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Lawvere–Tierney topology

In mathematics, a Lawvere–Tierney topology is an analog of a Grothendieck topology for an arbitrary 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 William Lawvere (1971) and Myles Tierney.

Contents

DefinitionEdit

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 ( ), preserves intersections ( ), and is idempotent ( ).

j-closureEdit

 
Commutative diagrams showing how j-closure operates. Ω and t are the subobject classifier. χs is the characteristic morphism of s as a subobject of A and   is the characteristic morphism of   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   of an object A with classifier  , then the composition   defines another subobject   of A such that s is a subobject of  , and   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:  
  • idempotence:  
  • preservation of intersections:  
  • preservation of order:  
  • stability under pullback:  .

ExamplesEdit

Grothendieck topologies on a small category C are essentially the same as Lawvere–Tierney topologies on the topos of presheaves of sets over C.

ReferencesEdit