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).

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.

A cospan K in a category C is a functor K : Λ^op → 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.