nLab
geometric morphism

Idea

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

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

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

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 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

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

$$f_{*}:E\to F$$f_* : E \to F

$$f^{*}:F\to E,$$f^* : F \to E \,,

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

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

Remarks

• 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 between Grothendieck toposes could equivalently be defined as a functor preserving finite limits and all small colimits.

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

$${\mathrm{Sh}}_j(E)\to E,$$Sh_j(E) \to E \,,

where ${\mathrm{Sh}}_j(E)$ 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

$$E\to E_G$$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 $f^{*}⊣f_{*}$, or construct ${\mathrm{Sh}}_j(F)$ where $j$ is the smallest Lawvere-Tierney topology on $F$ such that $f$ factors through ${\mathrm{Sh}}_j(F)$. In fact, surjections and embeddings form a 2-categorical orthogonal factorization system on the 2-category of topoi.

Examples

Global sections and constant sheaves

For every topos $E$, there is a geometric morphism

$$\Gamma :E\overleftarrow{\to }\mathrm{Set}:\mathrm{const}$$\Gamma : E \stackrel{\leftarrow}{\to} Set : const

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

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

and hence assigns to each object $A\in E$ its set of global elements $\Gamma (A)={\mathrm{Hom}}_E(*,A)$. If $E$ is a Grothendieck topos then one thinks of $A$ as a sheaf and of $\Gamma (A)$ as its set of global sections.

The left adjoint $\mathrm{const}:\mathrm{Set}\to E$ of the global section functor is the canonical Set-tensoring functor

$$\otimes :\mathrm{Set}\times E\to E$$\otimes : Set \times E \to E

applied to the terminal object

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

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

$$S\otimes *=\coprod _{s\in S}*.$$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

$${\mathrm{Hom}}_E(\mathrm{const}S,A)\simeq {\mathrm{Hom}}_E(S\otimes *,A)\simeq {\mathrm{Hom}}_{\mathrm{Set}}(S,{\mathrm{Hom}}_E(*,A)).$$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)

$$(\coprod _{s_1\in S_1}*)\times (\coprod _{s_2\in S_2}*)=\coprod _{s_1\in S_1}(*\times (\coprod _{s_2\in S_2}*))=\coprod _{s_1\in S_1}(\coprod _{s_2\in S_2}*)=\coprod _{s_1\in S_1}\coprod _{s_2\in S_2}*=\coprod _{s\in S_1\times S_2}*$$\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 let exact and we do have a geometric morphism.

Point of a topos

For $E$ a topos, a geometric morphism

$$x:\mathrm{Set}\to E$$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

$$k^{*}:(E/A)\to (E/B)$$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 ${\prod }_k:E/A\to E/B$. Therefore

$$({\Pi }_k,k^{*}):E/B\leftrightarrow E/B$$(\Pi_k, k^*) : E/B \leftrightarrow E/B

is a geometric morphism.

Sheafification

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

$$\mathrm{Sh}(C)\hookrightarrow \mathrm{PSh}(C).$$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 $X$ a topological space, write $\mathrm{Sh}(X):=\mathrm{Sh}(\mathrm{Op}(X))$ as usual for the topos given by the category of sheaves on the category of open subsets $\mathrm{Op}(X)$ with the standard coverage

Lemma

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

$$f_{*}:\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$$f_* : Sh(X) \to Sh(Y)

and the inverse image

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

constitute a geometric morphism

$$f:\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$$f : Sh(X) \to Sh(Y)

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

This map ${\mathrm{Hom}}_{\mathrm{Top}}(X,Y)\to \mathrm{GeomMor}(\mathrm{Sh}(X),\mathrm{Sh}(Y))$ 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 $\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$ is equivalent to the poset of continuous functons $X\to Y$ with the specialization ordering.) We follow page 348 of

• MacLane-Moerdijk, Sheaves in Geometry and Logic .

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

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

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

$$\begin{array}{ccc}{U}_Y& \to & *=Y\\ \downarrow & & \downarrow \\ *=Y& \to & \Omega \end{array}$$\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 $f^{*}:\mathrm{Obj}(\mathrm{Op}(Y))\to \mathrm{Obj}(\mathrm{Op}(X))$ 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 $\overline{f}:X\to Y$ of the sets underlying the topological spaces $X$ and $Y$ by setting

$$(\overline{f}(x)=y)\iff \forall V\ni y:x\in f^{*}(V).$$(\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 $Y$ 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 $y_1\ne y_2$ 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) $f^{*}(V_1)\cap f^{*}(V_2)=f^{*}(V_1\cap V_2)=\varnothing$, 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 $x\neg \in f^{*}(V_y)$, so that similarly to above we conclude with $x\neg \in {\cup }_{y\in Y}{f}^{*}(V_y)=f^{*}({\cup }_y{V}_y)=f^{*}(Y)=X$ 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 $\overline{f}$ 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 $\overline{f}:X\to Y$ is well defined and satisfies ${\overline{f}}^{-1}(U_Y)=f^{*}(U_Y)$ for every open set $U_Y\in \mathrm{Obj}(\mathrm{Op}(Y))$. 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 $A\in \mathrm{Sh}(X)$ as

$$\begin{array}{rl}{\overline{f}}_{*}(A):U_Y& \mapsto A({\overline{f}}^{-1}(U_Y))\\ & \simeq {\mathrm{Hom}}_{\mathrm{Sh}(X)}({\overline{f}}^{-1}(U_Y),A)\\ & ={\mathrm{Hom}}_{\mathrm{Sh}(X)}(f^{*}V,E)\\ & \simeq {\mathrm{Hom}}_{\mathrm{Sh}(X)}(V,f_{*}E)\\ & \simeq (f_{*}A)(U_Y)\end{array}$$\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 $\mathrm{Sh}(X)$ in the sense of point of a topos.

References

Geometric morphisms are the topic of section VII of

• Saunders MacLane and Ieke Moerdijk, Sheaves in Geometry and Logic .

Embeddings and surjections are discussed in section VII.4.