A Boolean category is a coherent category (such as a topos or pretopos) in which every subobject has a complement, i.e., for any monomorphism there is a monomorphism such that is initial and . Therefore, the subobject lattice of any object is a Boolean algebra.
Any Boolean category is, in particular, a Heyting category and therefore supports a full first-order internal logic. However, unlike that of an arbitrary Heyting category, the internal logic of a Boolean category satisfies the principle of excluded middle; it is first-order classical logic.
In addition, every Boolean category is a first order Boolean hyperdoctrine given by the subobject poset functor .
Last revised on November 16, 2022 at 19:56:35. See the history of this page for a list of all contributions to it.