topos theory

# Contents

## Idea

A Lawvere-Tierney topology on a topos defines naturally a certain closure operation on subobjects. A subobject inclusion is called dense (a dense monomorphism) if its closure is an isomorphism. In other words, a dense subobject of an object $B$ is a subobject whose closure is all of $B$.

## Definition

Let $E$ be a topos equipped with a Lawvere-Tierney topology $j : \Omega \to \Omega$.

For every subobject $A \hookrightarrow B$ in the topos classified by $char A : B \to \Omega$, let its closure

$\bar A \hookrightarrow B$

be the subobject classified by $char \bar A := B \stackrel{char A}{\to} \Omega \stackrel{j}{\to} \Omega$.

The monomorphism $A \hookrightarrow B$ is called a dense monomorphism if $\bar A = B$, that is if $\bar A \hookrightarrow B$ is an isomorphism.

## Relation to other concepts

### To local isomorphisms

Recall that when $E$ is a presheaf Grothendieck topos $E = PSh(S) = [S^{op}, Set]$ then Lawvere-Tierney topologies on $E$ are in bijection with Grothendieck topologies on $S$ (making $S$ a site). In this case there is the notion of local epimorphism and local isomorphism in $PSh(S)$ with respect to this topology.

We have in this case:

the dense monomorphisms in $PSh(S)$ are precisely the local isomorphisms that are at the same time ordinary monomorphisms.

### To sheafification

A presheaf $F \in PSh(S)$ is a sheaf with respect to the given topology if $Hom_{PSh(S)}(-, F)$ sends all dense monomorphisms to isomorphisms.

Since Lawvere-Tierney topologies make sense for every topos (not necessarily a presheaf Grothendieck topos) this provides a general notion of sheafification in a Lawvere-Tierney topology.

## References

Dense monomorphisms appear around p. 223 of

Revised on November 18, 2011 16:54:59 by Urs Schreiber (131.220.133.61)