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.

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.

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