nLab
classifying topos

Contents

Idea

The classifying topos for a given type of mathematical structure TT — for example the structures: “group”, “torsor”, “ring”, “category” etc. — is a (Grothendieck) topos S[T]S[T] such that geometric morphisms f:ES[T]f: E \to S[T] are the same as structures of this sort in the topos EE, i.e. groups internal to EE, torsors internal to EE, etc. In other words, a classifying topos is a representing object for the functor which sends a topos EE to the category of structures of the desired sort in EE.

In particular for EE a sheaf topos on a topological space XX and GG a (bare, i.e. discrete) group, a GG-torsor in EE is a GG-principal bundle over XX. There is a classifying topos denoted BGB G, such that the groupoid GBund(X)G Bund(X) of GG-principal bundles over XX is equivalent to geometric morphims Sh(X)BGSh(X) \to B G:

GBund(X)Topos(Sh(X),BG). G Bund(X) \simeq Topos(Sh(X), B G) \,.

This is evidently analogous to the notion of classifying space in topology, which for the discrete group GG is a topological space G\mathcal{B} G such that

π 0GBund(X)π 0Top(X,G). \pi_0 G Bund(X) \simeq \pi_0 Top(X, \mathcal{B}G) \,.

Hence one can think of classifying topoi as a grand generalization of the notion of classifying space in topology.

Definition

In a tautological way, every topos FF is the classifying topos for something, namely for the categories of geometric morphisms EFE \to F into it. The concept of geometric theory allows one to usefully interpret these categories as categories of certain structures in EE :

as decribed in Geometric theories – In terms of sheaf topoi, every sheaf topos FF is a completion S[T]S[T] of the syntactic category C TC_T of some geometric theory TT

FS[T]. F \simeq S[T] \,.

And structures of type TT in EE is what geometric morphisms EFE \to F classify.

So the classifying topos for the geometric theory TT is a Grothendieck topos S[T]S[T] equipped with a “universal model UU of TT”. This means that for any Grothendieck topos EE together with a model XX of TT in EE, there exists a unique (up to isomorphism) geometric morphism f:ES[T]f: E \to S[T] such that f *f^* maps the TT-model UU to the model XX. More precisely, for any Grothendieck topos EE, the category of TT-models in EE is equivalent to the category of geometric morphisms ES[T]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 TC_T is the syntactic category of TT, so that TT-models are the same as geometric functors out of C TC_T, then this universal model can be identified with a certain geometric functor

U:C TS[T]. U : C_T \to S[T] \,.

Its universality property means that any geometric functor

X:C TE X : C_T \to E

factors essentially uniquely as

X:C TUS[T]f *E X : C_T \stackrel{U}{\to} S[T] \stackrel{f^*}{\to} E

for UU the universal model and f *f^* the left adjoint part of a geometric morphism. More precisely, composition with UU defines an equivalence between the category of geometric morphisms ES[T]E\to S[T] and the category of geometric functors C TEC_T\to E.

More specifically, for any cartesian theory, regular theory or coherent theory 𝕋\mathbb{T} (which in ascending order are special cases of each other and all of geometric theories), the corresponding syntactic category 𝒞 𝕋\mathcal{C}_{\mathbb{T}} comes equipped with the structure of a syntactic site (𝒞,𝕋,J)(\mathcal{C},\mathbb{T}, J) (see there) and the classifying topos for 𝕋\mathbb{T} is the sheaf topos Sh(𝒞 𝕋,J)Sh(\mathcal{C}_{\mathbb{T}}, J).

Classifying toposes can also be defined over any base topos SS instead of Set. In that case “Grothendieck topos” is replaced by “bounded SS-topos.”

If the classifying topos of a geometric theory TT is a presheaf topos, one called TT a theory of presheaf type.

Background on the theory of theories

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 TT there is a category with finite products C fp[T]C_{fp}[T] – the syntactic category, which serves as a “classifying category” for TT, in that models of TT in the category of sets correspond to product-preserving functors f:C fp[T]Setf : C_{fp}[T] \to Set. More generally, for any category with finite products, say EE, models of TT in EE correspond to product-preserving functors f:C fp[T]Ef : 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 TT admits a classifying category C fl(T)C_{fl}(T): a finitely complete category such that models of TT in a category EE with finite limits correspond to functors f:C fl(T)Ef: 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]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:ES[T]f\colon E\to S[T], or specifically with their inverse image parts.

Each type of theory may be considered a 22-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 TT to the corresponding finite limits theory, we can take the opposite of the category of finitely presentable models of TT in SetSet, 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.

Properties

Geometric morphisms equivalent to morphisms of sites

The fact that classifying toposes are what they all comes down, if spelled out explicitly, to the fact that geometric morphism f:f : \mathcal{E} \to \mathcal{F} of toposes can be identified with certain morphism of sites C C_{\mathcal{E}}, C C_{\mathcal{F}} for these toposes, going the other way round, C C C_\mathcal{E} \leftarrow C_{\mathcal{F}}, and having certain properties. If here C C_\mathcal{F} is the syntactic site of some theory 𝕋\mathbb{T} and we choose C 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 𝒯\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.

Proposition

Let (𝒞,J)(\mathcal{C}, J) and (𝒟,K)(\mathcal{D}, K) be cartesian sites such that 𝒞\mathcal{C} is a small category, 𝒟\mathcal{D} is an essentially small category and the coverage KK is subcanonical.

Then a geometric morphism of the corresponding sheaf toposes

f:Sh(𝒟,K)Sh(𝒞,J) f : Sh(\mathcal{D}, K) \to Sh(\mathcal{C}, J)

is induced by a morphism of sites

(𝒟,K)(𝒞,J) (\mathcal{D}, K) \leftarrow (\mathcal{C}, J)

precisely if the inverse image of ff respects the Yoneda embeddings jj as

𝒟 𝒞 j 𝒟 j 𝒞 Sh(𝒟,K) f * Sh(𝒞,J). \array{ \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) } \,.

This appears as (Johnstone, lemma C2.3.8).

Proof

It suffices to observe that the factorization, if it exists, is a morphism of sites.

Corollary

Let (𝒞,J)(\mathcal{C},J) be a small cartesian site and let \mathcal{E} be any sheaf topos. Then we have an equivalence of categories

Topos(,Sh(𝒞,J))Site((𝒞,J),(,C)) Topos(\mathcal{E}, Sh(\mathcal{C}, J)) \simeq Site((\mathcal{C}, J), (\mathcal{E}, C))

between the geometric morphisms from \mathcal{E} to Sh(𝒞,J)Sh(\mathcal{C}, J) and the morphisms of sites from (𝒞,J)(\mathcal{C}, J) to the big site (,C)(\mathcal{E}, C) for CC the canonical coverage on \mathcal{E}.

This appears as (Johnstone, cor. C2.3.9).

Remark

This means that a sheaf topos Sh(𝒞,J)Sh(\mathcal{C},J) is the classifying topos for the theory of local algebras determined by the site (𝒞,J)(\mathcal{C},J).

Examples

We list and discuss explicit examples of classifying toposes.

For objects

The presheaf topos [FinSet,Set][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 EE any topos, a geometric morphism E[FinSet,Set]E \to [FinSet,Set] is equivalently just an object of EE.

For pointed objects

Similarly, the presheaf topos [FinSet *,Set][FinSet_*, Set] (where FinSet *FinSet_* is the category of finite pointed sets) classifies pointed objects; cf. this question and answer. This is the topos of “Γ\Gamma-sets”; see Gamma-space.

For groups

We discuss the finite product theory of groups. This theory has a classifying category C fp(Grp)C_{fp}(Grp). C fp(Grp)C_{fp}(Grp) is a category with finite products equipped with an object GG, the “walking group”, a morphism m:G×GGm: G \times G \to G describing multiplication, a morphism inv:GGinv : G \to G describing inverses, and a morphism i:1Gi: 1 \to G describing the identity element of GG, obeying the usual group axioms. For any category with finite products, say EE, a finite-product-preserving functor f:C fp(Grp)Ef: C_{fp}(Grp) \to E is the same as a group object in EE. For more details, see Lawvere theory.

We can promote C fp(Grp)C_{fp}(Grp) to a category with finite limits, C fl(Grp)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)C_{fp}(Grp) in Set, which is simply GrpGrp, and then take the full subcategory of finitely presentable groups. By Gabriel-Ulmer duality, the opposite of this is C fl(Grp)C_{fl}(Grp). For any category with finite products, say EE, a left exact functor f:C fp(Grp)Ef: C_{fp}(Grp) \to E is the same as a group object in EE.

We can further promote C fl(Grp)C_{fl}(Grp) to a Grothendieck topos by taking the category of presheaves. This gives the classifying topos for groups:

S[Grp]=Set C fl(Grp) op. S[Grp] = Set^{C_{fl}(Grp)^{op}} \, .

For any Grothendieck topos, say EE, a left exact left adjoint functor f *:S[T]E f^*: S[T] \to E is the same as a group object in EE.

For rings

The discussion above for groups can be repeated verbatim for rings, since they too are described by a finite products theory.

For (inhabited) linear orders

Proposition

The category of cosimplicial sets [Δ,Set][\Delta, Set] – hence the presheaf topos over the opposite category Δ op\Delta^{op} of the simplex category – is the classifying topos for inhabited linear orders.

This appears as (Moerdijk 95, prop. 5.4).

Proof

For ease of notation we discuss this in Set, hence we show that geometric morphisms SetPSh(Δ op)Set \to PSh(\Delta^{op}) are equivalently linear orders. Or, by Diaconescu's theorem, that flat functors

X:Δ opSet X : \Delta^{op} \to Set

are equivalently linear orders. Evidently, such a functor is in particular a simplicial set and we will show that XX 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 XX is a flat functor. Since (by the discussion there) this preserves all finite limits that exist in Δ op\Delta^{op}, equivalently that it sends the finite colimits that exist in Δ\Delta to limits in SetSet, it in particular sends the gluings of intervals

[n] [k] [0][l](n=k+l) [1] [0][1] [0] [0][1] \begin{aligned} [n] & \simeq [k] \coprod_{[0]} [l] \;\;\;\; (n = k + l) \\ & \simeq [1] \coprod_{[0]} [1] \coprod_{[0]} \cdots \coprod_{[0]} [1] \end{aligned}

in Δ\Delta to isomorphisms

X n X k× X 0X l X 1× X 0× X 0X 1. \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} \,.

This are the Segal relations that say that XX is the nerve of a category.

Moreover, since monomorphisms are characterized by pullbacks, FF being flat means that it sends jointly epimorphic families of morphisms in Δ\Delta to monomorphisms in SetSet. In particular, the epimorphic family { 0:[0][1], 1:[0][1]}\{\partial_0 : [0] \to [1], \partial_1 : [0] \to [1]\} is sent to an injection

(d 0,d 1):X 1X 0×X 0. (d_0, d_1) : X_1 \hookrightarrow X_0 \times X_0 \,.

Since X 1X_1 is the set of morphisms of the category that XX 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

  1. The category of elements is inhabited, hence the poset of which XX is the nerve is inhabited.

  2. For every two elements y,zX 0y, z \in X_0 there exist morphisms α,β:[0][k]\alpha, \beta : [0] \to [k] in Δ\Delta and wX kw \in X_k such that X(α):wyX(\alpha) : w \mapsto y and X(β):wzX(\beta) : w \mapsto z. Since XX is the nerve of a poset, this means that there is a totally ordered set w=(w 0w k)w = (w_0 \leq \cdots \leq w_k) and yy and zz are among its elements y=w α(0)y = w_{\alpha(0)}, z=w β(0)z = w_{\beta(0)}. Accordingly we have either yzy \leq z or zyz \leq y and hence XX is in fact the nerve of a total order.

  3. If y,zy,z are elements in the total order with yzy \leq z and zyz \leq y, this means that in the nerve we have elements (y,z)X 1(y,z) \in X_1 and (z,y)X 1(z,y) \in X_1 with d 0(y,z)=d 1(z,y)d_0(y,z) = d_1(z,y) and d 1(y,z)=d 1(z,y)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 α,β:[1][k]\alpha, \beta : [1] \to [k] in Δ\Delta and wX kw \in X_k such that

    1. X(α):w(y,z)X(\alpha) : w \mapsto (y,z) and X(β):w(z,y)X(\beta) : w \mapsto (z,y);

    2. 0α= 1β\partial_0 \circ \alpha = \partial_1 \circ \beta and 1α= 0β\partial_1 \circ \alpha = \partial_0 \circ \beta.

    Here the last condition in Δ\Delta can only hold if α=β=const i\alpha = \beta = const_{i}, hence if y=zy = z.

Conversely, assume that XX is the nerve of a linear order. We show that then it is a flat functor X:Δ opSetX : \Delta^{op} \to Set.

(…)

For intervals

Andre Joyal showed that Set Δ opSet^{{\Delta}^{op}}, the category of simplicial sets, is the classifying topos for linear intervals.

Specifically a geometric morphism from SetSet to Set Δ opSet^{{\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? 00, 11, subject to the following axioms:

  • (xx)\vdash (x \leq x)
  • y(xy)(yz)(xz)\exists_y (x \leq y) \wedge (y \leq z) \vdash (x \leq z)
  • (xy)(yx)(x=y)(x \leq y) \wedge (y \leq x) \vdash (x = y)
  • (xy)(yx)\vdash (x \leq y) \vee (y \leq x)
  • (0x)(x1)\vdash (0 \leq x) \wedge (x \leq 1)
  • (0=1)false(0 = 1) \vdash false

(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 Δ a opSet^{\Delta_{a}^{op}}, where Δ a\Delta_a, sometimes called the algebraist’s Delta or the augmented simplex category, is the category of all finite ordinals including the empty one.)

For abstract circles

The category of cyclic sets is the classifying topos for abstract circles (Moerdijk 96).

For local rings

Local rings

The classifying topos for local rings is the big Zariski topos of the scheme Spec()Spec(\mathbb{Z}). A local ring is a model of the geometric theory of commutative unital rings subject to the extra axioms

  • (0=1)false(0 = 1) \vdash false
  • x+y=1 z(xz=1) z(yz=1)x + y = 1 \vdash \exists_z (x z = 1) \vee \exists_z (y z = 1)

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.

Strict local rings

For SpecRSpec R an affine scheme, the étale topos Sh(X et)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

For principal bundles

Essentially every topos may be regarded as a classifying topos for certain torsors/principal bundles.

Over bare groups

For any (bare / discrete) group GG, write BG\mathbf{B}G for its delooping groupoid, the groupoid with a single object and GG as its endomorphisms. The presheaf topos

GSet:=PSh(BG) G Set := PSh(\mathbf{B}G)

of permutation representations (objects are sets equipped with a GG-action, morphisms are GG-equivariant maps between these) is the classifying topos for GG-torsors.

For example, if XX is a topological space, geometric morphisms from the sheaf topos Sh(X)Sh(X) of sheaves on (the category of open subsets of) XX to GSetG Set are the same as GG-principal bundles over XX

GBund(X)Topos(Sh(X),GSet). G Bund(X) \simeq Topos(Sh(X), G Set) \,.

This follows via Diaconescu's theorem, which asserts that geometric morphisms Sh(X)Sh(BG)Sh(X) \to Sh(\mathbf{B}G) are equivalent to flat functors

BGSh(X). \mathbf{B}G \to Sh(X) \,.

Such a flat functor picks a single sheaf on XX and encodes a GG-action on this sheaf such that this sheaf is the sheaf of sections of a GG-principal bundle on XX.

Theorem

Let GG be a (bare, discrete) group, write G\mathcal{B}G \in Top for the ordinary classifying space and BG\mathbf{B}G \in Grpd the one-object groupoid version of GG. There is a canonical geometric morphisms

PSh(BG)Sh(G). PSh(\mathbf{B}G) \to Sh(\mathcal{B}G) \,.

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).

In terms of geometric theories

A geometric theory TT whose models are GG-torsors can be described as follows. It has one sort, XX, and one unary operation g:XXg:X\to X for every element gGg\in G. It has algebraic axioms x1(x)=x\top\vdash_x \;1(x) = x and xg(h(x))=(gh)(x)\top\vdash_x \;g(h(x)) = (g h)(x), which make XX into a GG-set, and geometric axioms xX\top \vdash\; \exists x \in X (inhabited-ness), g(x)=x xg(x) = x \;\vdash_x \;\bot for all g1g\neq 1 (freeness), and x,y gGg(x)=y\top\vdash_{x,y}\; \bigvee_{g\in G}\; g(x) = y (transitivity).

Over topological groups

If GG is a general topological group we have a simplicial topological space G ×G^{\times \bullet}. The category Sh(G ×)Sh(G^{\times \bullet}) of sheaves on this simplicial space is a topos.

This is such that for XX a topological space, geometric morphisms Sh(X)Sh(G ×)Sh(X) \to Sh(G^{\times \bullet}) classifies topological GG-principal bundles on XX.

This idea admits generalizations to localic groups — and even to localic groupoids. For more details, see classifying topos of a localic groupoid .

The universal GG-bundle topos

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 GG a group and BG={gG}\mathbf{B}G = \{\bullet \stackrel{g \in G}{\to} \bullet\} in ∞Grpd its delooping groupoid, the universal GG-bundle is really just the point inclusion

* BG \array{ * \\ \downarrow \\ \mathbf{B}G }

in that for XBGX \to \mathbf{B}G a morphism, the corresponding GG-principal ∞-bundle in ∞ Grpd is the homotopy pullback

P * X BG. \array{ P &\to& * \\ \downarrow &{}^{\simeq}\swArrow& \downarrow \\ X &\to& \mathbf{B}G } \,.

We can send this morphism (*BG)(* \to \mathbf{B}G) in Grpd with

PSh():GrpdToposes PSh(-) : Grpd \to Toposes

to the 2-category of toposes? to get a geometric morphism

PSh(*)=Set p PSh(BG)=Set G. \array{ PSh(*) = Set \\ \downarrow^{\mathrlap{p}} \\ PSh(\mathbf{B}G) = Set^G } \,.

By the rules of morphisms of sites we have that the inverse image p *:PSh(BG)Setp^* : PSh(\mathbf{B}G) \to Set is precomposition with p:*BGp : * \to \mathbf{B}G, i.e. the functor that just forgets the GG-action on a set.

Its right adjoint direct image p *:SetPSh(BG)p_* : Set \to PSh(\mathbf{B}G) is the functor

p *:SS×G p_* : S \mapsto S \times G

which sends a set SS to the GG-set S×GS \times G equipped with the evident GG-action induced by that of GG on itself.

Because for (V,ρ)(V,\rho) any set with GG-action ρ\rho we have naturally

Hom Set(S,V)Hom Set G(S×G,(V,ρ)). Hom_{Set}(S,V) \simeq Hom_{Set^G}(S \times G, (V,\rho)) \,.

The object

p *(*)=GPSh(BG) p_*(*) = G \in PSh(\mathbf{B}G)

singled out this way in this way is the universal object in Set GSet^G, namely GG equipped with the canonical GG-action on itself.

It ought to be true that the topos-incarnation of the GG-principal bundle on a topological space XX classified by a geometric morphism Sh(X)PSh(BG)Sh(X) \to PSh(\mathbf{B}G) is the (2,1)(2,1)-pullback

𝒫 Set Sh(X) PSh(BG). \array{ \mathcal{P} &\to& Set \\ \downarrow &{}^{\simeq}\swArrow& \downarrow \\ Sh(X) &\to& PSh(\mathbf{B}G) } \,.

needs more discussion…

For general localic groupoids

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 BGB G for some localic groupoid GG. A useful discussion of this idea starts here.

For flat functors

As a special case of the above, any presheaf topos, i.e. any topos of the form Set C opSet^{C^{op}}, is the classifying topos for flat functors from CC (sometimes also called “CC-torsors”). In other words, geometric morphisms ESet C opE \to Set^{C^{op}} are the same as flat functors CEC \to E. This is Diaconescu's theorem. If CC has finite limits, then a flat functor CEC \to E is the same as a functor that preserves finite limits.

For geometric theories / cover-preserving flat functors on a site

Another way, apart from that above, of viewing any Grothendieck topos EE 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 EE.

Moreover, for any object XX of EE, there is a small site of definition for EE which includes XX, and thus for which XX is (part of) the universal object.

We have:

Proposition

Every sheaf topos has a cartesian site (𝒞,J)(\mathcal{C}, J) of definition.

This Sh(𝒞,J)Sh(\mathcal{C}, J) is the classifying topos for cover-preserving flat functors out of 𝒞\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).

For local algebras

As a special case or rather re-interpretation of the above, let 𝒯\mathcal{T} be any essentially algebraic theory and equip its syntactic category 𝒞 𝕋\mathcal{C}_{\mathbb{T}} with some coverage JJ. Then the sheaf topos Sh(𝒞 𝕋,J)Sh(\mathcal{C}_{\mathbb{T}}, J) is the classifying topos for local 𝕋\mathbb{T}-algebras :

for Sh(X)Sh(X) any sheaf topos a geometric morphism

𝒪:Sh(X)Sh(𝒞 𝕋,J) \mathcal{O} : Sh(X) \to Sh(\mathcal{C}_{\mathbb{T}}, J)

is

  1. a 𝕋\mathbb{T}-algebra in Sh(X)Sh(X), hence a sheaf of 𝕋\mathbb{T}-algebras over the site XX;

  2. 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 𝒢\mathcal{G} is the (∞,1)-category that plays role of the syntactic theory. For 𝒳\mathcal{X} an (∞,1)-topos, a model of this theory is a limits and covering-preserving (∞,1)-functor

𝒢𝒳. \mathcal{G} \to \mathcal{X} \,.

The Yoneda embedding followed by ∞-stackification

𝒢YPSh (,1)(𝒢)(¯)Sh (,1)(𝒢) \mathcal{G} \stackrel{Y}{\to} PSh_{(\infty,1)}(\mathcal{G}) \stackrel{\bar(-)}{\to} Sh_{(\infty,1)}(\mathcal{G})

constitutes a model of 𝒢\mathcal{G} in the (Cech) ∞-stack (∞,1)-topos Sh (,1)(𝒢)Sh_{(\infty,1)}(\mathcal{G}) and exhibits it as the classifying topos for such models (geometries):

This is Structured Spaces prop 1.4.2.

As a generalization of the notion of classifying space in topology

In view of the analogy between the classifying topos denoted BGB G, such that the groupoid GBund(X)G Bund(X) of GG-principal bundles over XX is equivalent to geometric morphims Sh(X)BGSh(X) \to B G:

GBund(X)Topos(Sh(X),BG) G Bund(X) \simeq Topos(Sh(X), B G) \,

and the notion of classifying space in topology, which for the discrete group GG is a topological space G\mathcal{B} G such that

π 0GBund(X)π 0Top(X,G) \pi_0 G Bund(X) \simeq \pi_0 Top(X, \mathcal{B}G) \,

we should expect there to be a topos analog of the total space, EGE G, for the classifying space. This analog is the generic G-torsor, which is an internal GG-torsor in the topos Set GSet^G. The important aspect of the space EGE G is that as a principal GG-bundle over G\mathcal{B} G, it is a universal element, i.e. the natural transformation Hom(X,G)GBdl(X)Hom(X, \mathcal{B}G) \to G Bdl(X) that it induces (by the Yoneda lemma) is the isomorphism which exhibits G\mathcal{B}G as the object representing the functor XGBdl(X)X \mapsto G Bdl(X). For the same Yoneda reasons, the classifying topos Sh(C T)Sh(C_T) of any geometric theory TT comes with a generic TT-model, which is a TT-model in Sh(C T)Sh(C_T) which represents the functor ETMod(E)E \mapsto T Mod(E) in the same way. For TT = the theory of GG-torsors, this generic model is the generic GG-torsor.

References

A standard textbook reference is section D3 of

Original articles 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 III.2-4 of

and

Revised on November 26, 2014 00:22:44 by Urs Schreiber (82.224.164.72)