Spahn Lawvere-Tierney operator

The closure operator

Definition

A Lawvere–Tierney topology in $E$ is (internally) a closure operator given by a left exact idempotent monad on the internal meet-semilattice $\Omega$.

This means that: a Lawvere–Tierney topology in $E$ is a morphism

$$j: \Omega \to \Omega$$

such that

1. $j true = true$, equivalently $\id_\Omega \leq j: \Omega \to \Omega$ (‘if $p$ is true, then $p$ is locally true’)

   $$\array{ * &\stackrel{true}{\to}& \Omega \\ & {}_{\mathllap{true}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }$$

2. $j j = j$ (‘$p$ is locally locally true iff $p$ is locally true’);

   $$\array{ \Omega &\stackrel{j}{\to}& \Omega \\ & {}_{\mathllap{j}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega }$$

3. $j \circ \wedge = \wedge \circ (j \times j)$ (‘$p \wedge q$ is locally true iff $p$ and $q$ are each locally true’)

   $$\array{ \Omega \times \Omega &\stackrel{\wedge}{\to}& \Omega \\ {}^{\mathllap{j \times j}}\downarrow && \downarrow^{\mathrlap{j}} \\ \Omega \times \Omega &\underset{\wedge}{\to}& \Omega } \,.$$

Here $\leq$ is the internal partial order on $\Omega$, and $\wedge: \Omega \times \Omega \to \Omega$ is the internal meet.

This appears for instance as (MacLaneMoerdijk, V 1.).