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.
The notion of a span is due to Nobuo Yoneda and Jean Bénabou.

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

  • If R is a relation between sets X and Y, then X ← R → Y is a span, where the maps are the projection maps and.
  • Any object yields the trivial span A ← A → A, where the maps are the identity.
  • More generally, let be a morphism in some category. There is a trivial span A ← A → B, where the left map is the identity on A, and the right map is the given map φ.
  • If M is a model category, with W the set of weak equivalences, then the spans of the form where the left morphism is in W, can be considered a generalised morphism. Note that this is not the usual point of view taken when dealing with model categories.

    Cospans

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.