# 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 $[0,1]$ regarded as a cobordism from pt to pt. The cospan cotrace on the interval glues the two ends of the interval together to produce a circle regarded as a cospan from $\emptyset$ to itself.

The concrete dual of a cospan, 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 cospan cotrace is the concept dual to that: the image of the cotrace of a cospan under mapping it into a target object is the span trace of the result of mapping the original cospan to that target object.

# Definition

For

$\array{ && T \\ & {}^{in}\nearrow && \nwarrow^{out} \\ \Sigma &&&& \Sigma }$

a cospan with identical left and right index object $\Sigma$, its cospan cotrace $cotr(T)$ is the composite of the result

$\array{ && T \\ & {}^{in \sqcup out}\nearrow && \nwarrow \\ \Sigma \sqcup \Sigma &&&& \emptyset }$

of dualizing one leg of the cospan with the cospan

$\array{ && \Sigma \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} \\ \emptyset &&&& \Sigma \sqcup \Sigma }$

i.e. the pushout

$\array{ &&&& \mathrm{cotr}T \\ &&& \nearrow && \nwarrow \\ && \Sigma &&&& T \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} && {}^{in \sqcup out}\nearrow && \nwarrow \\ \emptyset &&&& \Sigma \sqcup \Sigma &&&& \emptyset }$

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

$\array{ && cotr(T) \\ & {}^{}\nearrow && \nwarrow \\ \emptyset &&&& \emptyset } \,.$

## 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

$\array{ & \Sigma \\ & {}^{Id}\nearrow \uparrow^{Id} & \cdots \\ \Sigma & \Sigma & \cdots & \Sigma & \cdots }$

# Examples

## Cotracing topological interval to circle

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

Let

$\array{ && I \\ & {}\nearrow && \nwarrow^{1-\epsilon+(-)} \\ e &&&& e }$

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

$\array{ cotr(I) &\leftarrow& I \\ \uparrow && \uparrow^{in \sqcup out} \\ e &\stackrel{Id \sqcup Id}{\leftarrow}& e \sqcup e }$

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

$cotr(I) = S^1 \,.$

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)

## Cotracing category interval object to the natural numbers

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

Let

$\array{ && I \\ & {}^{pt \mapsto a}\nearrow && \nwarrow^{pt \mapsto b} \\ pt &&&& pt }$

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

Dualizing it to

$\array{ && I \\ & {}^{in \sqcup out}\nearrow && \nwarrow^{} \\ pt \sqcup pt &&&& \emptyset }$

corresponds to thinking of it as a “bent interval”

$\array{ pt \\ & \searrow \\ && \downarrow \\ & \swarrow \\ pt } \,.$

Accordingly, the co-span

$\array{ && pt \\ & {}^{}\nearrow && \nwarrow^{Id \sqcup Id} \\ \emptyset &&&& pt \sqcup pt }$

can be thought of as

$\array{ & pt \\ \nearrow \\ \nwarrow \\ & pt } \,.$

Gluing these two arcs together yields the cotrace, the pushout

$\array{ cotr(I) &\leftarrow& I \\ \uparrow && \uparrow^{in \sqcup out} \\ pt &\stackrel{Id \sqcup Id}{\leftarrow}& pt \sqcup pt } \,,$

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

$\array{ && \rightarrow \\ & \nearrow && \searrow \\ \uparrow &&&& \downarrow \\ & \nwarrow && \swarrow \\ && \leftarrow } \,,$

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

$cotr(I) = \mathbf{B} \mathbb{N} = \{\bullet \stackrel{n}{\to} \bullet | n \in \mathbb{N}\} \,.$

If instead we start with the standard interval object in groupoids, $I_{inv} = \{a \stackrel{\simeq}{\to} b\}$ 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

$cotr(I_{inv}) = \mathbf{B} \mathbb{Z} = \{\bullet \stackrel{n}{\to} \bullet | n \in \mathbb{Z}\} \,.$

# Remarks

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