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\begin{document}

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\section*{decorated cospan}

[[!redirects decorated cospans]]

\hypertarget{decorated_cospan}{}\section*{{Decorated cospan}}\label{decorated_cospan}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{categories_of_cospans}{Categories of cospans}\dotfill \pageref*{categories_of_cospans} \linebreak
\noindent\hyperlink{categories_of_decorated_cospans}{Categories of decorated cospans}\dotfill \pageref*{categories_of_decorated_cospans} \linebreak
\noindent\hyperlink{the_monoidal_and_hypergraph_structures}{The monoidal and hypergraph structures}\dotfill \pageref*{the_monoidal_and_hypergraph_structures} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Decorated cospans are a convenient formalism to deal with open networks, i.e. networks were some nodes are interpreted to be `inputs' and some other to be `outputs'. In fact, a natural way to do such labelling is to specify some function assigning the set of inputs to those nodes in the network tasked with receiving them, and analogously, a function assigning to outputs those nodes which provide them. This is, morally, a [[cospan]]: the network is the vertex, and inputs and outputs are the feet. However, the two things sit in `different categories': e.g., inputs and outputs may be sets of wires and sockets, while the network has a more complicated description. Nevertheless, the network is just a set of nodes with more information attached, namely the way those nodes are connected. The key insight here is that to specify inputs/outputs of the network, we don't really care about this additional information. Hence we can use a `cospan of nodes' (formally, a cospan in [[FinSet]]), and deal with the network structure later, as a \emph{decoration}.

Surprisingly, this formalism naturally captures also other operations such as sequential and parallel composition of networks.

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

\hypertarget{categories_of_cospans}{}\subsubsection*{{Categories of cospans}}\label{categories_of_cospans}

In a category $\mathbf C$ with finite colimits (or even just [[pushout|pushouts]]) [[cospans]] can be composed in a natural way. In fact, given $x \to s \leftarrow y$ and $y \to t \leftarrow z$, we get a new cospan $x \to p \leftarrow z$ by taking a pushout in the middle:

\begin{displaymath}
\itexarray{
     &&&& s +_y t 
     \\& 
    &&
      {}^{p_s}\nearrow
      && \nwarrow^{p_t}
    \\
     && s &&&& t
      \\
      & {}^{f}\nearrow
      && \nwarrow^{g}
       &
       & {}^{h}\nearrow
      && \nwarrow^{i}
     \\
     x
     &&&&
     y
     &&&&
     z
  }
\end{displaymath}
Therefore cospans form a compositional structure: a category $\mathrm{Cospan}(\mathbf C)$ where objects are the same of $\mathbf C$ but morphisms from $x$ to $y$ are replaced by cospans with feet $x$ and $y$. Since the pushout we choose to form the composite of two cospans is, in general, unique only up to unique isomorphism, we actually need to define a $2$-category. Morphisms of cospans $(x \to p \leftarrow y) \Rightarrow (x \to q \leftarrow y)$ are given by any $\mathbf C$-morphism $\eta : p \to q$ making the following commute

\begin{displaymath}
\itexarray{
    && p
    \\
    & {}^{f}\nearrow && \nwarrow^{g}
    \\
    x &&\downarrow^\eta&& z
    \\
    & {}_{f'}\searrow
    && \swarrow_{g'}
    \\
    && q
  }
\end{displaymath}
It is also customary to ignore the $2$-structure and simply work with $\mathrm{Cospan}(\mathbf C)$ as a $1$-category whose morphisms are \emph{isomorphism classes} of cospans.

\hypertarget{categories_of_decorated_cospans}{}\subsubsection*{{Categories of decorated cospans}}\label{categories_of_decorated_cospans}

\begin{udefn}
(\hyperlink{BrendanFong2015}{B. Fong 2015}) Let $\mathbf C$ be a [[finitely cocomplete category|category with finite colimits]], and

\begin{displaymath}
(F, \phi): (\mathbf C, +) \to (\mathbf D, \otimes)
\end{displaymath}
be a [[lax monoidal functor]]. A \textbf{decorated cospan}, or more precisely an \textbf{$F$-decorated cospan}, is a pair $(x \overset{i}\to n \overset{o}\leftarrow y,\, 1 \overset{s}\to Fn)$. We shall call the element $1 \overset{s}\to Fn$ the \textbf{decoration} of the decorated cospan. A morphism of decorated cospans

\begin{displaymath}
f : (x \overset{i}\to n \overset{o}\leftarrow y,\, 1 \overset{s}\to Fn) \to (x \overset{i}\to n' \overset{o}\leftarrow y,\, 1 \overset{s'}\to Fn')
\end{displaymath}
is a morphism of cospans such that the following commutes:

\begin{displaymath}
\itexarray{
 && Fn\\
 & \nearrow^s\\
 1 & & \downarrow^{Ff}\\
 & \searrow^{s'}\\
 && Fn'
}
\end{displaymath}
\end{udefn}
\begin{udef}
Given a cospan $x \to n \leftarrow y$ in $\mathbf C$, the \textbf{empty decoration on $n$} is the unique map

\begin{displaymath}
1 \overset{\phi_1}\longrightarrow F\varnothing \overset{F!}\longrightarrow Fn
\end{displaymath}
where $\varnothing$ is [[initial object|inital]] in $\mathbf C$ and $!$ denotes the universal morphism from such object.

\end{udef}
\begin{prop}
\label{}\hypertarget{}{}
(\hyperlink{BrendanFong2015}{B. Fong 2015}) There is a category $F\mathrm{Cospan}$ of $F$-decorated cospans, with objects the objects of $\mathbf C$ and morphisms isomorphism classes od $F$-decorated cospans. Composition in this category is given by the class of the pushout of two representatives:

\begin{displaymath}
\itexarray{
     &&&& n +_y m
     \\& 
    &&
      {}^{j_n}\nearrow
      && \nwarrow^{j_m}
    \\
     && n &&&& m
      \\
      & {}^{i_x}\nearrow
      && \nwarrow^{o_y}
       &
       & {}^{i_y}\nearrow
      && \nwarrow^{o_z}
     \\
     x
     &&&&
     y
     &&&&
     z
  }
\end{displaymath}
along with the decoration

\begin{displaymath}
1 \overset{\lambda^{-1}}\longrightarrow 1 \otimes 1 \overset{s \otimes s'}\longrightarrow Fn \otimes Fm \overset{\phi_{n,m}}\longrightarrow F(n+m) \overset{F[j_n,j_m]}\longrightarrow F(n +_y m).
\end{displaymath}
where $1 \overset{s}\to Fn$ and $1 \overset{s'}\to Fn$ are decorations of the first and second cospan, respectively.

\end{prop}
\begin{proof}
The identity morphism of an object $x$ in $F\mathrm{Cospan}$ is simply $x \overset{\mathrm{id}}\to x \overset{\mathrm{id}}\leftarrow x$ equipped with the empty decoration on $x$. The check that all relevant axioms are satisfied can be found in (\hyperlink{BrendanFong2015}{B. Fong 2015, Appendix A})

\end{proof}
\begin{uremark}
The composition of decorations is the key construction of the decorated cospans formalisms. In fact, returning to the analogy with open networks, it constructs the composite network of a given `link' of two networks. Hence the compositional structures of the cospans is leveraged to describe the compositional structure of a richer structure.

\end{uremark}
\begin{uremark}
Notice that the empty decoration gives a canonical way to decorate any cospan on $\mathbf C$. Indeed, it can be shown this defines a wide functor $\mathrm{Cospan}(\mathbf C) \embedsin F\mathrm{Cospan}$, as the composition of two empty-decorated cospans is again empty-decorated.

\end{uremark}
\hypertarget{the_monoidal_and_hypergraph_structures}{}\subsubsection*{{The monoidal and hypergraph structures}}\label{the_monoidal_and_hypergraph_structures}

In the presence of a braiding on $\mathbf D$ (the `decorating category'), the category of decorated cospans becomes not just [[symmetric monoidal category|symmetric monoidal]], but a full-blown [[hypergraph category]].

\begin{theorem}
\label{}\hypertarget{}{}
(\hyperlink{BrendanFong2015}{B. Fong, 2015}) Let $\mathbf C$ be a category with finite colimts, $(\mathbf D, \otimes)$ a [[braided monoidal category]] and $(F, \phi) : (\mathbf C, +) \to (\mathbf D, \otimes)$ a lax braided monoidal functor. Then we may equip $F\mathrm{Cospan}$ with a symmetric monoidal and hypergraph structure, such that there is a [[wide subcategory|wide embedding]] of hypergraph categories

\begin{displaymath}
\mathrm{Cospan}(\mathbf C) \embedsin F\mathrm{Cospan}.
\end{displaymath}
\end{theorem}
\begin{proof}
We define the monoidal product of objects $x$ and $y$ of $F\mathrm{Cospan}$ to be their coproduct $x+y$ in $\mathbf C$, and defined the monoidal product of decorated cospans $(x \overset{i}\to n \overset{o}\leftarrow y, 1 \overset{s}\to Fn)$ and $(x \overset{i}\to n' \overset{o}\leftarrow y, 1 \overset{s'}\to Fn')$ to be

\begin{displaymath}
\itexarray{
     && n + n'
      \\
      & {}^{i_x + i_{x'}}\nearrow
      && \nwarrow^{o_y + o_{y'}} &&,&& 1 \overset{\lambda^{-1}}\longrightarrow 1 \otimes 1 \overset{s \otimes s'}\longrightarrow Fn \otimes Fn' \overset{\phi_{n,n'}}\longrightarrow F(n+n')
     \\
     x+x'
     &&&&
     y+y'
  }
\end{displaymath}
The braiding in $\mathbf D$ can be now used to show this product is indeed functorial. Finally, we choose associator, unitors and braiding to be the images of those in $\mathrm{Cospan}(\mathbf C)$. The necessary checks are done in (\hyperlink{BrendanFong2015}{B. Fong 2015, Appendix A}).

The hypergraph structure is defined by equipping each object $x \in F\mathrm{Cospan}$ with the image of the special commutative [[Frobenius algebra|Frobenius monoid]] specified by the hypergraph structure of $\mathrm{Cospan}(\mathbf C)$. The fact that the `empty-decoration' embedding is an hypergraph functor is evident.

\end{proof}
If the monoidal unit of $(\mathbf D, \otimes)$ is the inital object, then each object admits only a decoration --- the empty one. This implies $\mathrm{Cospan}(\mathbf C)$ and $1_{\mathbf C}\mathrm{Cospan}$ are isomorphic as hypergraph categories.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[cospan]]
\item [[structured cospan]]
\item [[hypergraph category]]

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

Decorated cospans were first defined by Brendan Fong:

\begin{itemize}%
\item [[Brendan Fong]], \emph{Decorated Cospans}, (\href{http://arxiv.org/abs/1502.00872}{arXiv:1502.00872}).

\end{itemize}


\end{document}