Span (category theory)


In category theory, a span, roof or correspondence is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks, spans can be considered as morphisms in a category of fractions.

Formal definition

A span is a diagram of type i.e., a diagram of the form.
That is, let Λ be the category. Then a span in a category C is a functor S : Λ → C. This means that a span consists of three objects X, Y and Z of C and morphisms f : X → Y and g : X → Z: it is two maps with common domain.
The colimit of a span is a pushout.

Examples

A cospan K in a category C is a functor K : Λop → C; equivalently, a contravariant functor from Λ to C. That is, a diagram of type i.e., a diagram of the form.
Thus it consists of three objects X, Y and Z of C and morphisms f : Y → X and g : Z → X: it is two maps with common codomain.
The limit of a cospan is a pullback.
An example of a cospan is a cobordism W between two manifolds M and N, where the two maps are the inclusions into W. Note that while cobordisms are cospans, the category of cobordisms is not a "cospan category": it is not the category of all cospans in "the category of manifolds with inclusions on the boundary", but rather a subcategory thereof, as the requirement that M and N form a partition of the boundary of W is a global constraint.
The category nCob of finite-dimensional cobordisms is a dagger compact category. More generally, the category Span of spans on any category C with finite limits is also dagger compact.