# nLab span

Contents

This entry is about the notion of spans/correspondences which generalizes that of relations. For spans in vector spaces or modules, see linear span.

category theory

## Applications

#### 2-Category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## Definition

### In set theory

In set theory, a span or correspondence between sets $A$ and $B$ is a set $C$ with a function $R:C \to A \times B$ to the product set $A \times B$. A span between a set $A$ and $A$ itself is a directed pseudograph, which is used to define categories in set theory.

### In dependent type theory

In dependent type theory, there is a distinction between a span, a multivalued partial function, and a correspondence:

• A span between types $A$ and $B$ is a type $C$ with families of elements $x:C \vdash g(x):A$ and $x:C \vdash h(x):B$

• A multivalued partial function from type $A$ to type $B$ is a type family $x:A \vdash P(x)$ with a family of elements $x:A, p:P(x) \vdash f(x, p):B$

• A correspondence between types $A$ and $B$ is a type family $x:A, y:B \vdash R(x, y)$.

However, from any one of the above structures, one could get the other two structures, provided one has identity types and dependent pair types in the dependent type theory. Given a type family $x:A \vdash P(x)$, let $z:\sum_{x:A} P(x) \vdash \pi_1(z):A$ and $z:\sum_{x:A} P(x) \vdash \pi_2(z):P(\pi_1(z))$ be the dependent pair projections for the dependent pair type $\sum_{x:A} P(x)$.

• From every span one could get a multivalued partial function by defining the type family $x:A \vdash P(x)$ as $P(x) \coloneqq \sum_{y:C} g(y) =_A x$ and the family of elements $x:A, p:P(x) \vdash f(x, p):B$ as $f(x, p) \coloneqq h(\pi_1(x))$.

• From every multivalued partial function one could get a span by defining the type $C$ as $C \coloneqq \sum_{x:A} P(x)$ and the family of elements $x:C \vdash g(x):A$ as $g(x) \coloneqq \pi_1(x)$.

• From every multivalued partial function one could get a correspondence by defining the type family $x:A, y:B \vdash R(x, y)$ as $R(x, y) \coloneqq \sum_{p:P(x)} f(x, p) =_B y$.

• From every correspondence one could get a multivalued partial function by defining the type family $x:A \vdash P(x)$ as $P(x) \coloneqq \sum_{y:B} R(x, y)$, and the family of elements $x:A, p:P(x) \vdash h(x, p):B$ as $h(x, p) \coloneqq \pi_1(p)$

• From every span one could get a correspondence by defining the type family $x:A, y:B \vdash R(x, y)$ as $R(x, y) \coloneqq \sum_{z:C} (g(z) =_A x) \times (h(z) =_B y)$.

• From every correspondence one could get a span by defining the type $C$ as $C \coloneqq \sum_{x:A} \sum_{y:B} R(x, y)$, the family of elements $z:C \vdash g(z):A$ as $g(z) \coloneqq \pi_1(z)$, and the function $z:C \vdash h(z):B$ as $h(z) \coloneqq \pi_1(\pi_2(z))$

Given types $A$, $B$, and $C$ and spans $(D, x:D \vdash g_D(x):A, x:D \vdash h_D(x):B)$ between $A$ and $B$ and $(E, y:E \vdash g_E(y):B, y:E \vdash h_E(y):C)$ between $B$ and $C$, there is a span

$(E \circ D, z:E \circ D \vdash g_{E \circ D}(z):A, z:E \circ D \vdash h_{E \circ D}(z):C)$

defined by

$E \circ D \coloneqq \sum_{x:D} \sum_{y:E} h_D(x) =_B g_E(y)$
$z:E \circ D \vdash g_{E \circ D}(z) \coloneqq g_D(\pi_1(z))$
$z:E \circ D \vdash h_{E \circ D}(z) \coloneqq h_E(\pi_1(\pi_2(\pi_1(z))(z)))$

Given types $A$, $B$, and $C$ and correspondences $x:A, y:B \vdash R(x, y)$ and $y:B, z:C \vdash S(y, z)$, there is a correspondence $x:A, z:C \vdash (S \circ R)(x, z)$ defined by

$(S \circ R)(x, z) \coloneqq \sum_{y:B} R(x, y) \times S(y, z)$

### In category theory

In any category $C$, a span, or roof, or correspondence, from an object $x$ to an object $y$ is a diagram of the form

$\array{ && s \\ & {}^{f}\swarrow && \searrow^{g} \\ x &&&& y }$

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.

### Categories of spans

If the category $C$ has pullbacks, we can compose spans. Namely, given a span from $x$ to $y$ and a span from $y$ to $z$:

$\array{ && s &&&& t \\ & {}^{f}\swarrow && \searrow^{g} & & {}^{h}\swarrow && \searrow^{i} \\ x &&&& y &&&& z }$

we can take a pullback in the middle:

$\array{ &&&& s \times_y t \\& && {}^{p_s}\swarrow && \searrow^{p_t} \\ && s &&&& t \\ & {}^{f}\swarrow && \searrow^{g} & & {}^{h}\swarrow && \searrow^{i} \\ x &&&& y &&&& z }$

and obtain a span from $x$ to $z$:

$\array{ && s \times_y t \\ & {}^{f p_s}\swarrow && \searrow^{i p_t} \\ x &&&& z }$

This way of composing spans lets us define a bicategory Span$(C)$ with:

• objects of $C$ as objects
• spans as morphisms
• maps between spans as 2-morphisms

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 bicategory we obtain a pseudo-double category?.

## Properties

### The 1-category of spans

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 classes 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

### Universal property of the bicategory of spans

We discuss the universal property that characterizes 2-categories of spans.

For $C$ be a category with pullbacks, write

1. $Span_2(C) \coloneqq (Span(C))_{\sim2}$ for the weak 2-category of objects of $C$, spans as morphisms, and maps between spans as 2-morphisms,

2. $\eta_C: C \rightarrow Span_2(C)$ for the functor given by:

Now let

1. $K$ be any bicategory

2. $F, G \,\colon\, C \rightarrow K$ be functors such that every map in $C$ is sent to a map in $K$ possessing a right adjoint and satisfying the Beck-Chevalley Condition for any commutative square in $K$,

3. $\alpha \,\colon\, F \rightarrow G$ be a natural transformation.

Then:

###### Proposition

(universal property of the bicategory of spans)
The following holds:

1. $\eta_C$ is universal among such functors $F$, i.e. $F$ as above factors as $F = \hat{F} \circ \eta_C$ for a functor $\hat{F} \,\colon\, Span_2(C) \rightarrow K$ which is unique up to isomorphism.

2. There exists a unique lax natural transformation: $\hat{\alpha} \,\colon\, \hat{F} \rightarrow \hat{G}$ such that $\hat{\alpha} \eta_C = \alpha$.

3. Let $x, y$ be objects in $C$ and $f: x \rightarrow y$ be a morphism in $C$. If $(\alpha_x, \alpha_y)$ induce a pseudo-map of adjoints $F(f) \dashv (Ff)^* \rightarrow G(f) \dashv (Gf)^*$, then $\hat{\alpha}$ is a pseudonatural transformation

Furthermore, if we denote $Pbk$ as the 2-category of categories with pullbacks, pullback-preserving functors, and equifibered natural transformations and $BiCat$ as the tricategory of bicategories, $Span(-): Pbk \rightarrow BiCat$ is well-defined as a functor.

This is due to Hermida 1999.

### Limits and colimits

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).

### Relation to relations

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.

## Examples

A category of correspondences is a refinement of a category Rel of relations. See there for more.

## References

The $Span(C)$ construction was introduced by Jean Bénabou (as an example of a bicategory) in

• Jean Bénabou, Introduction to Bicategories, Lecture Notes in Mathematics 47, Springer (1967), pp.1-77. (doi)

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 due to

and further discussed in:

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

Last revised on January 11, 2023 at 19:42:09. See the history of this page for a list of all contributions to it.