Contents

topos theory

# Contents

## Idea

For $X$ and $Y$ topological spaces, a continuous function $X \to Y$ induces (in particular) two functors

• the direct image $f_* : Sh(X) \to Sh(Y)$

• the inverse image $f^* : Sh(Y) \to Sh(X)$

between the corresponding Grothendieck topoi of sheaves on $X$ and $Y$. These are such that:

• $f^*$ is left adjoint to $f_*$, so $f^*$ preserves all small colimits and $f_*$ preserves all small limits.

• furthermore, $f^*$ is left exact in that it preserves finite limits.

Morever, if $X$ and $Y$ are sober topological spaces every pair of functors with these properties comes uniquely from a continuous map $X \to Y$ (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 fact that a functor such as $f^*$ 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 $E$ and $F$ are toposes, a geometric morphism $f:E\to F$ consists of a pair of adjoint functors $(f^*,f_*)$

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

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

We say that

• $f_*$ is the direct image

• $f^*$ is the inverse image

of the geometric morphism.

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

###### Remark

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

## 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
$f : X \to Y$

is dually a morphism of frames (the “frames of open subsets” of $X$ and $Y$, respectively)

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

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

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

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

$f_* F : \mathcal{O}(Y)^{op} \stackrel{f^*}{\longrightarrow} \mathcal{O}(X)^{op} \stackrel{F}{\longrightarrow} Set \,,$

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

###### Proposition

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

$(f^* \dashv f_*) : Sh(X) \stackrel{\overset{f^*}{\longleftarrow}}{\underset{f_*}{\longrightarrow}} Sh(Y) \,.$

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

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.

###### Proposition

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

$Sh : Locale \hookrightarrow Topos$

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

### 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:

###### Proposition

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

###### Proposition

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

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

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

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

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

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 $Lan_f$ and $Ran_f$ along $f$ exists, and form with $f^*$ an adjoint triple

$[C,Set] \stackrel{\overset{Lan_f}{\longrightarrow}}{\stackrel{\overset{f^*}{\longleftarrow}}{\underset{Ran_f}{\longrightarrow}}} [D, Set] \,.$

Hence $f_! \simeq Lan_f$ and $f_* \simeq Ran_f$. 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 $C$, $D$ and $f$ such that $f^*$ 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 $C$ and $D$ to their opposite categories hence consider genuine presheaves on $C$ and $D$.

###### Proposition

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

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

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

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

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

###### Proof

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

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

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

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

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

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

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

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

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

We check the axioms on a filtered category:

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

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

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

$\array{ && d \\ & \swarrow && \searrow \\ f(c_1) && \stackrel{\overset{h_1}{\to}}{\underset{h_2}{\to}}& & f(c_2) }$

two parallel morphism, let $eq(h_1,h_2)$ be the equalizer of the underlying morphism in $C$. Since $f$ preserves equalizers we have an object $(d \to f(eq(h_1,h_2)))$ and a morphism to $(d \to f(c_1))$ that equalizes the above two morphisms.

### Surjections and embeddings

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

###### Proposition

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

$Sh_j(E) \to E \,,$

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

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

$E \to E_G$

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

Every geometric morphism $f:E\to F$ factors, uniquely up to equivalence, as a surjection followed by an embedding. There are two ways to produce this factorization: either construct $E_G$ where $G= f^*f_*$ is the comonad induced by the adjunction $f^*\dashv f_*$, or construct $Sh_j(F)$ where $j$ is the smallest Lawvere-Tierney topology on $F$ such that $f$ factors through $Sh_j(F)$. 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 $E$, there is a geometric morphism

$\Gamma \;\colon\; E \stackrel{\leftarrow}{\to} Set : const$

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

$\Gamma(-) = Hom_E({*}, -)$

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

The left adjoint $const \;\colon\; Set \to E$ of the global section functor is the canonical Set-tensoring functor

$\otimes \;\colon\; Set \times E \to E$

applied to the terminal object

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

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

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

This is called the constant object of $E$ on the set $S$. Notably when $E$ is a sheaf topos this is the constant sheaf on $S$.

The left adjointness is just the defining property of the tensoring

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

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

$\left( \coprod_{s_1 \in S_1} *\right) \times \left( \coprod_{s_2 \in S_2} *\right) = \coprod_{s_1 \in S_1} \left(* \times \left( \coprod_{s_2 \in S_2} *\right)\right) = \coprod_{s_1 \in S_1} \left( \coprod_{s_2 \in S_2} *\right) = \coprod_{s_1 \in S_1} \coprod_{s_2 \in S_2} * = \coprod_{s \in S_1 \times S_2} *$

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

### Point of a topos

For $E$ a topos, a geometric morphism

$x : Set \to E$

is called a point of a topos.

### Change-of-base

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

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

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

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

is a geometric morphism. Hence $(\coprod_k \dashv k^* \dashv \prod_k)$ is an essential geometric morphism.

### Sheafification

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

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

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

###### Lemma

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

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

and the inverse image

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

constitute a geometric morphism

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

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

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

###### Proof

That the induced pair $(f^*, f_*)$ 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 $Sh(X)\to Sh(Y)$ is equivalent to the poset of continuous functons $X\to Y$ with the specialization ordering (Elephant, Proposition C.1.4.5.). We follow MacLane-Moerdijk, page 348.

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

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

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

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

that characterizes the subobject $U_Y$ in the topos.

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

This yields a function $f^* : Obj(Op(Y)) \to Obj(Op(X))$ 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 $X\to Y$ of locales.

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

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

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

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

• there is at least one $y$ satisfying this equation: again by contradiction: if there were none then every $y \in Y$ has a neighbourhood $V_y$ with $x \not\in f^*(V_y)$, so that similarly to above we conclude with $x \not\in \cup_{y \in Y} f^*(V_y) = f^*(\cup_y V_y) = f^*(Y) = X$ 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 $\bar f$ 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 $\bar f : X \to Y$ is well defined and satisfies $\bar f^{-1}(U_Y) = f^*(U_Y)$ for every open set $U_Y \in Obj(Op(Y))$. 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 $A \in Sh(X)$ as

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

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.