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 formula_1 i.e., a diagram of the form formula_2.
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.
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 formula_8 i.e., a diagram of the form formula_9.
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.