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.

The notion of a span is due to Nobuo Yoneda (1954) and Jean Bénabou (1967).

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 ''X'' ← ''R'' → ''Y'' is a span, where the maps are the projection maps $X \times Y \overset X$ and $X \times Y \overset Y$.
* Any object yields the trivial span ''A'' ← ''A'' → ''A,'' 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'' ← ''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 $X \leftarrow Y \rightarrow Z,$ 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 : Λ^op → C; equivalently, a ''contravariant'' functor from Λ to C. That is, a diagram of type $\Lambda^\text = (-1 \rightarrow 0 \leftarrow +1),$ 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.

See also

* Binary relation
* Pullback (category theory)
* Pushout (category theory)
* Cobordism

References

* {{nlab, id=span
*Yoneda, Nobuo, On the homology theory of modules. ''J. Fac. Sci. Univ. Tokyo Sect. I'',7 (1954), 193–227.
*Bénabou, Jean, Introduction to Bicategories, Lecture Notes in Mathematics 47, Springer (1967), pp.1-77

Functors