cospan

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.

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 $V$.

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)

Last revised on May 19, 2021 at 00:09:24. See the history of this page for a list of all contributions to it.