nLab cospan

Definition

In category theory

In any category, a cospan is a diagram like this:

$$\array{ && a &&&& b \\ & && {}_{f}\searrow & & \swarrow_g && \\ &&&& c &&&& }$$

A cospan in the category $C$ is the same as a span in the opposite category $C^{op}$. So, all general facts about cospans in $C$ are general facts about spans in $C^{op}$, and the reader may turn to the entry on spans to learn more.

A cospan that admits a cone is called a quadrable cospan.

In synthetic $(\infty,1)$-category theory

In simplicial type theory, binary parametric observational type theory, and other type theoretic approaches to synthetic $(\infty,1)$-category theory, a cospan in a synthetic $(\infty,1)$-category $C$ can be defined as the following:

• a record consisting of objects (elements of types) $x:C$, $y:C$, and $z:C$ and morphisms (elements of hom types) $g:\mathrm{hom}_C(y, z)$ and $h:\mathrm{hom}_C(x, z)$

• a function $k:\Lambda_0^2 \to C$ from the (2,0)-horn type $\Lambda_0^2$ to $C$, where $\Lambda_0^2$ is defined in terms of the directed interval as the type of pairs of elements $i, j:\mathbb{I}$ such that $i = 1$ or $j = 1$.

$$\Lambda_0^2 \coloneqq \sum_{i:\mathbb{I}} \sum_{j:\mathbb{I}} (i = 1) \vee (j = 1)$$

The walking cospan

The walking cospan or (2,0)-horn category is the category $\Lambda_0^2$ which consists of three objects $0, 1, 2 \in \Lambda_0^2$ and two morphisms $h_{02}:\mathrm{hom}_{\Lambda_0^2}(0, 2)$ and $h_{12}:\mathrm{hom}_{\Lambda_0^2}(1, 2)$.

The category $\Lambda_0^2$ is also called the (2,0)-horn preorder or the (2,0)-horn poset because $\Lambda_0^2$ can be shown to be a poset.

Every cospan in a category $C$ is represented by a functor $F:\Lambda_0^2 \to C$ from the walking cospan to $C$.

In type theoretic approaches to synthetic $(\infty,1)$-category theory, the (2,0)-horn type $\Lambda_0^2$ defined above plays the role of the walking cospan.

Properties

Right quotients

Following the notion of a right quotient of two elements in a monoid, we can define in category the notion of a right quotient of a cospan in a category.

A right quotient of a cospan $g:\mathrm{hom}_A(y, z)$ and $h:\mathrm{hom}_A(x, z)$ is a morphism $f:\mathrm{hom}_A(x, y)$ such that $h$ is the unique composite of $f$ and $g$, $g \circ f = h$.

Given a morphism $f:\mathrm{hom}_A(x, y)$, the cospan $(f, \mathrm{id}_y)$ is always right divisible with right quotient $f$. A section is a right quotient of the span $(\mathrm{id}_y, f)$.

The bicategory of cospans

Cospans in a category $V$ with small colimits form a bicategory whose objects are the objects of $V$, whose morphisms are cospans between two objects, and whose 2-morphisms $\eta$ are commuting diagrams of the form

$$\array{ && S \\ & {}^{\sigma_{S}}\nearrow && \nwarrow^{\tau_S} \\ a &&\downarrow^\eta&& b \\ & {}_{\sigma_T}\searrow && \swarrow_{\tau_T} \\ && T } \,.$$

The category of cospans from $a$ to $b$ is naturally a category enriched in $V$: for

$$\array{ && S \\ & {}^{\sigma_{S}}\nearrow && \nwarrow^{\tau_S} \\ a &&&& b \\ & {}_{\sigma_T}\searrow && \swarrow_{\tau_T} \\ && T }$$

two parallel cospans in $V$, the $V$-object ${}_a[S,T]_b$ of morphisms between them is the pullback

$$\array{ {}_a[S,T]_b &\to& pt \\ \downarrow && \downarrow^{\sigma_T \times \tau_T} \\ [S,T] &\stackrel{\sigma_S^* \times \sigma_T^*}{\to}& [a \sqcup b, T] }$$

formed in analogy to the enriched hom of pointed objects. The initial object of $V$ is the coproduct of $a$ and $b$.

If $V$ has a terminal object, $pt$, then cospans from $pt$ to itself are bi-pointed objects in $V$.

Related concepts

• cospan cotrace

• multi-cospan

• Cospans in Algebraic Topology

• multispan

• the cograph of a function

• the cograph of a functor

• sink, cosink

• product, coproduct

• horn

• span, cospan

• composable pair

• composite

• commutative triangle

• isomorphism

• walking structure

  • the boolean domain $2 \coloneqq \partial \Delta^1$; i.e. the walking pair of objects

  • the directed interval category $I \coloneqq \Delta^1$; i.e. the walking morphism

  • the (2,0)-horn category $\Lambda_0^2$; i.e. the walking cospan

  • the (2,1)-horn category $\Lambda_1^2$; i.e. the walking composable pair

  • the (2,2)-horn category $\Lambda_2^2$; i.e. the walking span

  • the 2-simplex category $\Delta^2$; i.e. the walking commutative triangle

  • the walking parallel pair

  • the walking isomorphism

  • Adj; i.e. the walking adjunction

  • the walking equivalence

  • the walking adjoint equivalence

  • the walking 2-isomorphism with trivial boundary

  • the syntactic category of a theory $T$; i.e. the walking $T$-model

• In dependent type theory, many higher inductive types can be considered to be walking structures:

  • the unit type is the walking element

  • the boolean type is the walking pair of elements

  • the interval type is the walking identification

  • one definition of the circle type is the walking self-identification

  • another definition of the circle type is the walking parallel pair of identifications.

  • the suspension type of a type $I$ is the walking parallel $I$-indexed family of identifications.

  • the natural numbers type is the walking sequence

  • the integers type is the walking bi-infinite sequence

  • the walking bridge

  • the directed interval type is the walking morphism in the sense of element of a hom type

References

Topological cospans and their role as models for cobordisms are discussed in

• Marco Grandis, Collared cospans, cohomotopy and TQFT (Cospans in algebraic topology, II) (pdf)