Open local operator ($o(U)(p)=u\Rightarrow p$)

For a subterminal object classified by $u$, the open local operator is $p\mapsto (u\Rightarrow p)$. Its dense monos are exactly those whose image contains the pullback of that subterminal. It is complementary to the corresponding closed local operator.

The open local operator and closed local operator associated with $U$ have meet the identity and join the largest local operator. The meet identity is $(u\Rightarrow p)\wedge (u\vee p)=p$. Every mono factors through its union with $U$ as a closed-operator-dense mono followed by an open-operator-dense mono, proving that their join makes every mono dense.