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.

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

Formal definition

A span is a diagram of type i.e., a diagram of the form .

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

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 i.e., a diagram of the form .

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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