topos theory

# Contemts

## Definition

###### Definition

A geometric morphism between toposes $\left({f}^{*}⊣{f}_{*}\right):ℰ\to ℱ$ 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

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

for all $X\in ℱ$;

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

• $\left({f}^{*}⊣{f}_{*}\right)$ 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 $\left({f}^{*}\mathrm{faithful}\right)⇔\left(\mathrm{Id}\to {f}^{*}{f}_{*}\mathrm{is}\mathrm{mono}\right)$ is a general property of adjoint functors (see there).

The implication $\left({f}^{*}\mathrm{faithful}\right)⇒\left({f}^{*}\mathrm{induces}\phantom{\rule{thickmathspace}{0ex}}\mathrm{injection}\phantom{\rule{thickmathspace}{0ex}}\mathrm{on}\phantom{\rule{thickmathspace}{0ex}}\mathrm{subobjects}\right)$ 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 $U↪X$ be a subobject with characteristic morphism $\mathrm{char}U:X\to \Omega$ and consider the image

$\begin{array}{ccc}{f}^{*}U& \to & {f}^{*}*\simeq *\\ ↓& & ↓\\ {f}^{*}X& \stackrel{{f}^{*}\mathrm{char}U}{\to }& {f}^{*}\Omega \end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $U\le \stackrel{˜}{U}$ but ${f}^{*}\left(U\right)={f}^{*}\left(\stackrel{˜}{U}\right)$ then both corresponding pullback diagrams are sent by ${f}^{*}$ to the same such diagram. By faithfulness this implies that also

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

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

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

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

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

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

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

$X\to \mathrm{im}{\Delta }_{\varphi }\to X{×}_{X\prime }X$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

${f}^{*}\mathrm{im}{\Delta }_{\varphi }\simeq {f}^{*}\left(X{×}_{X\prime }X\right)\phantom{\rule{thinmathspace}{0ex}}.$f^* im \Delta_\phi \simeq f^* (X \times_{X'} X) \,.

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

(…)

## Properties

### Surjection/embedding factorization

###### Observation

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

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

$\begin{array}{ccc}ℰ& \stackrel{f}{\to }& ℱ\\ & {}_{F}↘& {↓}^{\simeq }\\ & & T\mathrm{CoAlg}\left(ℰ\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left({f}^{*}⊣{f}_{*}\right):\mathrm{Sh}\left(X\right)\to \mathrm{Sh}\left(Y\right)$ the corresponding geometric morphisms of sheaf toposes, $f$ is a surjection precisely if $\left({f}^{*}⊣{f}_{*}\right)$ 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)