### The closure operator +-- {: .num_defn #LTTopologyDef} ###### Definition A __Lawvere--Tierney topology__ in $E$ is ([[nLab:internalization|internally]]) a [[nLab:closure operator]] given by a [[nLab:exact functor|left exact]] [[nLab:idempotent monad]] on the internal meet-[[nLab:semilattice]] $\Omega$. This means that: a Lawvere--Tierney topology in $E$ is a [[nLab: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') $$ \array{ * &\stackrel{true}{\to}& \Omega \\ & {}_{\mathllap{true}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega } $$ 1. $j j = j$ (‘$p$ is locally locally true iff $p$ is locally true'); $$ \array{ \Omega &\stackrel{j}{\to}& \Omega \\ & {}_{\mathllap{j}}\searrow & \downarrow^{\mathrlap{j}} \\ && \Omega } $$ 1. $j \circ \wedge = \wedge \circ (j \times j)$ (‘$p \wedge q$ 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 [[nLab:partial order]] on $\Omega$, and $\wedge: \Omega \times \Omega \to \Omega$ is the internal [[nLab:meet]]. This appears for instance as ([MacLaneMoerdijk, V 1.](#MacLaneMoerdijk)).