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dense monomorphism
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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

Last revised on May 31, 2022 at 11:57:50.
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