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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeAug 1st 2017
• CommentRowNumber2.
• CommentAuthorPeter Heinig
• CommentTimeAug 1st 2017
• (edited Aug 1st 2017)

(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

• “output” and “input” so far are absent from quiver
• are not usual terms in combinatorics
• are of course usual terms in the more “operadic” literature.

There is a small and related-to-the-above Pandora’s box containing the more substantial issue

• whether to draw vertices in string diagrams as geometric shapes (a very common practice which partly seems a concession to typographical issues, but partly is done with good reason; it spoils the Poincaré-duality a bit, on the other hand is put to use in the “operadic literature” to distinguish said “outputs” and “inputs”.

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.

• CommentRowNumber3.
• CommentAuthorPeter Heinig
• CommentTimeAug 1st 2017
• (edited Aug 1st 2017)

(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.)

• CommentRowNumber4.
• CommentAuthorSam Staton
• CommentTimeJul 8th 2019

• CommentRowNumber5.
• CommentAuthorSam Staton
• CommentTimeJul 8th 2019

Included the result of Fong and Spivak characterizing hypergraph categories as lax monoidal presheaves. I feel compelled to remark that a presheaf $\mathbf{Cospan}_\Delta\to \mathbf{Set}$ is itself like a hypergraph: it associates to each list of vertices (objects) a set of edges.

• CommentRowNumber6.
• CommentAuthorjulesh
• CommentTimeJul 12th 2019

Some examples of hypergraph categories, plus a definition of spiders

• CommentRowNumber7.
• CommentAuthorjulesh
• CommentTimeJul 12th 2019

Some examples of hypergraph categories, plus a definition of spiders

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 12th 2019

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).

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJul 15th 2019

So I changed “quiver” to “directed graph”