nLab
geometric morphism

Context

Topos Theory

Could not include topos theory - contents

Contents

Idea

For XX and YY topological spaces, a continuous function XYX \to Y induces (in particular) two functors

between the corresponding Grothendieck topoi of sheaves on XX and YY. These are such that:

Morever, if XX and YY are sober topological spaces every pair of functors with these properties comes uniquely from a continuous map XYX \to Y (see the theorem below).

A geometric morphism between arbitrary topoi is the direct generalization of this situation.

Another motivation of the concept comes from the the fact that a functor such as f *f^* that preserves finite limits and arbitrary colimits (since it is a left adjoint) necessarily preserves all constructions in geometric logic. See also classifying topos.

Definition

Definition

If EE and FF are toposes, a geometric morphism f:EFf:E\to F consists of an pair of adjoint functors (f *,f *)(f^*,f_*)

f *:EF f_* : E \to F
EF:f *, E \leftarrow F : f^* \,,

such that the left adjoint f *:FEf^*:F \to E preserves finite limits.

We say that

of the geometric morphism.

If moreover the inverse image f *f^* has also a left adjoint f !:EFf_! : E \to F, then ff is an essential geometric morphism.

Remark

Since Grothendieck toposes satisfy the (dual) hypotheses of Freyd’s special adjoint functor theorem, any functor f *f^* between Grothendieck toposes which preserves all small colimits must have a right adjoint. Therefore, a geometric morphism f:EFf : E \to F between Grothendieck toposes could equivalently be defined as a functor EF:f *E \leftarrow F : f^* preserving finite limits and all small colimits.

Remark

In view of its definition in terms of a pair of adjoint functors, the direction of a geometric morphism is a convention. However, with the other convention it would better be called an algebraic morphism.

See Isbell duality for more on this duality between algebra and geometry.

See also (Johnstone, p. 162/163).

Properties

We discuss some general properties of geometric morphisms. The

also serves as a motivation or justification of the notion of geometric morphism. The

is a fairly straightforward generalization of that situation, reflecting the passage from (sheaf-) (0,1)-toposes to general (1,1)-toposes.

A somewhat subtle point about geometric morphisms of toposes is that there is also another sensible notion of topos homomorphisms: logical morphisms. In

aspects of the relation between the two concepts are discussed.

The reader wishing to learn about geometric morphisms systematically might want to first read the section on Geometric morphisms between presheaf toposes below, as much of the following discussion makes use of a few basic facts discussed there.

Relation to homomorphisms of locales

The definition of geometric morphisms may be motivated as being a categorification of the definition of morphisms of locales.

Recall that

Definition

A homomorphism of locales

f:XY f : X \to Y

is dually a morphism of frames (the “frames of open subsets” of XX and YY, respectively)

𝒪(X)𝒪(Y):f *. \mathcal{O}(X) \leftarrow \mathcal{O}(Y) : f^* \,.

This, in turn, is a functor (of posets) that

  1. preserves finite limits (called meets in this context);

  2. preserves arbitrary (small) colimits (called joins in this context).

Such a preservation of finite limits and arbitrary colimits is precisely what characterizes the inverse image part of a geometric morphism, and hence by the adjoint functor theorem already characterizes the full notion of geometric morphisms. Since a locale may equivalently be thought of as a (0,1)-topos, this means that geometric morphisms are direct generalization of the notion of locale homorphisms to 1-toposes.

The following says this in more precise fashion.

Definition

For f:XYf : X \to Y a homomorphism of locales, let

f *:Sh(X)Sh(Y) f_* : Sh(X) \to Sh(Y)

be the functor between their sheaf toposes that sends a sheaf F:𝒪(X) opSetF : \mathcal{O}(X)^{op} \to Set to the composite

f *F:𝒪(Y) opf *𝒪(X) opFSet, f_* F : \mathcal{O}(Y)^{op} \stackrel{f^*}{\to} \mathcal{O}(X)^{op} \stackrel{F}{\to} Set \,,

where f *f^* is the corresponding frame morphism as in def. 2.

Propositon

The functor f *f_* in def. 3 is the direct image part of a geometric morphism of sheaf toposes

(f *f *):Sh(X)f *f *Sh(Y). (f^* \dashv f_*) : Sh(X) \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} Sh(Y) \,.

Moreover, the corresponding inverse image functor f *f^* does restrict on representables to the frame morphism that we also denoted f *f^*.

In (Johnstone) this appears as lemma C1.4.1 and theorem C1.4.3.

Proof

Since a morphism of frames is a morphism of sites, as discussed there, this follows from the corresponding propositions in the section Morphisms of sites and geometric morphisms.

Propositon

The construction XSh(X)X \mapsto Sh(X) extends to a 2-functor

Sh:LocaleTopos Sh : Locale \hookrightarrow Topos

from the category Locale of locales to the 2-category Topos of toposes and geometric morphisms between them

See also at locale the section relation to toposes.

Relation to morphisms of sites

See at morphism of sites the section Relation to geometric morphisms .

Relation to logical morphisms

Proposition

Every geometric morphism whose direct image is a logical morphism is an equivalence.

This is a restatement of this proposition at logical morphism. See there for a proof.

But inverse images can be nontrivial logical morphisms:

Generally, a geometric morphism with logical inverse image is called an atomic geometric morphism. See there for more details.

Structure preserved by geometric morphisms

The inverse images of geometric morphisms preserves the structure of toposes in the sense of their characterization as categories with finite limits that are well-powered indexed categories with respect to the canonical indexing over themselves.

This appears in (Johnstone) as remark B2.2.7 based on example B1.3.17 and prop. B1.3.14. See at indexed category the section Well-poweredness.

Surjection/embedding factorization

Every geometric morphism factors, essentially uniquely, as a geometric surjection followed by a geometric embedding. See geometric surjection/embedding factorization for more on this.

Special classes of geometric morphisms

There are various special cases and types of classes of geometric morphisms. For instance

The following subsections describe some of these in more detail.

Between presheaf toposes

Let CC and DD be any two categories. We write C opC^{op} and D opD^{op} for their opposite categories and [C,Set][C, Set], [D,Set][D, Set] for the corresponding presheaf toposes over C opC^{op} and D opD^{op}, respectively.

Proposition

Every functor f:CDf : C \to D induces an (essential, even) geometric morphism

f:=(f *f *):[C,Set]f *f *f ![D,Set], f := (f^* \dashv f_*) : [C,Set] \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [D, Set] \,,

where f *=()ff^* = (-) \circ f is the functor given by precomposition presheaves with ff.

Moreover, for η:fg:CoD\eta : f \Rightarrow g : C \o D a natural transformation between two such functors there is an induced geometric transformation (f *f *)(g *g *)(f^* \dashv f_*) \Rightarrow (g^* \dashv g_*). This is compatible with composition in that it makes forming presheaf toposes a 2-functor

[,Set]:CatTopos [-,Set] : Cat \to Topos

from the 2-category Cat to the 2-category Topos.

This appears as (Johnstone, example A4.1.4).

Proof

Since categories of presheaves have all limits and colimits, the left and right Kan extensions Lan fLan_f and Ran fRan_f along ff exists, and form with f *f^* an adjoint triple

[C,Set]Ran ff *Lan f[D,Set]. [C,Set] \stackrel{\overset{Lan_f}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{Ran_f}{\to}}} [D, Set] \,.

Hence f !Lan ff_! \simeq Lan_f and f *Ran ff_* \simeq Ran_f. Notice that left adjoints and right adjoints to a functor are, if they exist, unique up to unique isomorphism.

Next we consider extra property on CC, DD and ff such that f *f^* induces also a second geometric morphism, going the other way round. This plays a role for the discussion of morphisms of sites. For that reason we pass now from CC and DD to their opposite categories hence consider genuine presheaves on CC and DD.

Proposition

Let CC and DD by categories with finite limits and let f:CDf : C \to D be a finite-limit preserving functor.

Then in the adjoint triple

(f !f *f *):[C op,Set]f *f *f ![D op,Set] (f_! \dashv f^* \dashv f_*) : [C^{op},Set] \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} [D^{op}, Set]

the left Kan extension f 1f_1 also preserves finite limits and hence in this case f *f^* is also the direct image of a geometric morphism going the other way round:

(f !f *):[D op,Set][C op,Set]. (f_! \dashv f^* ) : [D^{op},Set] \to [C^{op}, Set] \,.

This appears as (Johnstone, example A4.1.10).

Proof

Recall that for F:C opSetF : C^{op} \to Set a functor, the left Kan extension f !F:D opSetf_! F : D^{op} \to Set is computed over each object dDd \in D by the colimit

(f !F)(d)=lim ((d/f) opUC opFSet) (f_! F)(d) = \lim_\to \left( (d/f)^{op} \stackrel{U}{\to} C^{op} \stackrel{F}{\to} Set \right)

where (d/f)(d/f) is the comma category and

U:(d/f)C U : (d/f) \to C

is the evident forgetful functor. This is natural in FF and so (f !)(d)(f_! -)(d) is the functor

(f !)(d):[C op,Set]U *[(d/f) op,Set]lim Set. (f_! -)(d) : [C^{op}, Set] \stackrel{U^*}{\to} [(d/f)^{op}, Set] \stackrel{\lim_\to}{\to} Set \,.

By the above argument U *U^* has a left adjoint (the left Kan extension along UU) hence itself preserves all limits.

It then suffices to observe (see below) that by the fact that ff preserves finite limits we have that the categories (d/f) op(d/f)^{op} are filtered categories. Then by the fact (see there) that filtered colimits commute with finite limits, it follows that also lim \lim_\to preserves finite limits, and hence (f !)(d)(f_! -)(d) does. Since colimits of presheaves are computed objectwise, this shows that f !f_! preserves finite limits. This completes the proof.

Here is an explicit desciption of the filteredness of the comma category (d/f) op(d/f)^{op} for any object ff.

We check the axioms on a filtered category:

  • non-emptiness : There is an object in (d/f) op(d/f)^{op}: since ff by assumption preserves the terminal object, take the terminal morphism (df(*)=*)(d \to f(*) = *);

  • connectedness : for any two objects (dh 1f(c 1))(d \stackrel{h_1}{\to} f(c_1)) and (dh 2f(c 2))(d \stackrel{h_2}{\to} f(c_2)) form the product c 1×c 2c_1 \times c_2 and use that ff preserves this to produce the object (df(h 1),f(h 2)f(c 1)×f(c 2)f(c 1×c 2))(d \stackrel{f(h_1), f(h_2)}{\to} f(c_1) \times f(c_2) \simeq f(c_1 \times c_2)). Then the image under ff of the two projections provides the required span

    d f(h 1) (f(h 1),f(h 2)) f(h 2) f(c 1) f(p 1) f(c 1×c 2) f(p 2) f(c 2). \array{ && d \\ & {}^{\mathllap{f(h_1)}}\swarrow & \downarrow^{(f(h_1),f(h_2))} & \searrow^{\mathrlap{f(h_2)}} \\ f(c_1) &\stackrel{f(p_1)}{\leftarrow}& f(c_1 \times c_2) & \stackrel{f(p_2)}{\to} & f(c_2) } \,.
  • finally, for

    d f(c 1) h 2h 1 f(c 2) \array{ && d \\ & \swarrow && \searrow \\ f(c_1) && \stackrel{\overset{h_1}{\to}}{\underset{h_2}{\to}}& & f(c_2) }

    two parallel morphism, let eq(h 1,h 2)eq(h_1,h_2) be the equalizer of the underlying morphism in CC. Since ff preserves equalizers we have an object (df(eq(h 1,h 2)))(d \to f(eq(h_1,h_2))) and a morphism to (df(c 1))(d \to f(c_1)) that equalizes the above two morphisms.

Surjections and embeddings

A geometric morphism f:EFf : E \to F is a surjection if f *f^* is faithful. It is an embedding if f *f_* is fully faithful.

Proposition

Up to equivalence, every embedding of toposes is of the form

Sh j(E)E, Sh_j(E) \to E \,,

where Sh j(E)Sh_j(E) is the topos of sheaves with respect to a Lawvere-Tierney topology j:ΩΩj : \Omega \to \Omega on EE.

This means in particular that fully faithful geometric morphisms into Grothendieck topoi are an equivalent way of encoding a Grothendieck topology.

Proposition

Up to equivalence, every surjection of topoi is of the form

EE G E \to E_G

where E GE_G is the category of coalgebras for a finite-limit-preserving comonad on EE.

Every geometric morphism f:EFf:E\to F factors, uniquely up to equivalence, as a surjection followed by an embedding. There are two ways to produce this factorization: either construct E GE_G where G=f *f *G= f^*f_* is the comonad induced by the adjunction f *f *f^*\dashv f_*, or construct Sh j(F)Sh_j(F) where jj is the smallest Lawvere-Tierney topology on FF such that ff factors through Sh j(F)Sh_j(F). In fact, surjections and embeddings form a 2-categorical orthogonal factorization system on the 2-category of topoi.

Global sections and constant sheaves

For every Grothendieck topos EE, there is a geometric morphism

Γ:ESet:const \Gamma : E \stackrel{\leftarrow}{\to} Set : const

called the global sections functor. It is given by the hom-set out of the terminal object

Γ()=Hom E(*,) \Gamma(-) = Hom_E({*}, -)

and hence assigns to each object AEA\in E its set of global elements Γ(A)=Hom E(*,A)\Gamma(A) = Hom_E(*,A). If we think of AA as a sheaf, then Γ(A)\Gamma(A) is the set of global sections.

The left adjoint const:SetEconst : Set \to E of the global section functor is the canonical Set-tensoring functor

:Set×EE \otimes : Set \times E \to E

applied to the terminal object

const=()*:SetE const = (-)\otimes {*} : Set \to E

which sends a set SS to the coproduct of |S||S| copies of the terminal object

S*= sS*. S \otimes {*} = \coprod_{s \in S} {*} \,.

This is called the constant object of EE on the set SS. Notably when EE is a sheaf topos this is the constant sheaf on SS.

The left adjointness is just the defining property of the tensoring

Hom E(constS,A)Hom E(S*,A)Hom Set(S,Hom E(*,A)). Hom_E(const S, A) \simeq Hom_E(S \otimes {*},A) \simeq Hom_{Set}(S, Hom_E(*,A)) \,.

This left adjoint preserves products, using that colimits in a topos are stable by base change (see commutativity of limits and colimits)

( s 1S 1*)×( s 2S 2*)= s 1S 1(*×( s 2S 2*))= s 1S 1( s 2S 2*)= s 1S 1 s 2S 2*= sS 1×S 2* \left( \coprod_{s_1 \in S_1} *\right) \times \left( \coprod_{s_2 \in S_2} *\right) = \coprod_{s_1 \in S_1} \left(* \times \left( \coprod_{s_2 \in S_2} *\right)\right) = \coprod_{s_1 \in S_1} \left( \coprod_{s_2 \in S_2} *\right) = \coprod_{s_1 \in S_1} \coprod_{s_2 \in S_2} * = \coprod_{s \in S_1 \times S_2} *

and it preserves equalizers and therefore limits. So it is left exact and we do have a geometric morphism.

Point of a topos

For EE a topos, a geometric morphism

x:SetE x : Set \to E

is called a point of a topos.

Change-of-base

For EE any topos and k:BAk : B \to A any morphism in EE there is the change-of-base functor of over categories

k *:(E/A)(E/B) k^* : (E/A) \to (E/B)

by pullback. As described at dependent product this functor has both a left adjoint k:E/BE/A\coprod_k : E/B \to E/A as well as a right adjoint k:E/AE/B\prod_k : E/A \to E/B. Therefore

(Π k,k *):E/BE/B (\Pi_k, k^*) : E/B \leftrightarrow E/B

is a geometric morphism. Hence (Π kk * k)(\Pi_k \dashv k^* \dashv \coprod_k) is an essential geometric morphism.

Sheafification

A category of sheaves is a geometric embedding into a presheaf topos

Sh(C)PSh(C). Sh(C) \hookrightarrow PSh(C) \,.

Geometric morphisms of sheaf topoi

Geometric morphisms between localic topoi are equivalent to continuous maps of locales, which in turn are equivalent to continuous maps of topological spaces if you restrict to sober spaces.

Unrolling this: For XX a topological space, write Sh(X):=Sh(Op(X))Sh(X) := Sh(Op(X)) as usual for the topos given by the category of sheaves on the category of open subsets Op(X)Op(X) with the standard coverage

Lemma

For every continuous map f:XYf : X \to Y of sober topological spaces with the induced functor f 1:Op(Y)Op(X)f^{-1} : Op(Y) \to Op(X) of sites, the direct image

f *:Sh(X)Sh(Y) f_* : Sh(X) \to Sh(Y)

and the inverse image

f *:Sh(Y)Sh(X) f^* : Sh(Y) \to Sh(X)

constitute a geometric morphism

f:Sh(X)Sh(Y) f : Sh(X) \to Sh(Y)

(denoted by the same symbol, by convenient abuse of notation).

This map Hom Top(X,Y)GeomMor(Sh(X),Sh(Y))Hom_{Top}(X,Y) \to GeomMor(Sh(X),Sh(Y)) is an bijection of sets.

Proof

That the induced pair (f *,f *)(f^*, f_*) forms a geometric morphism is (or should eventually be) discussed at inverse image.

We now show that every geometric morphism of sheaf toposes arises this way from a continuous function, at least up to isomorphism. (In fact, more is true: the category of geometric morphisms Sh(X)Sh(Y)Sh(X)\to Sh(Y) is equivalent to the poset of continuous functons XYX\to Y with the specialization ordering.) We follow MacLane-Moerdijk, page 348.

One reconstructs the continuous map f:XYf : X \to Y from a geometric morphism f:Sh(X)Sh(Y)f : Sh(X) \to Sh(Y) as follows.

Write *=YSh(Y){*} = Y \in Sh(Y) for the sheaf on Op(Y)Op(Y) constant on the singleton set, the terminal object in Sh(Y)Sh(Y).

Notice that since the inverse image f *f^* preserves finite limits, every subobject U Y*U_Y \hookrightarrow {*} is taken by f *f^* to a subobject U XXU_X \hookrightarrow X, obtained by applying f *f^* to the pullback diagram

U Y *=Y *=Y Ω \array{ U_Y &\to& {*} = Y \\ \downarrow && \downarrow \\ {*} = Y &\to& \Omega }

that characterizes the subobject U YU_Y in the topos.

But, as the notation already suggests, the subobjects of X,YX,Y are just the open sets, i.e. the representable sheaves.

This yields a function f *:Obj(Op(Y))Obj(Op(X))f^* : Obj(Op(Y)) \to Obj(Op(X)) from open subsets to open subsets. By assumption, this preserves finite limits and arbitrary colimits, i.e. finite intersections and arbitrary unions of open sets. In other words, it is a frame homomorphism, and thus can be regarded as a morphism XYX\to Y of locales.

We can now use this to define a function f¯:XY\bar f : X \to Y of the sets underlying the topological spaces XX and YY by setting

(f¯(x)=y)Vy:xf *(V). (\bar f(x) = y) \Leftrightarrow \forall V \ni y: x \in f^*(V) \,.

This yields a well defined function for the following reasons (which for the moment we spell out in the case where YY is Hausdorff, although the result should hold —and furthermore, hold constructively— whenever YY is sober):

  • there is at most one yy satisfying this equation: if y 1y 2y_1 \neq y_2 both satisfy it, there are, by assumption of YY being Hausdorff, neighbourhoods V 1y 1V_1 \ni y_1 and V 2y 2V_2 \ni y_2 such that (using that f *f^* preserves limits hence intersections) f *(V 1)f *(V 2)=f *(V 1V 2)=f^*(V_1) \cap f^*(V_2) = f^*(V_1 \cap V_2) = \emptyset, which contradicts the assumption.

  • there is at least one yy satisfying this equation: again by contradiction: if there were none then every yYy \in Y has a neighbourhood V yV_y with x¬f *(V y)x \not\in f^*(V_y), so that similarly to above we conclude with x¬ yYf *(V y)=f *( yV y)=f *(Y)=Xx \not\in \cup_{y \in Y} f^*(V_y) = f^*(\cup_y V_y) = f^*(Y) = X again a contradiction.

Am I right that what we are really need of our space here is not necessarily that it be Hausdorff but simply that it be sober? (Then the nonconstructive aspects of the argument —which is what made me look at this— come in only because the theorem that a Hausdorff space must be sober is not constructively valid.) —Toby

Mike Shulman: Yes, that’s exactly right. All the complication defining f¯\bar f above is just an unrolled way of saying that geometric morphisms between localic topoi are equivalent to continuous maps of locales, which are equivalent to continuous functions if you have sober spaces. I think that should be clarified.

Toby: OK, I added a paragraph at the beginning of the example to clarify this. I still need to rewrite the argument immediately above to apply to sober spaces. (Everything else seems to go through exactly the same.)

So our function f¯:XY\bar f : X \to Y is well defined and satisfies f¯ 1(U Y)=f *(U Y)\bar f^{-1}(U_Y) = f^*(U_Y) for every open set U YObj(Op(Y))U_Y \in Obj(Op(Y)). In particular it is therefore a continuous map.

It remains to check that this map reproduces the geometric morphism that we started with. For that we compute its direct image on any sheaf ASh(X)A \in Sh(X) as

f¯ *(A):U Y A(f¯ 1(U Y)) Hom Sh(X)(f¯ 1(U Y),A) =Hom Sh(X)(f *V,E) Hom Sh(X)(V,f *E) (f *A)(U Y) \begin{aligned} \bar f_*(A) : U_Y &\mapsto A(\bar f^{-1}(U_Y)) \\ & \simeq Hom_{Sh(X)}(\bar f^{-1}(U_Y),A) \\ & = Hom_{Sh(X)}(f^* V, E) \\ & \simeq Hom_{Sh(X)}(V, f_* E) \\ & \simeq (f_* A)(U_Y) \end{aligned}
Corollary

The points xXx \in X of the topological space XX are in canonical bijection with the points of Sh(X)Sh(X) in the sense of point of a topos.

References

Geometric morphisms are the topic of section VII of

Embeddings and surjections are discussed in section VII.4.

Geometric morphisms are defined in section A4 of

The special classes of geometric morphisms are discussed in section C3.

Revised on November 29, 2014 14:45:37 by Adeel Khan (132.252.63.48)