Lawvere–Tierney topology

If E is a topos, then a topology on E is a morphism j from the subobject classifier Ω to Ω such that j preserves truth (${\displaystyle j\circ {\mbox{true}}={\mbox{true}}}$), preserves intersections (${\displaystyle j\circ \wedge =\wedge \circ (j\times j)}$), and is idempotent (${\displaystyle j\circ j=j}$).

Summarize Timeline Fact Check

Given a subobject ${\displaystyle s:S\rightarrowtail A}$ of an object A with classifier ${\displaystyle \chi _{s}:A\rightarrow \Omega }$, then the composition ${\displaystyle j\circ \chi _{s}}$ defines another subobject ${\displaystyle {\bar {s}}:{\bar {S}}\rightarrowtail A}$ of A such that s is a subobject of ${\displaystyle {\bar {s}}}$, and ${\displaystyle {\bar {s}}}$ is said to be the j-closure of s.

Some theorems related to j-closure are (for some subobjects s and w of A):

• inflationary property: ${\displaystyle s\subseteq {\bar {s}}}$
• idempotence: ${\displaystyle {\bar {s}}\equiv {\bar {\bar {s}}}}$
• preservation of intersections: ${\displaystyle {\overline {s\cap w}}\equiv {\bar {s}}\cap {\bar {w}}}$
• preservation of order: ${\displaystyle s\subseteq w\Longrightarrow {\bar {s}}\subseteq {\bar {w}}}$
• stability under pullback: ${\displaystyle {\overline {f^{-1}(s)}}\equiv f^{-1}({\bar {s}})}$.

• Grothendieck topologies on a small category C are essentially the same as Lawvere–Tierney topologies on the topos of presheaves of sets over C.
• Lawvere–Tierney topologies on the effective topos generalize the notion of an oracle in computability theory.^[1]

Summarize Timeline Fact Check

The statement that a morphism ${\displaystyle j:\Omega \to \Omega }$ is a Lawvere–Tierney topology can be expressed purely in the internal language of the elementary topos. Indeed, a rephrasing of the definition is that ${\displaystyle j}$ is a Lawvere–Tierney topology when the following three conditions are satisfied internally:

1. ${\displaystyle j(\top )=\top }$
2. ${\displaystyle \forall \,p:\Omega ,j(j(p))\Leftrightarrow j(p)}$
3. ${\displaystyle \forall \,p\,q:\Omega ,j(p\land q)\Leftrightarrow j(p)\land j(q)}$

An equivalent definition uses the following three conditions instead:^[2]

1. ${\displaystyle \forall \,p\,q:\Omega ,(p\Rightarrow q)\Rightarrow (j(p)\Rightarrow j(q))}$
2. ${\displaystyle \forall \,p:\Omega ,p\Rightarrow j(p)}$
3. ${\displaystyle \forall \,p:\Omega ,j(j(p))\Rightarrow j(p)}$

This reveals that a Lawvere–Tierney topology is the same as a monad on the set of truth values ${\displaystyle \Omega }$ partially ordered by implication, viewed as a posetal category. In particular, another equivalent definition is by the following conditions, which follow the standard convention of defining monads in functional programming by the ret and bind operators:

1. ${\displaystyle \forall \,p:\Omega ,p\Rightarrow j(p)}$
2. ${\displaystyle \forall \,p\,q:\Omega ,(p\Rightarrow j(q))\Rightarrow (j(p)\Rightarrow j(q))}$

The second condition can be informally stated as: to prove a proposition of the form ${\displaystyle j(q)}$, one can without loss of generality replace any hypothesis ${\displaystyle j(p)}$ with just ${\displaystyle p}$.