nLab
co-span

A co-span in a category VV is a diagram

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

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

Co-spans in a category VV with small co-limits form a bicategory whose objects are the objects of VV, whose morphisms are co-spans between two objects, and whose 2-morphisms η\eta are commuting diagrams of the form

S σ S τ S a η b σ T τ T T. \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 aa to bb is naturally a category enriched in VV: for

S σ S τ S a b σ T τ T T \array{ && S \\ & {}^{\sigma_{S}}\nearrow && \nwarrow^{\tau_S} \\ a &&&& b \\ & {}_{\sigma_T}\searrow && \swarrow_{\tau_T} \\ && T }

two parallel cospans in VV, the VV-object a[S,T] b{}_a[S,T]_b of morphisms between them is the pullback

a[S,T] b pt σ T×τ T [S,T] σ S *×σ T * [ab,T] \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 VV has a terminal object, ptpt, then co-spans from ptpt to itself are bi-pointed objects in VV.

Related concepts

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)