nLab
intersection

Context

Category theory

Limits and colimits

Contents

Definition

An intersection is a meet of subsets or (more generally) subobjects.

The dual notion is that of union/join.

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.

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.

Properties

Revised on May 20, 2017 13:21:28 by Urs Schreiber (92.218.150.85)