# Idea

One can naturally think of a cospan as the abstraction of a cobordism. For instance an interval object cospan models the standard topological interval $\left[0,1\right]$ regarded as a cobordism from pt to pt. The co-span co-trace on the interval glues the two ends of the interval together to produce a circle regarded as a cospan from $\varnothing$ to itself.

The concrete dual of a co-span, obtained by mapping it into some target object, is a span, which in the context of groupoidification and geometric function theory can be interpreted as a generalized linear map. On such a generalized linear map, there is a notion of trace, the span trace.

The co-span co-trace is the concept dual to that: the image of the co-trace of a co-span under mapping it into a target object is the span trace of the result of mapping the original co-span to that target object.

# Definition

For

$\begin{array}{ccc}& & T\\ & {}^{\mathrm{in}}↗& & {↖}^{\mathrm{out}}\\ \Sigma & & & & \Sigma \end{array}$\array{ && T \\ & {}^{in}\nearrow && \nwarrow^{out} \\ \Sigma &&&& \Sigma }

a cospan with identical left and right index object $\Sigma$, its co-span co-trace $\mathrm{cotr}\left(T\right)$ is the composite of the result

$\begin{array}{ccc}& & T\\ & {}^{\mathrm{in}\bigsqcup \mathrm{out}}↗& & ↖\\ \Sigma \bigsqcup \Sigma & & & & \varnothing \end{array}$\array{ && T \\ & {}^{in \sqcup out}\nearrow && \nwarrow \\ \Sigma \sqcup \Sigma &&&& \emptyset }

of dualizing one leg of the co-span with the co-span

$\begin{array}{ccc}& & \Sigma \\ & {}^{}↗& & {↖}^{\mathrm{Id}\bigsqcup \mathrm{Id}}\\ \varnothing & & & & \Sigma \bigsqcup \Sigma \end{array}$\array{ && \Sigma \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} \\ \emptyset &&&& \Sigma \sqcup \Sigma }

i.e. the pushout

$\begin{array}{ccccc}& & & & \mathrm{cotr}T\\ & & & ↗& & ↖\\ & & \Sigma & & & & T\\ & {}^{}↗& & {↖}^{\mathrm{Id}\bigsqcup \mathrm{Id}}& & {}^{\mathrm{in}\bigsqcup \mathrm{out}}↗& & ↖\\ \varnothing & & & & \Sigma \bigsqcup \Sigma & & & & \varnothing \end{array}$\array{ &&&& \mathrm{cotr}T \\ &&& \nearrow && \nwarrow \\ && \Sigma &&&& T \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} && {}^{in \sqcup out}\nearrow && \nwarrow \\ \emptyset &&&& \Sigma \sqcup \Sigma &&&& \emptyset }

regarded as a cospan from the initial object $\varnothing$ to $\varnothing$

$\begin{array}{ccc}& & \mathrm{cotr}\left(T\right)\\ & {}^{}↗& & ↖\\ \varnothing & & & & \varnothing \end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && cotr(T) \\ & {}^{}\nearrow && \nwarrow \\ \emptyset &&&& \emptyset } \,.

## Definition for multi-cospans

More generally, the trace of a multi-cospan over $n$ identical of its index objects $\Sigma$ is the composite with the multi-cospan

$\begin{array}{cc}& \Sigma \\ & {}^{\mathrm{Id}}↗{↑}^{\mathrm{Id}}& \cdots \\ \Sigma & \Sigma & \cdots & \Sigma & \cdots \end{array}$\array{ & \Sigma \\ & {}^{Id}\nearrow \uparrow^{Id} & \cdots \\ \Sigma & \Sigma & \cdots & \Sigma & \cdots }

# Examples

## Co-tracing topological interval to circle

Let the ambient category be Top, let $I=\left[0,1\right]$ be the standard topological interval and let $e:=\left[0,ϵ\right]$ be a small interval, for some $0<ϵ<1/2$ – to be thought here as a collar of the point $\mathrm{pt}$.

Let

$\begin{array}{ccc}& & I\\ & ↗& & {↖}^{1-ϵ+\left(-\right)}\\ e& & & & e\end{array}$\array{ && I \\ & {}\nearrow && \nwarrow^{1-\epsilon+(-)} \\ e &&&& e }

be the interval regarded as a collared cobordisms from the point to the point. Its cotrace, the pushout

$\begin{array}{ccc}\mathrm{cotr}\left(I\right)& ←& I\\ ↑& & {↑}^{\mathrm{in}\bigsqcup \mathrm{out}}\\ e& \stackrel{\mathrm{Id}\bigsqcup \mathrm{Id}}{←}& e\bigsqcup e\end{array}$\array{ cotr(I) &\leftarrow& I \\ \uparrow && \uparrow^{in \sqcup out} \\ e &\stackrel{Id \sqcup Id}{\leftarrow}& e \sqcup e }

is the result of gluing the ends of the interval to each other, i.e. the circle

$\mathrm{cotr}\left(I\right)={S}^{1}\phantom{\rule{thinmathspace}{0ex}}.$cotr(I) = S^1 \,.

Urs: This may require a bit more care with the topology involved. I still need to check the reference below for more details.

• Marco Grandis, Collared cospans, cohomotopy and TQFT (Cospans in Algebraic Topology II) (pdf)

## Co-tracing category interval object to the natural numbers

Let the ambient category be Cat, let $I=\left\{a\to b\right\}$ be the standard interval object in Cat and let $\mathrm{pt}=\left\{•\right\}$ be the terminal category.

Let

$\begin{array}{ccc}& & I\\ & {}^{\mathrm{pt}↦a}↗& & {↖}^{\mathrm{pt}↦b}\\ \mathrm{pt}& & & & \mathrm{pt}\end{array}$\array{ && I \\ & {}^{pt \mapsto a}\nearrow && \nwarrow^{pt \mapsto b} \\ pt &&&& pt }

be the standard interval object in Cat regarded in the standard way as a cospan from the point to the point.

Dualizing it to

$\begin{array}{ccc}& & I\\ & {}^{\mathrm{in}\bigsqcup \mathrm{out}}↗& & {↖}^{}\\ \mathrm{pt}\bigsqcup \mathrm{pt}& & & & \varnothing \end{array}$\array{ && I \\ & {}^{in \sqcup out}\nearrow && \nwarrow^{} \\ pt \sqcup pt &&&& \emptyset }

corresponds to thinking of it as a “bent interval”

$\begin{array}{c}\mathrm{pt}\\ & ↘\\ & & ↓\\ & ↙\\ \mathrm{pt}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ pt \\ & \searrow \\ && \downarrow \\ & \swarrow \\ pt } \,.

Accordingly, the co-span

$\begin{array}{ccc}& & \mathrm{pt}\\ & {}^{}↗& & {↖}^{\mathrm{Id}\bigsqcup \mathrm{Id}}\\ \varnothing & & & & \mathrm{pt}\bigsqcup \mathrm{pt}\end{array}$\array{ && pt \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} \\ \emptyset &&&& pt \sqcup pt }

can be thought of as

$\begin{array}{cc}& \mathrm{pt}\\ ↗\\ ↖\\ & \mathrm{pt}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ & pt \\ \nearrow \\ \nwarrow \\ & pt } \,.

Gluing these two arcs together yields the cotrace, the pushout

$\begin{array}{ccc}\mathrm{cotr}\left(I\right)& ←& I\\ ↑& & {↑}^{\mathrm{in}\bigsqcup \mathrm{out}}\\ \mathrm{pt}& \stackrel{\mathrm{Id}\bigsqcup \mathrm{Id}}{←}& \mathrm{pt}\bigsqcup \mathrm{pt}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ cotr(I) &\leftarrow& I \\ \uparrow && \uparrow^{in \sqcup out} \\ pt &\stackrel{Id \sqcup Id}{\leftarrow}& pt \sqcup pt } \,,

which is the result of gluing the ends of the interval object to each other, which here is the skeleton of the fundamental category of the directed circle

$\begin{array}{ccc}& & \to \\ & ↗& & ↘\\ ↑& & & & ↓\\ & ↖& & ↙\\ & & ←\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ && \rightarrow \\ & \nearrow && \searrow \\ \uparrow &&&& \downarrow \\ & \nwarrow && \swarrow \\ && \leftarrow } \,,

namely the monoid of natural numbers, regarded as a one-object category:

$\mathrm{cotr}\left(I\right)=Bℕ=\left\{•\stackrel{n}{\to }•\mid n\in ℕ\right\}\phantom{\rule{thinmathspace}{0ex}}.$cotr(I) = \mathbf{B} \mathbb{N} = \{\bullet \stackrel{n}{\to} \bullet | n \in \mathbb{N}\} \,.

If instead we start with the standard interval object in groupoids, ${I}_{\mathrm{inv}}=\left\{a\stackrel{\simeq }{\to }b\right\}$ with the nontrivial morphism from $a$ to $b$ being an isomorphism, then the co-trace in question is the skeleton of the fundamental groupoid of the ordinary topological circle

$\mathrm{cotr}\left({I}_{\mathrm{inv}}\right)=Bℤ=\left\{•\stackrel{n}{\to }•\mid n\in ℤ\right\}\phantom{\rule{thinmathspace}{0ex}}.$cotr(I_{inv}) = \mathbf{B} \mathbb{Z} = \{\bullet \stackrel{n}{\to} \bullet | n \in \mathbb{Z}\} \,.

# References

While the concept is obvious, it is apparently (?) not discussed yet in the (young) literature on the subject. On the blog the concept was mentioned in

Revised on January 26, 2009 15:44:36 by Urs Schreiber (134.100.222.156)