# Span (category theory)

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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.

## Contents

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

## Formal definition

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

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

• If R is a relation between sets X and Y (i.e. a subset of X×Y), then XRY is a span, where the maps are the projection maps ${\displaystyle X\times Y{\overset {\pi _{X}}{\to }}X}$ and ${\displaystyle X\times Y{\overset {\pi _{Y}}{\to }}Y}$.
• Any object yields the trivial span ${\displaystyle A=A=A;}$ formally, the diagram AAA, where the maps are the identity.
• More generally, let ${\displaystyle \phi \colon A\to B}$ be a morphism in some category. There is a trivial span A = AB; formally, the diagram AAB, 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 ${\displaystyle X\leftarrow Y\rightarrow Z,}$ 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

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

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.

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## References

• span in nLab
• Yoneda, Nobuo, On the homology theory of modules. J. Fac. Sci. Univ. Tokyo Sect. I,7 (1954), 193–227.
• Bénabou, Jean, Introduction to Bicategories, Lecture Notes in Mathematics 47, Springer (1967), pp.1-77