Lawvere–Tierney topology

Template:Short description In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary elementary topos, used to construct a topos of sheaves. A Lawvere–Tierney topology is also sometimes also called a local operator or coverage or topology or geometric modality. They were introduced by Template:Harvs and Myles Tierney.

Definition

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 ($j\circ \text{true}=\text{true}$), preserves intersections ($j\circ \wedge =\wedge \circ (j\times j)$), and is idempotent ($j\circ j=j$).

j-closure

Commutative diagrams showing how j-closure operates. Ω and t are the subobject classifier. χ_s is the characteristic morphism of s as a subobject of A and $j\circ {\chi }_s$ is the characteristic morphism of $\overline{s}$ which is the j-closure of s. The bottom two squares are pullback squares and they are contained in the top diagram as well: the first one as a trapezoid and the second one as a two-square rectangle.

Given a subobject $s:S\rightarrowtail A$ of an object A with classifier ${\chi }_s:A\to \mathrm{\Omega }$, then the composition $j\circ {\chi }_s$ defines another subobject $\overline{s}:\overline{S}\rightarrowtail A$ of A such that s is a subobject of $\overline{s}$, and $\overline{s}$ 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: $s\subseteq \overline{s}$
• idempotence: $\overline{s}\equiv \overline{\overline{s}}$
• preservation of intersections: $\overline{s\cap w}\equiv \overline{s}\cap \overline{w}$
• preservation of order: $s\subseteq w⟹\overline{s}\subseteq \overline{w}$
• stability under pullback: $\overline{f^{-1}(s)}\equiv f^{-1}(\overline{s})$.

Examples

• 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.Template:R

Internal point of view

The statement that a morphism $j:\mathrm{\Omega }\to \mathrm{\Omega }$ 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 $j$ is a Lawvere–Tierney topology when the following three conditions are satisfied internally:

1. $j(\mathrm{\top })=\mathrm{\top }$
2. $\mathrm{\forall }p:\mathrm{\Omega },j(j(p))\iff j(p)$
3. $\mathrm{\forall }pq:\mathrm{\Omega },j(p\wedge q)\iff j(p)\wedge j(q)$

An equivalent definition uses the following three conditions instead:Template:R

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

This reveals that a Lawvere–Tierney topology is the same as a monad on the set of truth values $\mathrm{\Omega }$ 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. $\mathrm{\forall }p:\mathrm{\Omega },p\Rightarrow j(p)$
2. $\mathrm{\forall }pq:\mathrm{\Omega },(p\Rightarrow j(q))\Rightarrow (j(p)\Rightarrow j(q))$

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

References

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