nLab Boolean category

Definition

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.

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.

In addition, every Boolean category $\mathcal{C}$ is a first order Boolean hyperdoctrine given by the subobject poset functor $\mathrm{sub}:\mathcal{C}^{op} \to BoolAlg$.

Related entries

• pretopos
• De Morgan Heyting category
• positive category

category	functor	internal logic	theory	hyperdoctrine	subobject poset	coverage	classifying topos
finitely complete category	cartesian functor	cartesian logic	essentially algebraic theory
lextensive category		disjunctive logic
regular category	regular functor	regular logic	regular theory	regular hyperdoctrine	infimum-semilattice	regular coverage	regular topos
coherent category	coherent functor	coherent logic	coherent theory	coherent hyperdoctrine	distributive lattice	coherent coverage	coherent topos
geometric category	geometric functor	geometric logic	geometric theory	geometric hyperdoctrine	frame	geometric coverage	Grothendieck topos
Heyting category	Heyting functor	intuitionistic first-order logic	intuitionistic first-order theory	first-order hyperdoctrine	Heyting algebra
De Morgan Heyting category		intuitionistic first-order logic with weak excluded middle			De Morgan Heyting algebra
Boolean category		classical first-order logic	classical first-order theory	Boolean hyperdoctrine	Boolean algebra
star-autonomous category		multiplicative classical linear logic
symmetric monoidal closed category		multiplicative intuitionistic linear logic
cartesian monoidal category		fragment $\{\&, \top\}$ of linear logic
cocartesian monoidal category		fragment $\{\oplus, 0\}$ of linear logic
cartesian closed category		simply typed lambda calculus