nLab
Boolean category

A Boolean category is a coherent category (such as a topos) in which every subobject has a complement, i.e., for any monic $A\hookrightarrow X$ there is a monic $B\hookrightarrow X$ such that $A\cap B$ is initial and $A\cup B=X$. Therefore, the lattice $\mathrm{Sub}(X)$ of subobjects 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.