Boolean category (Rev #4, changes) in nLab

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A Boolean category is a coherent category (such as a topos) in which every subobject has a complement, i.e., for any monic A↪XA\hookrightarrow X there is a monic B↪XB\hookrightarrow X such that A∩BA\cap B is initial and A∪B=XA\cup B = X. Therefore, the lattice Sub(X)Sub(X) of subobjects of any object XX 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.

Many-valued logics

Many-valued logics can be reduced to Boolean logic, ie. any coherent category is isomorphic to the Principle of Bivalence.