Span (category theory)

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In category theory a span 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.

[edit] Formal definition

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.

[edit] 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.
• Let φ:A → B be a morphism in some category. There is a trivial span A = A → B.
• If M is a model category, with W the set of weak equivalences, then the spans of the form

where the left morphism is in W, can be considered a generalised morphism. Note that this is not the usual point of view taken when dealing with model categories.

[edit] Cospans

A cospan K in a category C is a functor K:Λ^op → C (i.e. a contravariant functor from Λ to C). Thus it consists of three objects X, Y and Z of C and morphisms f:Y → X and g:Z → X.

An example of a cospan is a cobordism N between two manifolds M₁ and M₂. The two maps are the inclusions into N.