This entry is about the notion of spans/correspondences which generalizes that of relations. For spans in vector spaces or modules, see linear span.
Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
In any category $C$, a span, or roof, or correspondence, from an object $x$ to an object $y$ is a diagram of the form
where $s$ is some other object of the category. (The word “correspondence” is also sometimes used for a profunctor.)
This diagram is also called a ‘span’ because it looks like a little bridge; ‘roof’ is similar. The term ‘correspondence’ is prevalent in geometry and related areas; it comes about because a correspondence is a generalisation of a binary relation.
Note that a span with $f = 1$ is just a morphism from $x$ to $y$, while a span with $g = 1$ is a morphism from $y$ to $x$. So, a span can be thought of as a generalization of a morphism in which there is no longer any asymmetry between source and target.
A span in the opposite category $C^op$ is called a co-span in $C$.
A span that has a cocone is called a coquadrable span.
If the category $C$ has pullbacks, we can compose spans. Namely, given a span from $x$ to $y$ and a span from $y$ to $z$:
we can take a pullback in the middle:
and obtain a span from $x$ to $z$:
This way of composing spans lets us define a 2-category Span$(C)$ with:
This is a weak 2-category: it has a nontrivial associator: composition of spans is not strictly associative, because pullbacks are defined only up to canonical isomorphism. A naturally defined strict 2-category which is equivalent to $Span(C)$ is the strict 2-category of linear polynomial functors between slice categories of $C$.
(Note that we must choose a specific pullback when defining the composite of a pair of morphisms in $Span(C)$, if we want to obtain a bicategory as traditionally defined; this requires the axiom of choice. Otherwise we obtain a bicategory with ‘composites of morphisms defined only up to canonical iso-2-morphism’; such a structure can be modeled by an anabicategory or an opetopic bicategory?.)
By including functions as well, instead of a 2-category we obtain a double category.
Let $C$ be a category with pullbacks and let $Span_1(C) := (Span(C))_{\sim 1}$ be the 1-category of objects of $C$ and isomorphism class of spans between them as morphisms.
Then
Next assume that $C$ is a cartesian monoidal category. Then clearly $Span_1(C)$ naturally becomes a monoidal category itself, but more: then
(Dawson-Paré-Pronk 04) (…)
Since a category of spans/correspondences $Corr(\mathcal{C})$ is evidently equivalent to its opposite category, it follows that to the extent that limits exists they are also colimits and vice versa.
If the underlying category $\mathcal{C}$ is an extensive category, then the coproduct/product in $Corr(\mathcal{C})$ is given by the disjoint union in $\mathcal{C}$. (See also this MO discussion).
More generally, every van Kampen colimit in $\mathcal{C}$ is a (co)limit in $Corr(\mathcal{C})$ — and conversely, this property characterizes van Kampen colimits. (Sobocinski-Heindel 11).
Correspondences may be seen as generalizations of relations. A relation is a correspondence which is (-1)-truncated as a morphism into the cartesian product. See at relation and at Rel for more on this.
Spans in FinSet behave like the categorification of matrices with entries in the natural numbers: for $X_1 \leftarrow N \to X_2$ a span of finite sets, the cardinality of the fiber $X_{x_1, x_2}$ over any two elements $x_1 \in X_1$ and $x_2 \in X_2$ plays the role of the corresponding matrix entry. Under this identification composition of spans indeed corresponds to matrix multiplication. This implies that the category of spans of finite sets is equivalent to the Lawvere theory of commutative monoids, that is, to the category of finitely generated free commutative monoids.
The Burnside category is essentially the category of correspondences in G-sets for $G$ a finite group.
A cobordism $\Sigma$ from $\Sigma_{in}$ to $\Sigma_{out}$ is an example of a cospan $\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out}$ in the category of smooth manifolds. However, composition of cobordisms is not quite the pushpout-composition of these cospans: to make the composition be a smooth manifold again some extra technical aspects must be added (“collars”).
In prequantum field theory (see there for details), spans of stacks model trajectories of fields.
The category of Chow motives has as morphisms equivalence classes of linear combinations of spans of smooth projective varieties.
The Weinstein symplectic category has as morphisms Lagrangian correspondences between symplectic manifolds.
More generally symplectic dual pairs are correspondences between Poisson manifolds.
Cospans of homomorphisms of C*-algebras represent morphisms in KK-theory (by Cuntz’ result).
Correspondences of flag manifolds play a role as twistor correspondences, see at Schubert calculus – Correspondences and at horocycle correspondence
The Fourier-Mukai transform is a pull-push operation through correspondences equipped with objects in a cocycle given by an object in a derived category of quasi-coherent sheaves.
A hypergraph is a span from a set of vertices to a set of (hyper)edges.
A category of correspondences is a refinement of a category Rel of relations. See there for more.
The $Span(C)$ construction was introduced by Jean Bénabou (as an example of a bicategory) in
Bénabou cites an article by Yoneda (1954) for introducing the concept of span (in the category of categories).
An exposition discussing the role of spans in quantum theory:
The relationship between spans and bimodules is briefly discussed in
The relation to van Kampen colimits is discussed in
The universal property of categories of spans is discussed in
R. Dawson, Robert Paré, Dorette Pronk, Universal properties of Span, Theory and Appl. of Categories 13, 2004, No. 4, 61-85, TAC, MR2005m:18002
R. Dawson, Robert Paré, Dorette Pronk, The span construction, Theory Appl. Categ. 24 (2010), No. 13, 302–377, TAC MR2720187
The structure of a monoidal tricategory on spans in 2-categories is discussed in
Generally, an (∞,n)-category of spans is indicated in section 3.2 of