Contents

topos theory

# Contents

## Idea

The concept of dominance is a weakening of the concept of surjectivity for geometric morphisms. It generalizes the concept of a continuous function whose image is a dense subspace of its codomain in topology.

## Definition

A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ is called dominant if the following equivalent conditions hold:

• The direct image satisfies: $f_\ast(\emptyset_\mathcal{F})\cong\emptyset_\mathcal{E}$.

• The inverse image satisfies: from $f^\ast(Z)\cong\emptyset_\mathcal{F}$ follows $Z\cong\emptyset_\mathcal{E}$.

• The inverse image satisfies: from $f^\ast(Z)\cong\emptyset_\mathcal{F}$ follows $Z\cong\emptyset_\mathcal{E}$ for $Z$ a subterminal object.

## Examples

• A geometric morphism $f:Sh(X)\to Sh(Y)$ between the toposes of sheaves on two topological spaces $X, Y$ is dominant iff the corresponding continuous map $f:X\to Y$ has a dense image $f(X)$ in $Y$.

• A geometric embedding $i:\mathcal{E}\hookrightarrow\mathcal{F}$ is dominant precisely when it exhibits $\mathcal{E}$ as a dense subtopos.

## Properties

###### Proposition

If $f:\mathcal{F}\to\mathcal{E}$ is a surjection (i.e. $f^\ast$ is faithful) then $f$ is dominant.

Proof: Suppose $f^\ast(Z)\cong \emptyset_\mathcal{F}$ is initial, then $Hom_\mathcal{F}(\emptyset_\mathcal{F},f^\ast(Y))=Hom_\mathcal{F}(f^\ast(Z),f^\ast(Y))$ is a singleton for all $Y\in\mathcal{E}$ , but by faithfulness of $f^\ast$ this implies that $Hom_\mathcal{E}(Z,Y)$ is a singleton for all $Y\in\mathcal{E}$ , which says that $Z$ is initial in $\mathcal{E}$. $\qed$

###### Proposition

Let $f:\mathcal{F}\to\mathcal{E}$ be a geometric morphism and $i\circ s$ its surjection-inclusion factorization. $f:\mathcal{F}\to\mathcal{E}$ is dominant iff $i:Im(f)\hookrightarrow\mathcal{E}$ is a dense inclusion.

Proof: Suppose $i$ is dense, hence dominant. $s$ as a surjection is dominant as well, and so is their composition $f$.

Conversely, suppose $f:\mathcal{F}\to\mathcal{E}$ is dominant and $i^\ast(Z)\cong \emptyset_{Im(f)}$. Since $s^\ast$ preserves colimits, $\emptyset_\mathcal{F}\cong s^\ast (\emptyset_{Im(f)})\cong s^\ast\circ i^\ast (Z)=f^\ast(Z)$ but $f$ is dominant by assumption, therefore $Z\cong \emptyset_\mathcal{E}$, hence $i$ is dense. $\qed$

The following is a slight generalization of the (dense,closed)-factorization employing dominant geometric morphisms:

###### Proposition

Let $f:\mathcal{F}\to\mathcal{E}$ be a geometric morphism. Then $f$ factors as a dominant geometric morphism $d$ followed by a closed inclusion $c$.

Proof: Let $i\circ d_1$ be the surjection-inclusion factorization of $f$. Since $d_1$ is surjective, it is dominant (cf. above). Then we use the (dense,closed)-factorization to factor $i$ into $c\circ d_2$. Since both $d_i$ are dominant, so is $d:=d_2\circ d_1$ and $c\circ d$ yields the demanded factorization of $f$. $\qed$

## Remark

The concept can be generalized to morphisms in a topos $\mathcal{E}$ by calling $f:X\to Y$ dominant if the induced geometric morphism $f^\ast\dashv\prod_f:\mathcal{E}/X\to\mathcal{E}/Y$ is dominant, where $f^\ast$ denotes the pullback functor.