Lawvere–Tierney topology

In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary topos, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a local operator or coverage or topology or geometric modality. They were introduced by  and Myles Tierney.

Definition
If E is a topos, then a topology on E is a morphism j from the subobject classifier Ω to Ω such that j preserves truth ($$j \circ \mbox{true} = \mbox{true}$$), preserves intersections ($$j \circ \wedge = \wedge \circ (j \times j)$$), and is idempotent ($$j \circ j = j$$).

j-closure
Given a subobject $$s:S \rightarrowtail A$$ of an object A with classifier $$\chi_s:A \rightarrow \Omega$$, then the composition $$j \circ \chi_s$$ defines another subobject $$\bar s:\bar S \rightarrowtail A$$ of A such that s is a subobject of $$\bar s$$, and $$\bar s$$ is said to be the j-closure of s.

Some theorems related to j-closure are (for some subobjects s and w of A):
 * inflationary property: $$s \subseteq \bar s$$
 * idempotence: $$\bar s \equiv \bar \bar s$$
 * preservation of intersections: $$\overline{s \cap w} \equiv \bar s \cap \bar w $$
 * preservation of order: $$s \subseteq w \Longrightarrow \bar s \subseteq \bar w$$
 * stability under pullback: $$ \overline{f^{-1}(s)} \equiv f^{-1}(\bar s)$$.

Examples
Grothendieck topologies on a small category C are essentially the same as Lawvere–Tierney topologies on the topos of presheaves of sets over C.