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 (and satisfies a small number of other conditions), spans can be considered as morphisms in a category of fractions.

The notion of a span is due to Nobuo Yoneda (1954) and Jean Bénabou (1967).

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 &times; 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.