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 William Lawvere (1971) and Myles Tierney.

Definition[edit]

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[edit]

Commutative diagrams showing how j-closure operates. Ω and t are the subobject classifier. χ_s is the characteristic morphism of s as a subobject of A and $j\circ \chi _{s}$ is the characteristic morphism of ${\bar {s}}$ which is the j-closure of s. The bottom two squares are pullback squares and they are contained in the top diagram as well: the first one as a trapezoid and the second one as a two-square rectangle.

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: ${\overline {f^{-1}(s)}}\equiv f^{-1}({\bar {s}})$ .