# Contents

## Definition

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.

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.

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)