some} [[geometric theory]] $T$ \begin{displaymath} F \simeq S[T] \,. \end{displaymath} And structures of type $T$ in $E$ is what geometric morphisms $E \to F$ classify. So the \textbf{classifying [[topos]]} for the [[geometric theory]] $T$ is a [[Grothendieck topos]] $S[T]$ equipped with a ``universal model $U$ of $T$''. This means that for any Grothendieck topos $E$ together with a model $X$ of $T$ in $E$, there exists a unique (up to isomorphism) [[geometric morphism]] $f: E \to S[T]$ such that $f^*$ maps the $T$-model $U$ to the model $X$. More precisely, for any Grothendieck topos $E$, the category of $T$-models in $E$ is equivalent to the category of geometric morphisms $E \to S[T]$. The fact that a classifying topos is like the ambient [[set theory]] but equipped with that universal model is essentially the notion of \emph{[[forcing]]} in [[logic]]: the passage to the [[internal logic]] of the classifying topos \emph{forces} the universal model to exist. If $C_T$ is the [[syntactic category]] of $T$, so that $T$-models are the same as [[geometric category|geometric functors]] out of $C_T$, then this universal model can be identified with a certain geometric functor \begin{displaymath} U : C_T \to S[T] \,. \end{displaymath} Its universality property means that any geometric functor \begin{displaymath} X : C_T \to E \end{displaymath} factors essentially uniquely as \begin{displaymath} X : C_T \stackrel{U}{\to} S[T] \stackrel{f^*}{\to} E \end{displaymath} for $U$ the universal model and $f^*$ the [[left adjoint]] part of a [[geometric morphism]]. More precisely, composition with $U$ defines an equivalence between the category of geometric morphisms $E\to S[T]$ and the category of geometric functors $C_T\to E$. More specifically, for any [[cartesian theory]], [[regular theory]] or [[coherent theory]] $\mathbb{T}$ (which in ascending order are special cases of each other and all of geometric theories), the corresponding [[syntactic category]] $\mathcal{C}_{\mathbb{T}}$ comes equipped with the structure of a [[syntactic site]] $(\mathcal{C},\mathbb{T}, J)$ (see there) and the classifying topos for $\mathbb{T}$ is the [[sheaf topos]] $Sh(\mathcal{C}_{\mathbb{T}}, J)$. Classifying toposes can also be defined over any [[base topos]] $S$ instead of [[Set]]. In that case ``Grothendieck topos'' is replaced by ``[[bounded topos|bounded]] $S$-topos''. The \emph{general existence of classifying toposes} for geometric theories for bounded $S$-toposes is then intimately connected to the existence of the [[classifying topos for the theory of objects]] which in turn hinges on the existence of a [[natural number object]] in $S$. See \hyperlink{ForObjects}{below} and, for further details, [[classifying topos for the theory of objects]] or Blass (\hyperlink{blass}{1989}). If the classifying topos of a geometric theory $T$ is a [[presheaf topos]], one calls $T$ a \emph{[[theory of presheaf type]]}. \hypertarget{background_on_the_theory_of_theories}{}\subsection*{{Background on the theory of theories}}\label{background_on_the_theory_of_theories} The notion of \emph{classifying topos} is part of a trend, begun by [[Bill Lawvere|Lawvere]], of viewing a mathematical [[theory]] in [[logic]] as a [[category]] with suitable [[exact functor|exactness]] properties and which contains a ``generic model'', and a [[model]] of the theory as a functor which preserves those properties. This is described in more detail at [[internal logic]] and [[type theory]], but here are some simple examples to give the flavor. The original example is that of a `finite products theory': \begin{itemize}% \item \textbf{Finite products theory.} Roughly speaking, a `finite products theory', `[[Lawvere theory]]', or `[[algebraic theory]]' is a [[theory]] describing some mathematical structure that can be defined in an arbitrary category with finite [[product]]s. An example would be the theory of [[groups]]. As explained in the entry for [[Lawvere theory]], for each such theory $T$ there is a category with finite products $C_{fp}[T]$ -- the [[syntactic category]], which serves as a ``classifying category'' for $T$, in that models of $T$ in the category of sets correspond to product-preserving [[functor]]s $f : C_{fp}[T] \to Set$. More generally, for any category with finite products, say $E$, models of $T$ in $E$ correspond to product-preserving functors $f : C_{fp}[T] \to E$. \item \textbf{Finite limits theory.} Next up the line is the notion of `finite limits theory', sometimes called an [[essentially algebraic theory]]. This is roughly a theory describing some structure that can be defined in an arbitrary category with [[finite limit]]s (also called a [[finitely complete category]]). An example of a finite limits theory would be the theory of categories. (The notion of `category' requires finite limits, while the notion of `group' does not, because categories but not groups involve a \emph{partially defined} operation, namely composition of morphisms.) Every finite limits theory $T$ admits a [[syntactic category|classifying category]] $C_{fl}(T)$: a finitely complete category such that models of $T$ in a category $E$ with finite limits correspond to functors $f: C_{fl}(T) \to E$ that preserve finite limits. (Such functors are called [[left exact]], or `lex' for short.) \item \textbf{Geometric theory.} Further up the line, a [[geometric theory]] is roughly a theory which can be formulated in that fragment of first-order logic that deals in finite limits and arbitrary (small) colimits, plus certain [[familial regularity and exactness|exactness]] properties the details of which need not concern us. The point is that a category with finite limits, small colimits, and appropriate exactness is just a Grothendieck topos, and a functor preserving finite limits and small colimits is just the inverse image part of a geometric morphism. Just as in the previous two cases, any `geometric theory' has a classifying category $S[T]$ (which is now a Grothendieck topos) which possesses a ``generic object'' for that theory, and T-models in any other Grothendieck topos E can be identified with geometric morphisms $f\colon E\to S[T]$, or specifically with their inverse image parts. \end{itemize} Each type of theory may be considered a $2$-theory, or [[doctrine]]. Furthermore, each type of theory can be promoted to a theory ``further up the line'', by [[completion|freely adding]] the missing structure to the classifying category. This can always be done purely formally, but in a few cases this promotion also has other, more explicit descriptions. For instance, to go from a finite products theory $T$ to the corresponding finite limits theory, we can take the opposite of the category of [[finitely presentable object|finitely presentable models]] of $T$ in $Set$, thanks to [[Gabriel-Ulmer duality]]. Similarly, to go from a finite limits theory to the classifying topos of the corresponding geometric theory, we can take the category of presheaves on the classifying category of the finite limits theory. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{GeometricMorphismsAndMorphismsOfSites}{}\subsubsection*{{Geometric morphisms equivalent to morphisms of sites}}\label{GeometricMorphismsAndMorphismsOfSites} The fact that classifying toposes are what they are all comes down, if spelled out explicitly, to the fact that a [[geometric morphism]] $f : \mathcal{E} \to \mathcal{F}$ of toposes can be identified with a certain morphism of [[site]]s $C_{\mathcal{E}}$, $C_{\mathcal{F}}$ for these toposes, going the other way round, $C_\mathcal{E} \leftarrow C_{\mathcal{F}}$, and having certain properties. If here $C_\mathcal{F}$ is the [[syntactic site]] of some [[theory]] $\mathbb{T}$ and we choose $C_{\mathcal{E}}$ to be the [[canonical site]] of $\mathcal{E}$ (itself equipped with the [[canonical coverage]]) this makes manifest why the geometric morphisms in $\mathcal{F}$ correspond to [[model]]s of $\mathcal{T}$ in $\mathcal{E}$. We now say this in precise manner. In the following a \emph{[[cartesian site]]} means a [[site]] whose underlying category is [[finitely complete category|finitely complete]]. \begin{prop} \label{}\hypertarget{}{} Let $(\mathcal{C}, J)$ and $(\mathcal{D}, K)$ be [[cartesian site]]s such that $\mathcal{C}$ is a [[small category]], $\mathcal{D}$ is an [[essentially small category]] and the [[coverage]] $K$ is [[subcanonical coverage|subcanonical]]. Then a [[geometric morphism]] of the corresponding [[sheaf toposes]] \begin{displaymath} f : Sh(\mathcal{D}, K) \to Sh(\mathcal{C}, J) \end{displaymath} is induced by a [[morphism of sites]] \begin{displaymath} (\mathcal{D}, K) \leftarrow (\mathcal{C}, J) \end{displaymath} precisely if the [[inverse image]] of $f$ respects the [[Yoneda embedding]]s $j$ as \begin{displaymath} \itexarray{ \mathcal{D } &\leftarrow& \mathcal{C} \\ {}^{\mathllap{j_{\mathcal{D}}}}\downarrow && \downarrow^{\mathrlap{j_{\mathcal{C}}}} \\ Sh(\mathcal{D}, K) &\stackrel{f^*}{\leftarrow}& Sh(\mathcal{C}, J) } \,. \end{displaymath} \end{prop} This appears as (\hyperlink{Johnstone}{Johnstone, lemma C2.3.8}). \begin{proof} It suffices to observe that the factorization, if it exists, is a morphism of sites. \end{proof} \begin{cor} \label{SheafToposesAreClassifyingForTheirTheoryOfLocalAlgegras}\hypertarget{SheafToposesAreClassifyingForTheirTheoryOfLocalAlgegras}{} Let $(\mathcal{C},J)$ be a [[small category|small]] [[cartesian site]] and let $\mathcal{E}$ be any [[sheaf topos]]. Then we have an [[equivalence of categories]] \begin{displaymath} Topos(\mathcal{E}, Sh(\mathcal{C}, J)) \simeq Site((\mathcal{C}, J), (\mathcal{E}, C)) \end{displaymath} between the [[geometric morphism]]s from $\mathcal{E}$ to $Sh(\mathcal{C}, J)$ and the morphisms of [[sites]] from $(\mathcal{C}, J)$ to the [[big site]] $(\mathcal{E}, C)$ for $C$ the [[canonical coverage]] on $\mathcal{E}$. \end{cor} This appears as (\hyperlink{Johnstone}{Johnstone, cor. C2.3.9}). \begin{remark} \label{}\hypertarget{}{} This means that a sheaf topos $Sh(\mathcal{C},J)$ is the classifying topos for the theory of [[local algebras]] determined by the [[site]] $(\mathcal{C},J)$. \end{remark} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} We list and discuss explicit examples of classifying toposes. \hypertarget{ForObjects}{}\subsubsection*{{For objects}}\label{ForObjects} The [[presheaf topos]] $[FinSet, Set]$ on the [[opposite category]] of [[FinSet]] is the classifying topos for the [[theory of objects]], sometimes called the ``object classifier''. This is not to be confused with the notion of an [[object classifier]] in an [[(∞,1)-topos]] and maybe better called in full the \emph{[[classifying topos for the theory of objects]]}. For $E$ any [[topos]], a [[geometric morphism]] $E \to [FinSet,Set]$ is equivalently just an [[object]] of $E$. \hypertarget{ForPointedObjects}{}\subsubsection*{{For pointed objects}}\label{ForPointedObjects} Similarly, the presheaf topos $[FinSet_*, Set]$ (where $FinSet_*$ is the category of finite [[pointed sets]]) classifies [[pointed objects]]; cf. \href{http://mathoverflow.net/questions/85600/what-do-gamma-sets-classify}{this question} and answer. This is the topos of ``$\Gamma$-sets''; see [[Gamma-space]]. \hypertarget{for_groups}{}\subsubsection*{{For groups}}\label{for_groups} We discuss the [[finite product theory]] of [[group]]s. This theory has a classifying category $C_{fp}(Grp)$. $C_{fp}(Grp)$ is a category with finite products equipped with an object $G$, the ``[[walking]] group'', a morphism $m: G \times G \to G$ describing multiplication, a morphism $inv : G \to G$ describing inverses, and a morphism $i: 1 \to G$ describing the identity element of $G$, obeying the usual group axioms. For any category with finite products, say $E$, a finite-product-preserving functor $f: C_{fp}(Grp) \to E$ is the same as a [[group object]] in $E$. For more details, see [[Lawvere theory]]. We can promote $C_{fp}(Grp)$ to a category with finite limits, $C_{fl}(Grp)$, by adjoining all finite limits. As mentioned above, one way to do this is to take the category of models of $C_{fp}(Grp)$ in Set, which is simply $Grp$, and then take the full subcategory of [[finitely presentable object|finitely presentable]] groups. By [[Gabriel-Ulmer duality]], the opposite of this is $C_{fl}(Grp)$. For any category with finite products, say $E$, a left exact functor $f: C_{fl}(Grp) \to E$ is the same as a [[group object]] in $E$. We can further promote $C_{fl}(Grp)$ to a [[Grothendieck topos]] by taking the category of [[presheaves]]. This gives the classifying topos for groups: \begin{displaymath} S[Grp] = Set^{C_{fl}(Grp)^{op}} \, . \end{displaymath} For any Grothendieck topos, say $E$, a left exact left adjoint functor $f^*: S[Grp] \to E$ is the same as a [[group object]] in $E$. \hypertarget{for_rings}{}\subsubsection*{{For rings}}\label{for_rings} The discussion above for groups can be repeated verbatim for rings, since they too are described by a finite products theory. \hypertarget{ForLinearOrders}{}\subsubsection*{{For (inhabited) linear orders}}\label{ForLinearOrders} \begin{prop} \label{}\hypertarget{}{} The category of [[cosimplicial sets]] $[\Delta, Set]$ -- hence the [[presheaf topos]] over the [[opposite category]] $\Delta^{op}$ of the [[simplex category]] -- is the classifying topos for [[inhabited set|inhabited]] [[linear orders]]. \end{prop} This appears as (\hyperlink{Moerdijk95}{Moerdijk 95, prop. 5.4}). \begin{proof} For ease of notation we discuss this in [[Set]], hence we show that [[geometric morphisms]] $Set \to PSh(\Delta^{op})$ are equivalently [[linear orders]]. Or, by [[Diaconescu's theorem]], that [[flat functors]] \begin{displaymath} X : \Delta^{op} \to Set \end{displaymath} are equivalently linear orders. Evidently, such a functor is in particular a [[simplicial set]] and we will show that $X$ being flat is equivalent to this simplicial set being the [[nerve]] of an [[inhabited set|inhabited]] linear order regarded as a [[category]] (a [[(0,1)-category]]). First assume that $X$ is a [[flat functor]]. Since (by the discussion there) this preserves all [[finite limits]] that exist in $\Delta^{op}$, equivalently that it sends the finite [[colimits]] that exist in $\Delta$ to limits in $Set$, it in particular sends the gluings of intervals \begin{displaymath} \begin{aligned} [n] & \simeq [k] \coprod_{[0]} [l] \;\;\;\; (n = k + l) \\ & \simeq [1] \coprod_{[0]} [1] \coprod_{[0]} \cdots \coprod_{[0]} [1] \end{aligned} \end{displaymath} in $\Delta$ to [[isomorphisms]] \begin{displaymath} \begin{aligned} X_n & \simeq X_k \times_{X_0} X_l \\ & \simeq X_1 \times_{X_0} \cdots \times_{X_0} X_1 \end{aligned} \,. \end{displaymath} This are the [[Segal space|Segal relations]] that say that $X$ is the [[nerve]] of a [[category]]. Moreover, since [[monomorphisms]] are characterized by [[pullbacks]], $F$ being flat means that it sends jointly epimorphic families of morphisms in $\Delta$ to monomorphisms in $Set$. In particular, the epimorphic family $\{\partial_0 : [0] \to [1], \partial_1 : [0] \to [1]\}$ is sent to an injection \begin{displaymath} (d_0, d_1) : X_1 \hookrightarrow X_0 \times X_0 \,. \end{displaymath} Since $X_1$ is the set of [[morphisms]] of the category that $X$ is the nerve of, this means that there is at most one morphism in this category from any one object to any other. Hence this category is a [[poset]]. Finally to show that this poset is an inhabited linear order, we use the fact that a functor is flat precisely if its [[category of elements]] [[filtered category|cofiltered]]. This means \begin{enumerate}% \item The category of elements is inhabited, hence the poset of which $X$ is the nerve is inhabited. \item For every two elements $y, z \in X_0$ there exist morphisms $\alpha, \beta : [0] \to [k]$ in $\Delta$ and $w \in X_k$ such that $X(\alpha) : w \mapsto y$ and $X(\beta) : w \mapsto z$. Since $X$ is the nerve of a poset, this means that there is a totally ordered set $w = (w_0 \leq \cdots \leq w_k)$ and $y$ and $z$ are among its elements $y = w_{\alpha(0)}$, $z = w_{\beta(0)}$. Accordingly we have either $y \leq z$ or $z \leq y$ and hence $X$ is in fact the nerve of a [[total order]]. \item If $y,z$ are elements in the total order with $y \leq z$ and $z \leq y$, this means that in the nerve we have elements $(y,z) \in X_1$ and $(z,y) \in X_1$ with $d_0(y,z) = d_1(z,y)$ and $d_1(y,z) = d_1(z,y)$. By co-filtering, there exists a [[cone]] over this diagram in the category of elements, hence morphisms $\alpha, \beta : [1] \to [k]$ in $\Delta$ and $w \in X_k$ such that \begin{enumerate}% \item $X(\alpha) : w \mapsto (y,z)$ and $X(\beta) : w \mapsto (z,y)$; \item $\partial_0 \circ \alpha = \partial_1 \circ \beta$ and $\partial_1 \circ \alpha = \partial_0 \circ \beta$. \end{enumerate} Here the last condition in $\Delta$ can only hold if $\alpha = \beta = const_{i}$, hence if $y = z$. \end{enumerate} Conversely, assume that $X$ is the nerve of a linear order. We show that then it is a flat functor $X : \Delta^{op} \to Set$. (\ldots{}) \end{proof} \hypertarget{ForIntervals}{}\subsubsection*{{For intervals}}\label{ForIntervals} [[Andre Joyal]] showed that $Set^{{\Delta}^{op}}$, the category of [[simplicial sets]], is the classifying topos for [[linear intervals]]. Specifically a [[geometric morphism]] from $Set$ to $Set^{{\Delta}^{op}}$ is an [[linear interval]] in [[Set]], meaning a [[totally ordered set]] with distinct [[top]] and [[bottom]] elements. In general, a linear interval is a model for the one-sorted [[geometric theory]] whose [[signature]] consists of a binary [[relation]] $\leq$ and two [[constant|constants]] $0$, $1$, subject to the following [[axiom|axioms]]: \begin{itemize}% \item $\vdash (x \leq x)$ \item $\exists_y (x \leq y) \wedge (y \leq z) \vdash (x \leq z)$ \item $(x \leq y) \wedge (y \leq x) \vdash (x = y)$ \item $\vdash (x \leq y) \vee (y \leq x)$ \item $\vdash (0 \leq x) \wedge (x \leq 1)$ \item $(0 = 1) \vdash false$ \end{itemize} (Joyal calls this a \textbf{strict} linear interval; by removing the hypothesis of distinct top and bottom, one arrives at a weaker notion he calls ``linear interval''. Linear intervals in this sense are classified by the topos $Set^{\Delta_{a}^{op}}$, where $\Delta_a$, sometimes called the algebraist's Delta or the augmented [[simplex category]], is the category of all finite ordinals including the empty one.) The generic such interval is $\Delta^1 \in Set^{{\Delta}^{op}}$; see [[generic interval]] for more details and references. \hypertarget{for_abstract_circles}{}\subsubsection*{{For abstract circles}}\label{for_abstract_circles} The category of [[cyclic sets]] is the classifying topos for [[abstract circles]] (\hyperlink{Moerdijk96}{Moerdijk 96}). \hypertarget{for_local_rings}{}\subsubsection*{{For local rings}}\label{for_local_rings} \hypertarget{local_rings}{}\paragraph*{{Local rings}}\label{local_rings} The classifying topos for [[local ring|local rings]] is the [[big Zariski topos]] of the [[scheme]] $Spec(\mathbb{Z})$. A \textbf{local ring} is a model of the geometric theory of commutative unital rings subject to the extra axioms \begin{itemize}% \item $(0 = 1) \vdash false$ \item $x + y = 1 \vdash \exists_z (x z = 1) \vee \exists_z (y z = 1)$ \end{itemize} In a topos of sheaves over a [[sober space]], a local ring is precisely what algebraic geometers usually call a ``sheaf of local rings'': namely, a sheaf of rings all of whose [[stalk|stalks]] are local. See [[locally ringed topos]]. This is a special case of the case of \hyperlink{CoverPreservingFLatFunctors}{Cover-preserving flat functors} below. \hypertarget{StrictLocalRings}{}\paragraph*{{Strict local rings}}\label{StrictLocalRings} For $Spec R$ an [[affine scheme]], the [[étale topos]] $Sh(X_{et})$ classifies ``[[strict local ring|strict local R-algebras]]''. The [[point of a topos|points of this topos]] are \emph{[[strict Henselian ring|strict Henselian R-algebras]]} (\hyperlink{Hakim}{Hakim, III.2-4}) and (\hyperlink{Wraith}{Wraith}). See also \href{http://mathoverflow.net/questions/48690/what-does-an-etale-topos-classify}{this MO discussion} \hypertarget{PrincipalBund}{}\subsubsection*{{For principal bundles}}\label{PrincipalBund} Essentially every topos may be regarded as a classifying topos for certain [[torsor]]s/[[principal bundle]]s. \hypertarget{BareGTor}{}\paragraph*{{Over bare groups}}\label{BareGTor} For any (bare / discrete) [[group]] $G$, write $\mathbf{B}G$ for its [[delooping]] groupoid, the groupoid with a single object and $G$ as its endomorphisms. The presheaf topos \begin{displaymath} G Set := PSh(\mathbf{B}G) \end{displaymath} of [[permutation representation]]s (objects are [[set]]s equipped with a $G$-[[action]], morphisms are $G$-equivariant maps between these) is the classifying topos for $G$-[[torsor]]s. For example, if $X$ is a [[topological space]], geometric morphisms from the [[sheaf topos]] $Sh(X)$ of [[sheaves]] on (the [[category of open subsets]] of) $X$ to $G Set$ are the same as $G$-[[principal bundle]]s over $X$ \begin{displaymath} G Bund(X) \simeq Topos(Sh(X), G Set) \,. \end{displaymath} This follows via [[Diaconescu's theorem]], which asserts that [[geometric morphism]]s $Sh(X) \to Sh(\mathbf{B}G)$ are equivalent to [[flat functor]]s \begin{displaymath} \mathbf{B}G \to Sh(X) \,. \end{displaymath} Such a flat functor picks a single sheaf on $X$ and encodes a $G$-action on this sheaf such that this sheaf is the sheaf of [[section]]s of a $G$-[[principal bundle]] on $X$. \begin{theorem} \label{}\hypertarget{}{} Let $G$ be a (bare, discrete) group, write $\mathcal{B}G \in$ [[Top]] for the ordinary [[classifying space]] and $\mathbf{B}G \in$ [[Grpd]] the one-object groupoid version of $G$. There is a canonical [[geometric morphism]]s \begin{displaymath} PSh(\mathbf{B}G) \to Sh(\mathcal{B}G) \,. \end{displaymath} This is a \emph{weak homotopy equivalence} of toposes, in that it induces isomorphisms on [[geometric homotopy groups in an (∞,1)-topos|geometric homotopy groups]] of the terminal object. \end{theorem} This is (\hyperlink{Moerdijk95}{Moerdijk 95, theorem 1.1, proven in chapter IV}). \hypertarget{BareGTorGeomTheo}{}\paragraph*{{In terms of geometric theories}}\label{BareGTorGeomTheo} A [[geometric theory]] $T$ whose [[model]]s are $G$-torsors can be described as follows. It has one sort, $X$, and one unary operation $g:X\to X$ for every element $g\in G$. It has algebraic axioms $\top\vdash_x \;1(x) = x$ and $\top\vdash_x \;g(h(x)) = (g h)(x)$, which make $X$ into a $G$-set, and geometric axioms $\top \vdash\; \exists x \in X$ (inhabited-ness), $g(x) = x \;\vdash_x \;\bot$ for all $g\neq 1$ (freeness), and $\top\vdash_{x,y}\; \bigvee_{g\in G}\; g(x) = y$ (transitivity). \hypertarget{TopGTor}{}\paragraph*{{Over topological groups}}\label{TopGTor} If $G$ is a general [[topological group]] we have a [[simplicial object|simplicial]] [[topological space]] $G^{\times \bullet}$. The category $Sh(G^{\times \bullet})$ of [[sheaves on a simplicial topological space|sheaves on this simplicial space]] is a topos. This is such that for $X$ a topological space, geometric morphisms $Sh(X) \to Sh(G^{\times \bullet})$ classifies topological $G$-principal bundles on $X$. This idea admits generalizations to [[localic groups]] --- and even to [[localic groupoids]]. For more details, see [[classifying topos of a localic groupoid]] . \hypertarget{UniversalBundle}{}\paragraph*{{The universal $G$-bundle topos}}\label{UniversalBundle} At [[generalized universal bundle]] and [[principal ∞-bundle]] it is discussed that the principal bundle classified by a morphims into a classifying object is its [[homotopy fiber]], and how the universal bundle is a replacement of the point such that its ordinary pullback models that [[homotopy pullback]]. Concretely, for $G$ a [[group]] and $\mathbf{B}G = \{\bullet \stackrel{g \in G}{\to} \bullet\}$ in [[∞Grpd]] its [[delooping]] [[groupoid]], the universal $G$-bundle is really just the point inclusion \begin{displaymath} \itexarray{ * \\ \downarrow \\ \mathbf{B}G } \end{displaymath} in that for $X \to \mathbf{B}G$ a morphism, the corresponding $G$-[[principal ∞-bundle]] in [[∞Grpd]] is the [[homotopy pullback]] \begin{displaymath} \itexarray{ P &\to& * \\ \downarrow &{}^{\simeq}\swArrow& \downarrow \\ X &\to& \mathbf{B}G } \,. \end{displaymath} We can send this morphism $(* \to \mathbf{B}G)$ in [[Grpd]] with \begin{displaymath} PSh(-) : Grpd \to Toposes \end{displaymath} to the [[2-category of toposes]] to get a [[geometric morphism]] \begin{displaymath} \itexarray{ PSh(*) = Set \\ \downarrow^{\mathrlap{p}} \\ PSh(\mathbf{B}G) = Set^G } \,. \end{displaymath} By the rules of morphisms of [[site]]s we have that the [[inverse image]] $p^* : PSh(\mathbf{B}G) \to Set$ is precomposition with $p : * \to \mathbf{B}G$, i.e. the functor that just forgets the $G$-action on a set. Its [[right adjoint]] [[direct image]] $p_* : Set \to PSh(\mathbf{B}G)$ is the functor \begin{displaymath} p_* : S \mapsto S \times G \end{displaymath} which sends a set $S$ to the $G$-set $S \times G$ equipped with the evident $G$-action induced by that of $G$ on itself. Because for $(V,\rho)$ any set with $G$-action $\rho$ we have naturally \begin{displaymath} Hom_{Set}(S,V) \simeq Hom_{Set^G}(S \times G, (V,\rho)) \,. \end{displaymath} The object \begin{displaymath} p_*(*) = G \in PSh(\mathbf{B}G) \end{displaymath} singled out this way in this way is the universal object in $Set^G$, namely $G$ equipped with the canonical $G$-action on itself. It ought to be true that the topos-incarnation of the $G$-principal bundle on a topological space $X$ classified by a [[geometric morphism]] $Sh(X) \to PSh(\mathbf{B}G)$ is the $(2,1)$-pullback \begin{displaymath} \itexarray{ \mathcal{P} &\to& Set \\ \downarrow &{}^{\simeq}\swArrow& \downarrow \\ Sh(X) &\to& PSh(\mathbf{B}G) } \,. \end{displaymath} \begin{quote}% needs more discussion\ldots{} \end{quote} \hypertarget{ForLocalicGroupoids}{}\subsubsection*{{For general localic groupoids}}\label{ForLocalicGroupoids} In fact, \emph{any} [[Grothendieck topos]] can be thought of as a classifying topos for some [[localic groupoid]]. This is related to the discussion above, since Joyal and Tierney showed that any Grothendieck topos is equivalent to the $B G$ for some [[localic groupoid]] $G$. A useful discussion of this idea starts \href{http://golem.ph.utexas.edu/category/2007/10/geometric_representation_theor_2.html#c012724}{here}. \hypertarget{for_flat_functors}{}\subsubsection*{{For flat functors}}\label{for_flat_functors} As a special case of the above, any [[presheaf topos]], i.e. any topos of the form $Set^{C^{op}}$, is the classifying topos for [[flat functors]] from $C$ (sometimes also called ``$C$-[[torsor]]s''). In other words, geometric morphisms $E \to Set^{C^{op}}$ are the same as [[flat functor]]s $C \to E$. This is [[Diaconescu's theorem]]. If $C$ has finite limits, then a flat functor $C \to E$ is the same as a functor that preserves finite limits. \hypertarget{CoverPreservingFLatFunctors}{}\subsubsection*{{For geometric theories / cover-preserving flat functors on a site}}\label{CoverPreservingFLatFunctors} Another way, apart from that \hyperlink{ForLocalicGroupoids}{above}, of viewing any [[Grothendieck topos]] $E$ as a classifying topos is to start with a small [[site]] of definition for it. Any such site gives rise to a [[geometric theory]] called the theory of [[cover-preserving functor|cover-preserving]] flat functors on that site. The classifying topos of this theory is again $E$. Moreover, for any object $X$ of $E$, there is a small site of definition for $E$ which includes $X$, and thus for which $X$ is (part of) the universal object. We have: \begin{prop} \label{EveryToposIsAClassifyingToposForLocalAlgebras}\hypertarget{EveryToposIsAClassifyingToposForLocalAlgebras}{} Every [[sheaf topos]] has a [[cartesian site]] $(\mathcal{C}, J)$ of definition. This $Sh(\mathcal{C}, J)$ is the classifying topos for cover-preserving [[flat functor]]s out of $\mathcal{C}$. Every category of such functors is the category of [[model]]s of some geometric theory, and for every geometric theory there is such a cartesian site. \end{prop} This appears as (\hyperlink{Johnstone}{Johnstone, remark D3.1.13}). \hypertarget{LocalAlgebras}{}\subsubsection*{{For local algebras}}\label{LocalAlgebras} As a special case or rather re-interpretation of the \hyperlink{CoverPreservingFLatFunctors}{above}, let $\mathcal{T}$ be any [[essentially algebraic theory]] and equip its [[syntactic category]] $\mathcal{C}_{\mathbb{T}}$ with some [[coverage]] $J$. Then the [[sheaf topos]] $Sh(\mathcal{C}_{\mathbb{T}}, J)$ is the classifying topos for \emph{local $\mathbb{T}$-algebras} : for $Sh(X)$ any [[sheaf topos]] a [[geometric morphism]] \begin{displaymath} \mathcal{O} : Sh(X) \to Sh(\mathcal{C}_{\mathbb{T}}, J) \end{displaymath} is \begin{enumerate}% \item a $\mathbb{T}$-[[algebra over an algebraic theory|algebra]] in $Sh(X)$, hence a [[sheaf]] of $\mathbb{T}$-algebras over the site $X$; \item such that this sheaf of algebras is local as seen by the respective topologies. \end{enumerate} See [[locally algebra-ed topos]] for more on this. By prop. \ref{EveryToposIsAClassifyingToposForLocalAlgebras} we have that every [[sheaf topos]] is the classifying topos of \emph{some} theory of local algebras. The [[vertical categorification]] of this situation to the context of [[(∞,1)-category]] theory is the notion of [[structured (∞,1)-topos]] and of [[geometry (for structured (∞,1)-toposes)]]: The geometry $\mathcal{G}$ is the [[(∞,1)-category]] that plays role of the syntactic theory. For $\mathcal{X}$ an [[(∞,1)-topos]], a model of this theory is a limits and covering-preserving [[(∞,1)-functor]] \begin{displaymath} \mathcal{G} \to \mathcal{X} \,. \end{displaymath} The [[Yoneda embedding]] followed by [[∞-stackification]] \begin{displaymath} \mathcal{G} \stackrel{Y}{\to} PSh_{(\infty,1)}(\mathcal{G}) \stackrel{\bar(-)}{\to} Sh_{(\infty,1)}(\mathcal{G}) \end{displaymath} constitutes a model of $\mathcal{G}$ in the (Cech) [[∞-stack]] [[(∞,1)-topos]] $Sh_{(\infty,1)}(\mathcal{G})$ and exhibits it as the classifying topos for such models (geometries): This is \emph{[[Structured Spaces]]} \href{http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.0459v1.pdf#page=26}{prop 1.4.2}. \hypertarget{as_a_generalization_of_the_notion_of_classifying_space_in_topology}{}\subsection*{{As a generalization of the notion of classifying space in topology}}\label{as_a_generalization_of_the_notion_of_classifying_space_in_topology} In view of the analogy between the classifying topos denoted $B G$, such that the [[groupoid]] $G Bund(X)$ of $G$-[[principal bundle]]s over $X$ is equivalent to geometric morphims $Sh(X) \to B G$: \begin{displaymath} G Bund(X) \simeq Topos(Sh(X), B G) \, \end{displaymath} and the notion of [[classifying space]] in [[topology]], which for the discrete group $G$ is a [[topological space]] $\mathcal{B} G$ such that \begin{displaymath} \pi_0 G Bund(X) \simeq \pi_0 Top(X, \mathcal{B}G) \, \end{displaymath} we should expect there to be a topos analog of the total space, $E G$, for the classifying space. This analog is the \emph{generic G-torsor}, which is an internal $G$-torsor in the topos $Set^G$. The important aspect of the space $E G$ is that as a principal $G$-bundle over $\mathcal{B} G$, it is a \emph{universal element}, i.e. the natural transformation $Hom(X, \mathcal{B}G) \to G Bdl(X)$ that it induces (by the [[Yoneda lemma]]) is the isomorphism which exhibits $\mathcal{B}G$ as the object representing the functor $X \mapsto G Bdl(X)$. For the same Yoneda reasons, the classifying topos $Sh(C_T)$ of any geometric theory $T$ comes with a \emph{generic $T$-model}, which is a $T$-model in $Sh(C_T)$ which represents the functor $E \mapsto T Mod(E)$ in the same way. For $T$ = the theory of $G$-torsors, this generic model is the generic $G$-torsor. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[representable functor]] \item [[classifying topos for the theory of objects]] \item [[classifying space]], [[classifying stack]], [[moduli space]], [[moduli stack]], [[derived moduli space]] \item [[universal principal bundle]], [[universal principal infinity-bundle]] \item [[classifying morphism]] \item [[classifying (infinity,1)-topos]] \item [[natural number object]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} Standard textbook references for classifying topoi of theories \begin{itemize}% \item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]} , Oxford UP 2002. (In particular, sections B4.2 pp.424-432, D3.2 pp.901-910) \item [[Francis Borceux]], \emph{Handbook of categorical algebra}, (in series Enc. Math. Appl.) vol. 3, Categories and sheaves, Cambridge Univ. Press 1994, Ch.4 Classifying toposes \item [[Saunders Mac Lane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}, Springer Heidelberg 1994. (chap. VIII) \end{itemize} The relation between the existence of natural number objects and classifying toposes is discussed in \begin{itemize}% \item [[Andreas Blass]], \emph{Classifying topoi and the axiom of infinity} , Algebra Universalis \textbf{26} (1989) pp.341-345. \end{itemize} The study of classifying spaces of topological categories is described in the monograph \begin{itemize}% \item [[Ieke Moerdijk]], \emph{Classifying spaces, classifying topoi}, Lec. Notes Math. 1616, Springer Verlag 1995 \end{itemize} The original theory for a general algebraic theory is developed in \begin{itemize}% \item M. Makkai, G. Reyes, \emph{First-order categorical logic}, Lecture Notes in Mathematics 611, Springer 1977. \end{itemize} The results for the continuous groupoids include \begin{itemize}% \item [[Ieke Moerdijk]], The classifying topos of a continuous groupoid I, \emph{Trans. A.M.S.} \textbf{310} (1988), 629-668. \item [[Ieke Moerdijk]], \emph{The classifying topos of a continuous groupoid II}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, \textbf{31} no. 2 (1990), 137-168. (\href{http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1990__31_2_137_0}{web}) \item [[Ieke Moerdijk]], \emph{Classifying spaces and classifying topoi}, Lecture Notes in Math. \textbf{1616}, Springer-Verlag, New York, 1995. \item [[Ieke Moerdijk]], \emph{Cyclic sets as a classifying topos}, 1996 ([[MoerdijkCyclic.pdf:file]]) \end{itemize} Classifying toposes as [[locally algebra-ed (infinity,1)-toposes]] are discussed in section 1.4 of \begin{itemize}% \item [[Jacob Lurie]], \emph{[[Structured Spaces]]} \end{itemize} The [[étale topos]] as a classifying topos for [[strict local rings]] is discussed in \begin{itemize}% \item [[Monique Hakim]], \emph{Topos annel\'e{}s et sch\'e{}mas relatifs}, Sec. III.2-4 \item [[Gavin Wraith]], \emph{Generic Galois theory of local rings} \end{itemize} [[Nikolai Durov]] has introduced somewhat a generalization of topos called [[vectoid]] and quite flexible notion of a classifying vectoid in \begin{itemize}% \item [[Nikolai Durov]], \emph{Classifying vectoids and generalisations of operads}, \href{http://arxiv.org/abs/1105.3114}{arxiv/1105.3114}, the translation of `` '', Trudy MIAN, vol. 273 \end{itemize} \hypertarget{ReferencesRelationToForcing}{}\subsubsection*{{Relation to forcing}}\label{ReferencesRelationToForcing} Reviews of the interpretation of \emph{[[forcing]]} as the passge to classifying toposes include \begin{itemize}% \item [[Andreas Blass]], Andrej Ščedrov, \emph{Classifying topoi and finite forcing} (\href{http://deepblue.lib.umich.edu/bitstream/handle/2027.42/25225/0000666.pdf;jsessionid=8D65C672D70D8E24A5ACD366C1815921?sequence=1}{pdf}) \item Andrej Ščedrov, \emph{Forcing and classifying topoi}, Memoirs of the American Mathematical Society 1984; 93 pp \end{itemize} For more see \begin{itemize}% \item [[David Roberts]], \emph{Class forcing and topos theory}, talk at \emph{\href{https://indico.math.cnrs.fr/event/747/}{Topos at l'IHES}} 2015 (\href{https://www.dropbox.com/s/vk1efw7lsvta80p/Roberts_IHES.pdf?dl=0}{talk notes pdf}, \href{https://youtu.be/4AaSySq8-GQ}{video recording}) \end{itemize} [[!redirects Another page]] [[!redirects classifying toposes]] [[!redirects classifying topoi]] [[!redirects classifying locale]] [[!redirects classifying locales]] \end{document}