nLab structured cospan

Idea

The notion of a structured cospan is a modification of the concept of decorated cospans, introduced to provide an improved definition of isomorphism classes of decorated cospans.

To better understand this motivation, notice that the decoration of the target of an isomorphism of decorated cospans is completely determined by the decoration of the source and the chosen isomorphism of vertices.

More clearly, let

$$\array{ && c && &&&&&& F(c)\\ & {}^{i}\nearrow && \nwarrow^{o} &&&&&& {}^d\nearrow\\ a && \downarrow^f && b &&&& 1 & & \downarrow^{F(f)} \\ & {}^{i'}\searrow && \swarrow^{o'} &&&&&& {}^{d'}\searrow \\ && c' &&&&&&&& F(c') }$$

be an isomorphism of decorated cospans, where $d$ and $d'$ are the decorations of the source and the target, respectively (i.e., the ‘top’ and the ‘bottom’ cospan in the left diagram, respectively). Then, since the right diagram sits in $\mathbf{Set}$, it commutes on the nose. Thus the decoration $d'$ is already determined by the data of bijection $c \to c'$, together with the decoration on the source.

Therefore, this definition of isomorphism means we are lacking a degree of freedom, namely the freedom to specify an isomorphism for the decorations as well. For example, when open graphs are treated with decorated cospans, the decoration $d$ of a cospan $a \to c \leftarrow b$ is a directed graph with $c$ as vertex set. An isomorphism of such decorated cospans simply renames the source and target of each edge in $d$. However, isomorphism of directed graphs can be more than just relabeling of the vertices; hence we end up distinguishing open graphs merely by frivolous details like the specific names we give to edges.

Decorated cospans solve this problem by moving the cospans to the ‘decorating category’, meaning that the data of an isomorphism of cospans now is an arrow between the decorations. In the example of open graphs, we now have to specify an isomorphism of quivers $d \to d'$ instead of getting this from the (poorer) isomorphism of their vertices.

Definition

Let $\mathbf A$ be a category admitting finite coproducts, $\mathbf X$ a category admitting finite colimits and $L : \mathbf A \to \mathbf X$ a functor preserving finite coproducts. Then the symmetric monoidal double category of structured cospans over $L$ is the category $_L\mathrm{Csp}(\mathbf X)$ which has

• objects given by objects of $\mathbf A$,
• vertical 1-morphisms given by morphisms of $\mathbf A$
• horizontal 1-morphisms given by structured cospans:
  $$\array{ && x \\ & {}^{i}\nearrow && \nwarrow^{o} \\ L(a) &&&& L(b) }$$

  which are composed through the obvious pushout,

• 2-morphisms given by commutative diagrams of the form
  $$\array{ L(a) & \leftarrow^i & x & \rightarrow^o & L(b)\\ \downarrow^{L(f)} & & \downarrow^h & & \downarrow^{L(g)}\\ L(a') & \leftarrow^{i'} & x' & \rightarrow^{o'} & L(b')\\ }$$

  which are composed horizontally in the obvious way.

(Assuming the existence and preservations of coproducts are unnecessary simply to get a double category, without monoidal structure.)

A structured cospan is then a ($1$-)morphism in such a category, that is, a cospan in $\mathbf X$ with the additional data of the functor $L$ and the two preimages of the feet.

Examples

• By taking $L : \mathbf{Set} \to \mathbf{Graph}$ to be the discrete graph functor, i.e. the functor assigning to a set $V$ the edgeless graph with vertex set $V$, we get a category $_L\mathrm{Csp}(\mathbf{Graph})$ which models open graphs.

Related concepts

• cospan
• decorated cospan
• hypergraph category

References

Structured cospan categories were invented by John C. Baez, Kenny Courser, and Christina Vasilakopoulou. An introductory talk was given by Courser at the 4th Symposium on Compositional Structures:

• John C. Baez, Kenny Courser, Structured cospans, Theory and Applications of Categories 35 48 (2020) 1771-1822 $[$arXiv:1911.04630, tac:35-48$]$

• John C. Baez, Kenny Courser, Structured cospans talk notes (2019) (pdf)