nLab Span

Definition

$Span$ is the bicategory, or double category, whose

• objects are sets
• (horizontal) morphisms are spans
• 2-cells are morphisms of spans
• (in the double category case) vertical morphisms are functions.

More generally, we have $Span(C)$ consists of spans in $C$ for any category $C$ with pullbacks.

Properties

$Span$ is a sort of classifying object for arbitrary functors; see displayed category.

Categorifications

There is an (infinity,n)-category of correspondences.