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The classifying topos for a given type of mathematical structure $T$ — for example the structures: “group”, “torsor”, “ring”, “category” etc. — is a (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 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 bundles over $X$ is equivalent to geometric morphims $Sh(X) \to B G$:
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
Hence one can think of classifying topoi as a grand generalization of the notion of classifying space in topology.
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 categories of certain structures in $E$ :
as decribed in Geometric theories – In terms of sheaf topoi, every sheaf topos $F$ is a completion $S[T]$ of the syntactic category $C_T$ of some geometric theory $T$
And structures of type $T$ in $E$ is what geometric morphisms $E \to F$ classify.
So the 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 forcing in logic: the passage to the internal logic of the classifying topos forces the universal model to exist.
If $C_T$ is the syntactic category of $T$, so that $T$-models are the same as geometric functors out of $C_T$, then this universal model can be identified with a certain geometric functor
Its universality property means that any geometric functor
factors essentially uniquely as
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 $S$-topos.”
If the classifying topos of a geometric theory $T$ is a presheaf topos, one called $T$ a theory of presheaf type.
The notion of classifying topos is part of a trend, begun by Lawvere, of viewing a mathematical theory in logic as a category with suitable 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’:
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 products. 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 functors $f : C_{fp}[T] \to Set$. More generally, for any category with finite products, say $E$, models of $T$ in $E$ correspond to product-preserving functors $f : C_{fp}[T] \to E$.
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 limits (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 partially defined operation, namely composition of morphisms.) Every finite limits theory $T$ admits a 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.)
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 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.
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 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 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.
The fact that classifying toposes are what they all comes down, if spelled out explicitly, to the fact that geometric morphism $f : \mathcal{E} \to \mathcal{F}$ of toposes can be identified with certain morphism of sites $C_{\mathcal{E}}$, $C_{\mathcal{F}}$ 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 models of $\mathcal{T}$ in $\mathcal{E}$.
We now say this in precise manner. In the following a cartesian site means a site whose underlying category is finitely complete.
Let $(\mathcal{C}, J)$ and $(\mathcal{D}, K)$ be cartesian sites such that $\mathcal{C}$ is a small category, $\mathcal{D}$ is an essentially small category and the coverage $K$ is subcanonical.
Then a geometric morphism of the corresponding sheaf toposes
is induced by a morphism of sites
precisely if the inverse image of $f$ respects the Yoneda embeddings $j$ as
This appears as (Johnstone, lemma C2.3.8).
It suffices to observe that the factorization, if it exists, is a morphism of sites.
Let $(\mathcal{C},J)$ be a small cartesian site and let $\mathcal{E}$ be any sheaf topos. Then we have an equivalence of categories
between the geometric morphisms 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 $\mathcal{E}$.
This appears as (Johnstone, cor. C2.3.9).
This means that a sheaf topos $Sh(\mathcal{C},J)$ is the classifying topos for the theory of local algebras determined by the site $(\mathcal{C},J)$.
We list and discuss explicit examples of classifying toposes.
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 classifying topos for the theory of objects.
For $E$ any topos, a geometric morphism $E \to [FinSet,Set]$ is equivalently just an object of $E$.
Similarly, the presheaf topos $[FinSet_*, Set]$ (where $FinSet_*$ is the category of finite pointed sets) classifies pointed objects; cf. this question and answer. This is the topos of “$\Gamma$-sets”; see Gamma-space.
We discuss the finite product theory of groups. 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 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:
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 $E$.
The discussion above for groups can be repeated verbatim for rings, since they too are described by a finite products theory.
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 linear orders.
This appears as (Moerdijk 95, prop. 5.4).
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
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 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
in $\Delta$ to isomorphisms
This are the 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
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 cofiltered.
This means
The category of elements is inhabited, hence the poset of which $X$ is the nerve is inhabited.
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.
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
$X(\alpha) : w \mapsto (y,z)$ and $X(\beta) : w \mapsto (z,y)$;
$\partial_0 \circ \alpha = \partial_1 \circ \beta$ and $\partial_1 \circ \alpha = \partial_0 \circ \beta$.
Here the last condition in $\Delta$ can only hold if $\alpha = \beta = const_{i}$, hence if $y = z$.
Conversely, assume that $X$ is the nerve of a linear order. We show that then it is a flat functor $X : \Delta^{op} \to Set$.
(…)
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 constants? $0$, $1$, subject to the following axioms:
(Joyal calls this a 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 category of cyclic sets is the classifying topos for abstract circles (Moerdijk 96).
The classifying topos for local rings is the big Zariski topos of the scheme $Spec(\mathbb{Z})$. A local ring is a model of the geometric theory of commutative unital rings subject to the extra axioms
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 stalks are local. See locally ringed topos. This is a special case of the case of Cover-preserving flat functors below.
For $Spec R$ an affine scheme, the étale topos $Sh(X_{et})$ classifies “strict local R-algebras”. The points of this topos are strict Henselian R-algebras? (Hakim, III.2-4) and (Wraith).
See also this MO discussion
Essentially every topos may be regarded as a classifying topos for certain torsors/principal bundles.
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
of permutation representations (objects are sets equipped with a $G$-action, morphisms are $G$-equivariant maps between these) is the classifying topos for $G$-torsors.
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 bundles over $X$
This follows via Diaconescu's theorem, which asserts that geometric morphisms $Sh(X) \to Sh(\mathbf{B}G)$ are equivalent to flat functors
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 sections of a $G$-principal bundle on $X$.
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 morphisms
This is a weak homotopy equivalence of toposes, in that it induces isomorphisms on geometric homotopy groups of the terminal object.
This is (Moerdijk 95, theorem 1.1, proven in chapter IV).
A geometric theory $T$ whose models 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).
If $G$ is a general topological group we have a simplicial topological space $G^{\times \bullet}$. The category $Sh(G^{\times \bullet})$ of 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 .
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
in that for $X \to \mathbf{B}G$ a morphism, the corresponding $G$-principal ∞-bundle in ∞ Grpd is the homotopy pullback
We can send this morphism $(* \to \mathbf{B}G)$ in Grpd with
to the 2-category of toposes? to get a geometric morphism
By the rules of morphisms of sites 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
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
The object
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
needs more discussion…
In fact, 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 here.
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 “$C$-torsors”). In other words, geometric morphisms $E \to Set^{C^{op}}$ are the same as flat functors $C \to E$. 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.
Another way, apart from that 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 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:
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 functors out of $\mathcal{C}$.
Every category of such functors is the category of models of some geometric theory, and for every geometric theory there is such a cartesian site.
This appears as (Johnstone, remark D3.1.13).
As a special case or rather re-interpretation of the 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 local $\mathbb{T}$-algebras :
for $Sh(X)$ any sheaf topos a geometric morphism
is
a $\mathbb{T}$-algebra in $Sh(X)$, hence a sheaf of $\mathbb{T}$-algebras over the site $X$;
such that this sheaf of algebras is local as seen by the respective topologies.
See locally algebra-ed topos for more on this.
By prop. 3 we have that every sheaf topos is the classifying topos of 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
The Yoneda embedding followed by ∞-stackification
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 Structured Spaces prop 1.4.2.
In view of the analogy between the classifying topos denoted $B G$, such that the groupoid $G Bund(X)$ of $G$-principal bundles over $X$ is equivalent to geometric morphims $Sh(X) \to B G$:
and the notion of classifying space in topology, which for the discrete group $G$ is a topological space $\mathcal{B} G$ such that
we should expect there to be a topos analog of the total space, $E G$, for the classifying space. This analog is the 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 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 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.
classifying space, classifying stack, moduli space, moduli stack, derived moduli space
universal principal bundle, universal principal infinity-bundle
Standard textbooks reference for classifying topoi of theories
The study of classifying spaces of topological categories is described in the monograph
The original theory for a general algebraic theory is developed in
The results for the continuous groupoids include
Ieke Moerdijk, The classifying topos of a continuous groupoid I, Trans. A.M.S. 310 (1988), 629-668.
Ieke Moerdijk, The classifying topos of a continuous groupoid II, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 31 no. 2 (1990), 137-168. (web)
Ieke Moerdijk, Classifying spaces and classifying topoi, Lecture Notes in Math. 1616, Springer-Verlag, New York, 1995.
Ieke Moerdijk, Cyclic sets as a classifying topos, 1996 (pdf)
Classifying toposes as locally algebra-ed (infinity,1)-toposes are discussed in section 1.4 of
The étale topos as a classifying topos for strict local rings is discussed in
Nikolai Durov has introduced somewhat a generalization of topos called vectoid and quite flexible notion of a classifying vectoid in