co-span

A co-span in a category $V$ is a diagram

$\array{
&& S
\\
& \nearrow && \nwarrow
\\
a &&&& b
}$

in $V$, i.e. a span in the opposite category $V^{op}$.

Co-spans in a category $V$ with small co-limits form a bicategory whose objects are the objects of $V$, whose morphisms are co-spans 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 co-spans 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.

If $V$ has a terminal object, $pt$, then co-spans 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)

Revised on May 17, 2013 23:56:36
by Urs Schreiber
(89.204.154.16)