nLab
surjective geometric morphism

Definition

Definition

A geometric morphism between toposes $(f^{*}⊣f_{*}):ℰ\to ℱ$ is surjective or a geometric surjection if it satisfies the following equivalent criteria:

• its inverse image functor $f^{*}$ is faithful (in contrast to the direct image being full and faithful as for a geometric embedding );

• its inverse image functor $f^{*}$ is conservative;

• the components $X\to f_{*}{f}^{*}X$ of the adjunction unit are monomorphisms, for all $X\in ℱ$;

• $f^{*}$ induces a injective homomorphism of subobject lattices

  $$\mathrm{Sub}(X)\hookrightarrow \mathrm{Sub}(f^{*}X)$$Sub(X) \hookrightarrow Sub(f^* X)

  for all $X\in ℱ$;

• $f^{*}$ reflects the order on subobjects;

• $(f^{*}⊣f_{*})$ is a comonadic adjunction.

The equivalence of these condition appears for instance as MacLaneMoerdijk, VII 4. lemma 3 and prop. 4.

Proof

We discuss the equivalence of these conditions:

The equivalence $(f^{*}\mathrm{faithful})\iff (\mathrm{Id}\to f^{*}{f}_{*}\mathrm{is}\mathrm{mono})$ is a general property of adjoint functors (see there).

The implication $(f^{*}\mathrm{faithful})\Rightarrow (f^{*}\mathrm{induces}\mathrm{injection}\mathrm{on}\mathrm{subobjects})$ works as follows:

first of all $f^{*}$ does indeed preserves subobjects: since it respects pullbacks and since monomorphisms are characterized as those morphisms whose domain is stable under pullback along themselves.

To see that $f^{*}$ induces an injective function on subobjects let $U\hookrightarrow X$ be a subobject with characteristic morphism $\mathrm{char}U:X\to \Omega$ and consider the image

$$\begin{array}{ccc}{f}^{*}U& \to & f^{*}*\simeq *\\ \downarrow & & \downarrow \\ f^{*}X& \stackrel{f^{*}\mathrm{char}U}{\to }& f^{*}\Omega \end{array}.$$\array{ f^* U &\to& f^* * \simeq * \\ \downarrow && \downarrow \\ f^* X &\stackrel{f^* char U}{\to}& f^* \Omega } \,.

of the pullback diagram that exhibits $U$ as a subobject. Since $f^{*}$ preserves pullbacks, this is still a pullback diagram.

If now $U\le \tilde{U}$ but $f^{*}(U)=f^{*}(\tilde{U})$ then both corresponding pullback diagrams are sent by $f^{*}$ to the same such diagram. By faithfulness this implies that also

$$\begin{array}{ccc}\tilde{U}& \to & *\\ \downarrow & & \downarrow \\ X& \stackrel{\mathrm{char}U}{\to }& \Omega \end{array}$$\array{ \tilde U &\to& * \\ \downarrow && \downarrow \\ X &\stackrel{char U}{\to}& \Omega }

commutes, and hence that also $\tilde{U}\subset U$, so that in fact $\tilde{U}\simeq U$.

Next we consider the implication $(f^{*}\mathrm{induces}\mathrm{injection}\mathrm{on}\mathrm{subobjects})\Rightarrow (f^{*}\mathrm{is}\mathrm{conservative})$.

Assume $f^{*}(X\stackrel{\simeq }{\to }X\prime )$ is an isomorphism. We have to show that then $\varphi$ is an isomorphism. Consider the image factorization $X\to \mathrm{im}(\varphi )\hookrightarrow X\prime$. Since $f$ preserves pushouts and pullbacks, it preserves epis and monos and so takes this to the image factorization

$$f^{*}X\to f^{*}(\mathrm{im}\varphi )\stackrel{\simeq }{\to }{f}^{*}X\prime$$f^* X \to f^* (im \phi) \stackrel{\simeq}{\to} f^* X'

of $f^{*}\varphi$, where now the second morphism is an iso, because $f^{*}\varphi$ is assumed to be an iso. By the assumption that $f^{*}$ is injective on subobjects it follows that also $\mathrm{im}\varphi \simeq X\prime$ and thus that $\varphi$ is an epimorphism.

It remains to show that $\varphi$ is also a monomorphism. For that it is sufficient to show that in the pullback square

$$\begin{array}{ccc}X{\times }_{X\prime }X& \to & X\\ \downarrow & & {\downarrow }^{\varphi }\\ X& \stackrel{\varphi }{\to }& X\prime \end{array}$$\array{ X \times_{X'} X &\to& X \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ X &\stackrel{\phi}{\to}& X' }

we have $X{\times }_{X\prime }X\simeq X$. Write $\Delta :X\to X{\times }_{X\prime }X$ for the diagonal and let

$$X\to \mathrm{im}{\Delta }_{\varphi }\to X{\times }_{X\prime }X$$X \to im \Delta_\phi \to X \times_{X'} X

be its image factorization. Doing the same for $f^{*}\varphi$, which we have seen is a monomorphism, and using that $f^{*}$ preserves the pullback, we get

$$f^{*}\mathrm{im}{\Delta }_{\varphi }\simeq f^{*}(X{\times }_{X\prime }X).$$f^* im \Delta_\phi \simeq f^* (X \times_{X'} X) \,.

Now using again the assumption that $f^{*}$ is injective on subobjects, this implies $\mathrm{im}{\Delta }_{\varphi }=X{\times }_{X\prime }X$ and hence that $\varphi$ is a monomorphism.

(…)

The statement about the comonadic adjunction we discuss below as prop. 2.

Properties

Surjection/embedding factorization

Observation

For $T:ℰ\to ℰ$ a left exact comonad the cofree algebra functor

$$F:ℰ\to T\mathrm{CoAlg}(ℰ)$$F : \mathcal{E} \to T CoAlg(\mathcal{E})

to the topos of coalgebras is a geometric surjection.

Proof

By the discussion at topos of coalgebras the inverse image is the forgetful functor to the underlying $ℰ$-objects. This is clearly a faithful functor.

Proposition

Up to equivalence, every geometric surjection is of this form.

This appears for instance as (MacLaneMoerdijk, VII 4., prop 4).

Proof

With observation 1 we only need to show that if $f:ℰ\to ℱ$ is surjective, then there is $T$ such that

$$\begin{array}{ccc}ℰ& \stackrel{f}{\to }& ℱ\\ & {}_F\searrow & {\downarrow }^{\simeq }\\ & & T\mathrm{CoAlg}(ℰ)\end{array}.$$\array{ \mathcal{E} &\stackrel{f}{\to}& \mathcal{F} \\ & {}_{\mathllap{F}}\searrow & \downarrow^{\mathrlap{\simeq}} \\ && T CoAlg(\mathcal{E}) } \,.

For this, take $T:=f^{*}{f}_{*}$. This is a left exact functor by definition of geometric morphism. By assumption on $f$ and using the equivalent definition of def. 1 we have that $f^{*}$ is a conservative functor. This means that the conditions of the monadicity theorem are met, so so $f^{*}$ is a comonadic functor.

For more on this see geometric surjection/embedding factorization .

Examples

Proposition

For $f:X\to Y$ a continuous function between topological spaces and $(f^{*}⊣f_{*}):\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$ the corresponding geometric morphisms of sheaf toposes, $f$ is a surjection precisely if $(f^{*}⊣f_{*})$ is a surjective geometric morphism.

Related concepts

• geometric morphism

  • geometric surjection

  • geometric embedding

References

Section VII. 4. of

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