nLab
co-span

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

$$\begin{array}{ccc}& & S\\ & \nearrow & & \nwarrow \\ a& & & & b\end{array}$$\array{ && S \\ & \nearrow && \nwarrow \\ a &&&& b }

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

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

$$\begin{array}{ccc}& & S\\ & {}^{{\sigma }_S}\nearrow & & {\nwarrow }^{{\tau }_S}\\ a& & {\downarrow }^{\eta }& & b\\ & {}_{{\sigma }_T}\searrow & & {\swarrow }_{{\tau }_T}\\ & & T\end{array}.$$\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

$$\begin{array}{ccc}& & S\\ & {}^{{\sigma }_S}\nearrow & & {\nwarrow }^{{\tau }_S}\\ a& & & & b\\ & {}_{{\sigma }_T}\searrow & & {\swarrow }_{{\tau }_T}\\ & & T\end{array}$$\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

$$\begin{array}{ccc}{}_a[S,T{]}_b& \to & \mathrm{pt}\\ \downarrow & & {\downarrow }^{{\sigma }_T\times {\tau }_T}\\ [S,T]& \stackrel{{\sigma }_S^{*}\times {\sigma }_T^{*}}{\to }& [a\bigsqcup b,T]\end{array}$$\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, $\mathrm{pt}$, then co-spans from $\mathrm{pt}$ to itself are bi-pointed objects in $V$.

Related concepts

• co-span co-trace

• multi-cospan

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)