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 $A↪B$ in the topos classified by $\mathrm{char}A:B\to \Omega$, let its closure

$\overline{A}↪B$\bar A \hookrightarrow B

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

The monomorphism $A↪B$ is called a dense monomorphism if $\overline{A}=B$, that is if $\overline{A}↪B$ is an isomorphism.

## Relation to other concepts

### To local isomorphisms

Recall that when $E$ is a presheaf Grothendieck topos $E=\mathrm{PSh}\left(S\right)=\left[{S}^{\mathrm{op}},\mathrm{Set}\right]$ 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 $\mathrm{PSh}\left(S\right)$ with respect to this topology.

We have in this case:

the dense monomorphisms in $\mathrm{PSh}\left(S\right)$ are precisely the local isomorphisms that are at the same time ordinary monomorphisms.

### To sheafification

A presheaf $F\in \mathrm{PSh}\left(S\right)$ is a sheaf with respect to the given topology if ${\mathrm{Hom}}_{\mathrm{PSh}\left(S\right)}\left(-,F\right)$ 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)