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 $A\hookrightarrow X$ there is a monomorphism $B\hookrightarrow X$ such that $A\cap B$ is initial and $A\cup B = X$. Therefore, the subobject lattice $Sub(X)$ of any object $X$ 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 $\mathcal{C}$ is a first order Boolean hyperdoctrine given by the subobject poset functor $\mathrm{sub}:\mathcal{C}^{op} \to BoolAlg$.

Last revised on November 16, 2022 at 19:56:35. See the history of this page for a list of all contributions to it.