# nLab complete Heyting algebra

A complete Heyting algebra is a Heyting algebra which is also a complete lattice; that is, it is a poset with arbitrary limits and colimits, which is also cartesian closed.

By the adjoint functor theorem, one can demonstrate that every frame is a complete Heyting algebra, and vice versa, so far as the underlying poset goes. However, morphisms of frames needn’t preserve exponentials or infinitary meets, as would most naturally be required of morphisms of complete Heyting algebras. Also, when considering large lattices which are only small-complete, then frames and complete Heyting algebras are different objects.

Revised on February 24, 2010 03:35:18 by Toby Bartels (75.88.94.75)