Span (category theory)

Formal definition

A span is a diagram of type \Lambda = (-1 \leftarrow 0 \rightarrow +1), i.e., a diagram of the form Y \leftarrow X \rightarrow Z.

That is, let Λ be the category (-1 ← 0 → +1). 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 (i.e. a subset of X × Y), then X ← R → Y is a span, where the maps are the projection maps X \times Y \overset{\pi_X}{\to} X and X \times Y \overset{\pi_Y}{\to} Y.
• Any object yields the trivial span A ← A → A, where the maps are the identity.
• More generally, let \phi\colon A \to B 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 X \leftarrow Y \rightarrow Z, where the left morphism is in W, can be considered a generalised morphism (i.e., where one "inverts the weak equivalences"). 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 \Lambda^\text{op} = (-1 \rightarrow 0 \leftarrow +1), i.e., a diagram of the form Y \rightarrow X \leftarrow Z.

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(C) of spans on any category C with finite limits is also dagger compact.

References