Lawvere-Tierney topology

A Lawvere-Tierney topology on a topos is a morphism $j:\Omega \to \Omega$ satisfying, internally,It is also called a local operator.

The pullback of the truth point along a local operator $j$ is the subobject $J\hookrightarrow \Omega$. A monomorphism with characteristic map $\chi$ is j-dense exactly when $\chi$ factors through $J$. Its internal order is inherited from the Heyting algebra of truth values. Having a least global element is an internal condition over all parameter objects, not just a condition on global truth values.

For a subterminal truth value $u$, the quasi-closed local operator is relative double negation $q(U)(p)=((p\Rightarrow u)\Rightarrow u)$. It has bottom value $q(U)(0)=u$. Its fixed truth values are Boolean with relative bottom $u$, inherited meet, join closed by $q(U)$ and complement $p\Rightarrow u$. Therefore its sheaf topos is a Boolean topos.

In $\mathcal{E}/\Omega$, the generic truth mono $\top :1\hookrightarrow \Omega$ defines a quasi-closed Boolean subtopos. Its composite to $\mathcal{E}$ is surjective. If a predicate $p(x)$ pulled back from $\mathcal{E}$ becomes dense, then $((p(x)\Rightarrow u)\Rightarrow u)=1$ for the generic truth $u$. Substitution $u=p(x)$ yields $p(x)=1$, proving reflection of invertible monos and hence faithfulness.

For a subterminal object classified by $u$, the closed local operator is $p\mapsto u\vee p$. Its closed monos contain the pullback of that subterminal, hence are exactly the monos dense for the corresponding open local operator.

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.

For $j=o(U)$, the dense monomorphism classifier consists of the truth values above $u$, so has least element $u$. Conversely, if it has least element $u$, density of a monomorphism is exactly containment of the pulled-back subterminal $U$. For a subobject $S\subseteq X$, the mono $S\to \overline{S}$ is dense, so $U\cap \overline{S}\subseteq S$ and $j(p)\le u\Rightarrow p$. Since $j(u)=\top$, applying $j$ to $u\wedge (u\Rightarrow p)\le p$ gives $u\Rightarrow p\le j(p)$. Hence equality holds.

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.

If a subobject $A^{\prime }\hookrightarrow A$ has characteristic map $\chi :A\to \Omega$, its $j$-closure is classified by $j\chi$. This closure operation is inflationary, idempotent, and stable under pullback.

A mono is j-closed if it equals its closure under the local operator $j$. Its characteristic map factors through the closed-subobject classifier ${\Omega }_j$. A j-closed subobject of a j-sheaf is a j-sheaf; a mono that is both closed and j-dense is an isomorphism.

A monomorphism is $j$-dense when its $j$-closure is its whole codomain. It is $j$-closed when it equals its closure. Dense monomorphisms are stable under pullback, and a monomorphism that is both dense and closed is an isomorphism.

An object $X$ is a $j$-sheaf when every map $A^{\prime }\to X$ defined on the domain of a j-dense monomorphism $A^{\prime }\hookrightarrow A$ extends uniquely to a map $A\to X$.

An object is j-separated when two maps into it which agree on a j-dense monomorphism are equal. Equivalently its diagonal is j-closed. Quotienting an arbitrary object by the j-closure of its diagonal gives its separated reflection, used in constructing the sheaf reflector for a local operator.

The closed-subobject classifier is the equalizerIt classifies $j$-closed subobjects and is itself a j-sheaf. The local operator factors as a split epimorphism $q:\Omega \twoheadrightarrow {\Omega }_j$ followed by the canonical inclusion $i:{\Omega }_j\hookrightarrow \Omega$.

The full subcategory of $j$-sheaves is reflective. Its reflector $L$ sends every $j$-dense monomorphism to an isomorphism and preserves finite limits.

For a local operator $j$, the following are equivalent: $L$ preserves the subobject classifier; $L(q):L\Omega \to {\Omega }_j$ is an isomorphism; the canonical inclusion $i:{\Omega }_j\hookrightarrow \Omega$ is $j$-dense; and every monomorphism $A^{\prime }\hookrightarrow A$ factors as a $j$-closed mono $A^{\prime }\hookrightarrow A^{\prime \prime }$ followed by a $j$-dense mono $A^{\prime \prime }\hookrightarrow A$.