Lawvere–Tierney topology

Remarks

Example: the identity $\id_\Omega$ satisfies all three equations. Example: the double negation $\neg\neg \colon \Omega \to \Omega$ is a Lawvere–Tierney topology in every topos. Non-example: in $\mathbf{Set}$, with $\Omega = \set{0, 1}$ and $\top = 1$, the negation $\neg$ sends $1$ to $0$, so $\neg \circ \top \ne \top$ and clause 1 fails. An object $F$ is a $j$-sheaf when, for every monomorphism $m \colon A \to B$ whose characteristic morphism satisfies $j \circ \chi_m = \top \circ {!}$, where ${!} \colon B \to 1$ is the unique morphism, every morphism $A \to F$ extends uniquely along $m$. The $j$-sheaves form a topos $\mathrm{Sh}_j(\mathcal{E})$, and the inclusion has a left adjoint that preserves finite limits. Another name for a Lawvere–Tierney topology is a local operator.