In mathematics, a Lawvere–Tierney topology is an analog of a

Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is ca ...

for an arbitrary

topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally, on a site). Topoi behave much like the category of sets and possess a notio ...

, 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 and Myles Tierney.

Definition

If ''E'' is a topos, then a topology on ''E'' is a morphism ''j'' from the

subobject classifier In mathematics, especially in category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object ''X'' in the category correspond to the morphisms from ''X'' to Ω. In typical examples, ...

Ω to Ω such that ''j'' preserves truth (

j \circ \mbox = \mbox

), preserves intersections (

j \circ \wedge = \wedge \circ (j \times j)

), and is idempotent (

j \circ j = j

''j''-closure

Given a subobject

s:S \rightarrowtail A

of an object ''A'' with classifier

\chi_s:A \rightarrow \Omega

, then the composition

j \circ \chi_s

defines another subobject

\bar s:\bar S \rightarrowtail A

of ''A'' such that ''s'' is a subobject of

\bar s

, and

\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:

s \subseteq \bar s

* idempotence:

\bar s \equiv \bar \bar s

* preservation of intersections:

\overline \equiv \bar s \cap \bar w

* preservation of order:

s \subseteq w \Longrightarrow \bar s \subseteq \bar w

* stability under pullback:

\overline \equiv f^(\bar 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''.

References

* * * {{DEFAULTSORT:Lawvere-Tierney topology Topos theory Closure operators