nLab (0,1)-category theory - contents

(0,1)-category theory: logic, order theory

(0,1)-category

• relation between preorders and (0,1)-categories

• preordered set

  • presemilattice

  • prelattice

  • total preorder

  • Heyting prealgebra

  • opposite preorder

• partially ordered set (directed set, total order, linear order)

  • top, true,

  • bottom, false

  • monotone function

  • implication

  • filter, interval

  • opposite poset

• lattice, semilattice

  • meet, logical conjunction, and

  • join, logical disjunction, or

  • compact element

• lattice of subobjects

• complete lattice, algebraic lattice

• distributive lattice, completely distributive lattice, canonical extension

• hyperdoctrine

  • first-order, Boolean, coherent, tripos

(0,1)-topos

• Heyting algebra

  • regular element

  • Boolean algebra

• frame, locale

Theorems

• Stone duality