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).

## Contents

* 1 Formal definition
* 2 Examples
* 3 Cospans
* 4 See also
* 5 References

## Formal definition

A span is a diagram of type Λ=(−1<-0->+1), i.e., a diagram of the form Y<-X->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×Y->πXX and X×Y->πYY.
* Any object yields the trivial span A ← A → A, where the maps are the identity.
* More generally, let ϕ:A->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<-Y->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 Λop=(−1->0<-+1), i.e., a diagram of the form Y->X<-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.

## See also

* Binary relation
* Pullback (category theory)
* Pushout (category theory)
* Cobordism

## References

* span in nLab
* Yoneda, Nobuo (1954). "On the homology theory of modules". J. Fac. Sci. Univ. Tokyo Sect. I 7: 193–227.
* Bénabou, Jean (1967). "Introduction to Bicategories". 47. Springer. pp. 1–77. doi:10.1007/BFb0074299. ISBN 978-3-540-35545-8.

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