# nLab co-span

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

$\begin{array}{ccc}& & S\\ & ↗& & ↖\\ a& & & & b\end{array}$\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

$\begin{array}{ccc}& & S\\ & {}^{{\sigma }_{S}}↗& & {↖}^{{\tau }_{S}}\\ a& & {↓}^{\eta }& & b\\ & {}_{{\sigma }_{T}}↘& & {↙}_{{\tau }_{T}}\\ & & T\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}& & S\\ & {}^{{\sigma }_{S}}↗& & {↖}^{{\tau }_{S}}\\ a& & & & b\\ & {}_{{\sigma }_{T}}↘& & {↙}_{{\tau }_{T}}\\ & & T\end{array}$\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 ${}_{a}\left[S,T{\right]}_{b}$ of morphisms between them is the pullback

$\begin{array}{ccc}{}_{a}\left[S,T{\right]}_{b}& \to & \mathrm{pt}\\ ↓& & {↓}^{{\sigma }_{T}×{\tau }_{T}}\\ \left[S,T\right]& \stackrel{{\sigma }_{S}^{*}×{\sigma }_{T}^{*}}{\to }& \left[a\bigsqcup b,T\right]\end{array}$\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 $V$.

# 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)

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