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Created hypergraph category.
(Small terminological thing that could be made smoother in future, nothing wrong: it’s about the words “output” and “input”, used within the Idea-section in
An ordinary string diagram is a quiver, where the inputs and outputs of a vertex describe objects appearing in a tensor-product decomposition of the domain and codomain of a morphism
There is a small and related-to-the-above Pandora’s box containing the more substantial issue
We should not open this box here.
For the time being my suggestion is only to make a few judicious terminological changes in the “Idea”-section of hypergraph category and/or in quiver. I would suggest one if I had made up my mind on this.)
EDIT: one issue is that simply saying “in-neighbour” and “out-neighbour” instead of “input” and “output” is not an option, since people in operad-theory really mean the parts of the edges sticking out of the respective vertices.
But it seems to me that formalizing “input” and “ouput” is a natural application of the usual concept of “bidirected graphs”, that I started to document in the unfinished article directed graph.
In particular the half-edges (not yet mentioned in directed graph) that are usual in the literature on bidirected graphs seem naturally suited to be put to use in string diagrams.
(re 2: suggestion: maybe an explict reference to the nice illustration at the beginning of Section 1.3.2 of Fong’s thesis could be clarifyingly added to hypergraph category, or maybe even a screenshot of said illustration, with full attribution.)
Let’s not write
An ordinary string diagram is a quiver
because “quiver” is a concept with an attitude and it’s not the attitude in question here.
Probably you want to say
An ordinary string diagram is a graph
possibly with some adjectives added (for most of which the relevant entries should exist, too).
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