nLab
cospan

Contents

Definition

In any category, a cospan is a diagram like this:

a b f g c \array{ && a &&&& b \\ & && {}_{f}\searrow & & \swarrow_g && \\ &&&& c &&&& }

A cospan in the category CC is the same as a span in the opposite category C opC^{op}. So, all general facts about cospans in CC are general facts about spans in C opC^{op}, and the reader may turn to the entry on spans to learn more.

A cospan that admits a cone is called a quadrable cospan.

Cospans in a category VV with small colimits form a bicategory whose objects are the objects of VV, whose morphisms are cospans 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 cospans 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 cospans from ptpt to itself are bi-pointed objects in VV.

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 August 5, 2017 01:59:09 by David Corfield (91.88.197.172)