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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 BB is a subobject whose closure is all of BB.

Definition

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

For every subobject ABA \hookrightarrow B in the topos classified by charA:BΩchar A : B \to \Omega, let its closure

A¯B \bar A \hookrightarrow B

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

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

Relation to other concepts

To local isomorphisms

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

We have in this case:

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

To sheafification

A presheaf FPSh(S)F \in PSh(S) is a sheaf with respect to the given topology if Hom PSh(S)(,F)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)