nLab intersection

Definition

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 $A\hookrightarrow X$ and $B\hookrightarrow X$ can be computed by a pullback of the cospan $A\to X \leftarrow B$. Dually, the cointersection of two epimorphisms is their pushout.

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.

Properties

• interactions of images and pre-images with unions and intersections

  • the image of an intersection is contained in the intersection of the images

  • pre-images preserve intersections,

Related concepts

• product

• union

• intersection product, intersection theory

• brane intersection