A port graph
A special type of prop where morphisms are open, directed, acyclic port graphs \((V, in, out, i)\)
\(V\) is a set of vertices
functions \(V \xrightarrow{in, out} \mathbb{N}\) give the in degree and out degree of each vertex
A bijection \(\bar m \sqcup O \xrightarrow{i} I \sqcup \bar n\), where \(I = \{(v,i)\ |\ v \in V, 1 \leq i \leq in(v)\}\) and \(O=\{(v,i)\ |\ v \in V, 1 \leq i \leq out(v))\}\) are the vertex inputs and vertex outputs, respectively.
Furthermore, an acyclicity condition:
Use the bijection \(i\) to construct the internal flow graph: a graph with an arrow \(u \xrightarrow{e^{u,i}_{v,j}} v\) for every \(i,j \in \mathbb{N}\) such that \(i(u,i)=(v,j)\)
This graph must be acyclic
A \((2,3)\) port-graph. [Draw this]
Since the overall type is \((2,3)\) we know we need a box with two overall inputs and three overall outputs.
There are three internal boxes, meaning the vertex set is \(\{a, b, c\}\).
\(in(a)=1\) and \(out(a)=3\) which tells us to draw one port on the LHS and three on the RHS.
The bijection \(i\) tells us how the ports are connected by wires.
The acyclic internal flow graph is shown below [TODO Draw this]
A category PG whose objects are natural numbers and morphisms are port graphs
We can compose a \((m,n)\) port graph with a \((n, p)\) port graph \((V \sqcup V',[in,in'],[out,out'], i'')\)
\(V \sqcup V' \xrightarrow{[in,in']} \mathbb{N}\) maps elements of \(V\) according to \(in\) and elements of \(V'\) according to \(in'\) (likewise for out).
The bijection \(\bar m \sqcup O \sqcup O' \xrightarrow{i''} I \sqcup I' \sqcup \bar p\) is defined as \(i''(x)=\begin{cases}i(x) & i(x) \in I \\ i'(i(x))& i(x) \in \bar n \\ i'(x) & x \in O' \end{cases}\)
The identity port graph on \(n\) is \((\varnothing, !, !, id_{\bar n})\) where \(\varnothing \xrightarrow{!} \mathbb{N}\) is a unique function.
This category is a prop.
The monoidal product of two portgraphs... TODO
Describe how port graph composition looks, with respect to the visual representation of Example 5.14, and give a nontrivial example
Todo