Span (category theory)

In , a span, roof or correspondence is a generalization of the notion of relation between two of a . When the category has all (and satisfies a small number of other conditions), spans can be considered as morphisms in a . The notion of a span is due to Nobuo Yoneda (1954) and Jean Bénabou (1967). A span is a 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 . If R is a relation between sets X and Y (i.e. a subset of X × Y), then X ← R → Y is a span, where the maps are the projection maps and . Any object yields the trivial span A ← A → A, where the maps are the identity. More generally, let be a morphism in some category. There is a trivial span A ← A → B, where the left map is the identity on A, and the right map is the given map φ. If M is a , 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 (i.e., where one "inverts the weak equivalences"). Note that this is not the usual point of view taken when dealing with model categories. 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 of a cospan is a . 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 thereof, as the requirement that M and N form a partition of the boundary of W is a global constraint.