# 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 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*.

## 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, where the maps are the projection maps and . - Any object yields the trivial span formally, the diagram
*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;*formally, the diagram*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 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 (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 : Λ^{op} → **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 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.