nLab intersection



Category theory

Limits and colimits



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.

In a finitely complete category, the intersection of two monomorphisms AXA\hookrightarrow X and BXB\hookrightarrow X can be computed by a pullback of the cospan AXBA\to X \leftarrow B. Dually, the cointersection of two epimorphisms is their pushout.

The nullary intersection of the subsets of XX is XX 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.


Last revised on April 4, 2023 at 20:40:35. See the history of this page for a list of all contributions to it.