nLab surjective geometric morphism

Context

Topos Theory

Could not include topos theory - contents

Contemts

Definition

Definition

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

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 $(f^* faithful) \Leftrightarrow (Id \to f^* f_* is mono)$ is a general property of adjoint functors (see there).

The implication $(f^* faithful) \Rightarrow (f^* induces\;injection\;on\;subobjects)$ 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 $char U : X \to \Omega$ and consider the image

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

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

If now $U \leq \tilde U$ 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

$\array{ \tilde U &\to& * \\ \downarrow && \downarrow \\ X &\stackrel{char U}{\to}& \Omega }$

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

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

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

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

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

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

$\array{ X \times_{X'} X &\to& X \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ X &\stackrel{\phi}{\to}& X' }$

we have $X \times_{X'} X \simeq X$. Write $\Delta : X \to X \times_{X'} X$ for the diagonal and let

$X \to im \Delta_\phi \to X \times_{X'} X$

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

$f^* im \Delta_\phi \simeq f^* (X \times_{X'} X) \,.$

Now using again the assumption that $f^*$ is injective on subobjects, this implies $im \Delta_\phi = X \times_{X'} X$ and hence that $\phi$ is a monomorphism.

(…)

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

Properties

Surjection/embedding factorization

Observation

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

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

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 $\mathcal{E}$-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 $f : \mathcal{E} \to \mathcal{F}$ is surjective, then there is $T$ such that

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

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 $(f^* \dashv f_*) : Sh(X) \to Sh(Y)$ the corresponding geometric morphisms of sheaf toposes, $f$ is a surjection precisely if $(f^* \dashv f_*)$ is a surjective geometric morphism.

References

Section VII. 4. of

Revised on December 4, 2012 17:16:54 by Urs Schreiber (131.174.40.191)