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.

Formal definition

A span is a diagram of type $\Lambda =(-1\leftarrow 0\rightarrow +1),$  i.e., a diagram of the form $Y\leftarrow X\rightarrow Z$ .

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.

Examples

• If R is a relation between sets X and Y (i.e. a subset of X × Y), then XRY is a span, where the maps are the projection maps $X\times Y{\overset {\pi _{X}}{\to }}X$  and $X\times Y{\overset {\pi _{Y}}{\to }}Y$ .
• Any object yields the trivial span $A=A=A;$  formally, the diagram AAA, where the maps are the identity.
• More generally, let $\phi \colon A\to B$  be a morphism in some category. There is a trivial span A = AB; formally, the diagram AAB, where the left map is the identity on A, and the right map is the given map φ.
• If M is a model category, with W the set of weak equivalences, then the spans of the form $X\leftarrow Y\rightarrow Z,$  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.

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 $\Lambda ^{\text{op}}=(-1\rightarrow 0\leftarrow +1),$  i.e., a diagram of the form $Y\rightarrow X\leftarrow Z$ .

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.

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.