nLab
Boolean category

Contents

Contents

Definition

A Boolean category is a coherent category (such as a topos) in which every subobject has a complement, i.e., for any monomorphism AXA\hookrightarrow X there is a monomorphism BXB\hookrightarrow X such that ABA\cap B is initial and AB=XA\cup B = X. Therefore, the subobject lattice Sub(X)Sub(X) of any object XX is a Boolean algebra.

Properties

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.

Last revised on June 5, 2019 at 16:15:01. See the history of this page for a list of all contributions to it.