An intersection is a meet of subsets or (more generally) subobjects. Dually, a cointersection is a union/join of cosubobjects.

This includes the traditional set-theoretic intersection of subsets of some ambient set, as well as intersections of completely arbitrary sets (which are subsets of the universe) in material set theory.

The nullary intersection of the subsets of $X$ is $X$ itself. A binary intersection is the intersection of two sets, and a finitary intersection is the intersection of finitely many sets. Finitary intersections may be built out of binary and nullary intersections.