nLab intersection

Idea

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 which can be constructed as arbitrary sets (which are subsets of the universe) in material set theory.

Definition

In material set theory

In material set theory such as ZFC, the intersection of two sets $A$ and $B$ is

$$A\cap B = \{ x \mid x\in A \;\text{ and }\; x\in B \}.$$

This makes sense for any two sets, but in practice it is usually used when $A$ and $B$ are both subsets of some ambient set $X$.

In category theory

In a finitely complete category, the intersection of two monomorphisms $A\hookrightarrow X$ and $B\hookrightarrow X$ is the pullback of the cospan $A\to X \leftarrow B$. This is, in particular, also a meet in the poset of subobjects of $X$.

In a category that lacks all pullbacks, there is some question as to whether an intersection of subobjects should be defined as a pullback or as a meet in the poset of subobjects; the former implies the latter but not in general conversely.

Dually, the cointersection of two epimorphisms is their pushout.

Other arities

The nullary intersection of the subsets of $X$ is $X$ itself. Note that this does not make sense in material set theory until we fix an ambient set, since “the universe” is not a set.

A binary intersection is the intersection of two sets or subobjects, as defined above, and a finitary intersection is the intersection of finitely many sets or subobjects. By induction, intersections may be built out of binary and nullary intersections.

One can also define infinitary intersections, which can be constructed categorically as wide pullbacks.

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