Span (category theory)

Generalization of a notion in 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

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A span is a diagram of type  Λ = ( − 1 <- 0 -> + 1 ) , {\displaystyle \Lambda =(-1\leftarrow 0\rightarrow +1),} i.e., a diagram of the form  Y <- X -> Z {\displaystyle 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

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* 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 -> π X X {\displaystyle X\times Y{\overset {\pi _{X}}{\to }}X} and  X × Y -> π Y Y {\displaystyle 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  ϕ : A -> B {\displaystyle \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 <- Y -> Z , {\displaystyle 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

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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  Λ op = ( − 1 -> 0 <- + 1 ) , {\displaystyle \Lambda ^{\text{op}}=(-1\rightarrow 0\leftarrow +1),} i.e., a diagram of the form  Y -> X <- Z {\displaystyle 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.

## See also

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

## References

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* span at the 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". Reports of the Midwest Category Seminar. Lecture Notes in Mathematics. Vol. 47\. Springer. pp. 1–77\. doi:10.1007/BFb0074299. ISBN 978-3-540-35545-8.