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\begin{document}

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\section*{cospan cotrace}

[[!redirects co-span co-trace]]

\hypertarget{idea}{}\section*{{Idea}}\label{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 \emph{cospan cotrace} on the interval glues the two ends of the interval together to produce a \emph{circle} regarded as a cospan from $\emptyset$ to itself.

The [[duality|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 \emph{trace}, the [[span trace]].

The \emph{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.

\hypertarget{definition}{}\section*{{Definition}}\label{definition}

For

\begin{displaymath}
\itexarray{
    && T
    \\
    & {}^{in}\nearrow && \nwarrow^{out}
    \\
    \Sigma &&&& \Sigma
  }
\end{displaymath}
a [[cospan]] with identical left and right index object $\Sigma$, its \textbf{cospan cotrace} $cotr(T)$ is the composite of the result

\begin{displaymath}
\itexarray{
    && T
    \\
    & {}^{in \sqcup out}\nearrow && \nwarrow
    \\
    \Sigma \sqcup \Sigma &&&& \emptyset
  }
\end{displaymath}
of dualizing one leg of the cospan with the cospan

\begin{displaymath}
\itexarray{
    && \Sigma
    \\
    & {}^{}\nearrow && \nwarrow^{Id \sqcup Id}
    \\
    \emptyset &&&& \Sigma \sqcup \Sigma
  }
\end{displaymath}
i.e. the [[pushout]]

\begin{displaymath}
\itexarray{
    &&&&
    \mathrm{cotr}T
    \\
    &&& \nearrow && \nwarrow 
    \\
    && \Sigma &&&& T
    \\
    & {}^{}\nearrow && \nwarrow^{Id \sqcup Id}
    && {}^{in \sqcup out}\nearrow && \nwarrow
    \\
    \emptyset &&&& \Sigma \sqcup \Sigma &&&& \emptyset
  }
\end{displaymath}
regarded as a cospan from the initial object $\emptyset$ to $\emptyset$

\begin{displaymath}
\itexarray{
    && cotr(T)
    \\
    & {}^{}\nearrow && \nwarrow
    \\
    \emptyset &&&& \emptyset
  }
  \,.
\end{displaymath}
\hypertarget{definition_for_multicospans}{}\subsection*{{Definition for multi-cospans}}\label{definition_for_multicospans}

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

\begin{displaymath}
\itexarray{
    & \Sigma
    \\
    & {}^{Id}\nearrow \uparrow^{Id} & \cdots 
    \\
    \Sigma & \Sigma & \cdots & \Sigma & \cdots
  }
\end{displaymath}
\hypertarget{examples}{}\section*{{Examples}}\label{examples}

\hypertarget{cotracing_topological_interval_to_circle}{}\subsection*{{Cotracing topological interval to circle}}\label{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 \emph{[[collar]]} of the point $pt$.

Let

\begin{displaymath}
\itexarray{
     && I
     \\
     & {}\nearrow && \nwarrow^{1-\epsilon+(-)}
     \\
     e &&&& e
  }
\end{displaymath}
be the interval regarded as a collared cobordisms from the point to the point. Its cotrace, the pushout

\begin{displaymath}
\itexarray{
     cotr(I) &\leftarrow& I
     \\
     \uparrow && \uparrow^{in \sqcup out}
     \\
     e &\stackrel{Id \sqcup Id}{\leftarrow}& e \sqcup e
  }
\end{displaymath}
is the result of gluing the ends of the interval to each other, i.e. the circle

\begin{displaymath}
cotr(I) = S^1
  \,.
\end{displaymath}
[[Urs Schreiber|Urs]]: This may require a bit more care with the topology involved. I still need to check the reference below for more details.

See also

\begin{itemize}%
\item Marco Grandis, \emph{Collared cospans, cohomotopy and TQFT (Cospans in Algebraic Topology II)} (\href{http://www.dima.unige.it/~grandis/wCub2.pdf}{pdf})

\end{itemize}
\hypertarget{cotracing_category_interval_object_to_the_natural_numbers}{}\subsection*{{Cotracing category interval object to the natural numbers}}\label{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

\begin{displaymath}
\itexarray{
     && I
     \\
     & {}^{pt \mapsto a}\nearrow && \nwarrow^{pt \mapsto b}
     \\
     pt &&&& pt
  }
\end{displaymath}
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{displaymath}
\itexarray{
     && I
     \\
     & {}^{in \sqcup out}\nearrow && \nwarrow^{}
     \\
     pt \sqcup pt &&&& \emptyset
  }
\end{displaymath}
corresponds to thinking of it as a ``bent interval''

\begin{displaymath}
\itexarray{
    pt
    \\
    & \searrow
    \\
    && \downarrow
    \\
    & \swarrow
    \\
    pt
  }
  \,.
\end{displaymath}
Accordingly, the co-span

\begin{displaymath}
\itexarray{
     && pt
     \\
     & {}^{}\nearrow && \nwarrow^{Id \sqcup Id}
     \\
     \emptyset &&&& pt \sqcup pt
  }
\end{displaymath}
can be thought of as

\begin{displaymath}
\itexarray{
    & pt
    \\
    \nearrow
    \\
    \nwarrow
    \\
    & pt
  }
  \,.
\end{displaymath}
Gluing these two arcs together yields the cotrace, the pushout

\begin{displaymath}
\itexarray{
     cotr(I) &\leftarrow& I
     \\
     \uparrow && \uparrow^{in \sqcup out}
     \\
     pt &\stackrel{Id \sqcup Id}{\leftarrow}& pt \sqcup pt
  }
  \,,
\end{displaymath}
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 space|directed circle]]

\begin{displaymath}
\itexarray{
    && \rightarrow 
    \\
    & \nearrow && \searrow
    \\
    \uparrow &&&& \downarrow
    \\
    & \nwarrow && \swarrow
    \\
    && \leftarrow
  }
  \,,
\end{displaymath}
namely the [[monoid]] of natural numbers, regarded as a one-object category:

\begin{displaymath}
cotr(I) = \mathbf{B} \mathbb{N} = \{\bullet \stackrel{n}{\to} \bullet | n \in \mathbb{N}\}
  \,.
\end{displaymath}
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 space|topological circle]]

\begin{displaymath}
cotr(I_{inv}) = \mathbf{B} \mathbb{Z} = \{\bullet \stackrel{n}{\to} \bullet | n \in \mathbb{Z}\}
  \,.
\end{displaymath}
\hypertarget{remarks}{}\section*{{Remarks}}\label{remarks}

\begin{itemize}%
\item The dual notion is that of [[span trace]]

\end{itemize}
\hypertarget{references}{}\section*{{References}}\label{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

\begin{itemize}%
\item Urs Schreiber, \href{http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html#c021537}{(co)-traces}

\end{itemize}


\end{document}