nLab
Heyting category

Contents

Definition

A Heyting category (called a logos in Freyd-Scedrov) is a coherent category in which each base change functor f *:Sub(Y)Sub(X)f^*:Sub(Y)\to Sub(X) has a right adjoint f\forall_f (the universal quantifier, in addition to the left adjoint existential quantifier that exists in any regular category).

It follows that each poset of subobjects Sub(X)Sub(X) is a Heyting algebra and the base-change functors f *f^* are Heyting algebra homomorphisms. The Heyting implication can be defined as follows: if AA and BB are subobjects of XX, where the subobject monomorphism from AA to XX is mm, (AB):= m(AB)(A\Rightarrow B) := \forall_m(A\wedge B), where ABA\wedge B is considered as a subobject of AA. Any Heyting category has an internal logic which is full (typed) first-order intuitionistic logic. The extra right adjoint f\forall_f provided by the above axiom gives the universal quantifier in this logic.

A Heyting functor is a coherent functor betwen Heyting categories which additionally preserves the right adjoints f\forall_f. Such functors also preserve the internal interpretation of first-order logic.

Examples

Any topos, and in fact any well-powered infinitary coherent category, is a Heyting category. Any Boolean category is also a Heyting category.

References

Revised on October 10, 2013 07:11:21 by Urs Schreiber (82.113.121.201)