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Stream: theory: category theory

Topic: A site of Joyal-Street string diagrams?

Amar Hadzihasanovic (Sep 02 2022 at 14:11):

Going back to Joyal-Street's classic The geometry of tensor calculus, I believe the following should be a way of packaging their results (I'll only consider the case of planar diagrams). Namely, I think that it should be possible to define a site $\mathcal{S}$ whose

• objects are J&S's string diagrams i.e. progressive plane graphs,
• morphisms are certain embeddings of plane graphs, so that the isomorphisms are exactly the continuous deformations/planar isotopies allowed by J&S, and the non-isomorphisms include at least the inclusions of factors of a “tensor decomposition” or a “vertical decomposition”, and
• the topology is given by the obvious covering families of embeddings,

together with a functor to the category of strict monoidal categories, sending each such diagram to the free strict monoidal category on its nodes&wires.

Such a functor should send the covering cocones to colimits so it should extend to a functor on the topos of sheaves on this site, with a right adjoint nerve functor. Then “diagrammatic reasoning à la Joyal-Street” would, essentially, be working in this nerve.

I don't know what the notion of morphism of progressive plane graphs that would make this work is. I am not against working this out myself, but to avoid duplicating work: is anyone aware of this (or something essentially equivalent) having been done?

Tom Hirschowitz (Sep 02 2022 at 15:07):

Probably not familiar enough with J&S, but, e.g., monoidal categories, as algebras for a familial monad, have a nerve (= embed fully into some presheaf cat with the image characterised by a Segal condition). Does this have anything to do with your proposal? (Just trying… :innocent: )

Jules Hedges (Sep 02 2022 at 15:10):

Not an answer: Do you have some motivation, eg. what kind of things could you learn about string diagrams and/or monoidal categories by studying the sheaf topos on this thing?

Amar Hadzihasanovic (Sep 02 2022 at 15:28):

It's for an expository paper, I'm not trying to prove anything new.

Amar Hadzihasanovic (Sep 02 2022 at 15:30):

I would like to give an overview of different "formal interpretations of diagrammatic reasoning" in terms of different nerve functors, and it would be nice to have Joyal-Street in that form.

Tom Hirschowitz (Sep 02 2022 at 16:20):

Oooops, sorry the monad is pointwise analytic, but not familial.

Morgan Rogers (he/him) (Sep 02 2022 at 19:04):

It seems like you would need the morphisms to come equipped with extra data? Or maybe you mean something more precise when you say "certain embeddings of plane graphs". By this I mean that if you're taking these diagrams as a subcategory of topological spaces, you won't be able to categorically exclude the homeomorphisms of graphs that aren't well-behaved (you need some constraint on the complement of the graph in the plane too)

Amar Hadzihasanovic (Sep 02 2022 at 19:54):

Yeah, there's already extra data in the definition of plane graph; it's a topological graph, so a space with the structure of a 1d cell complex, together with an embedding into the plane

Amar Hadzihasanovic (Sep 02 2022 at 19:56):

Presumably morphisms would be commutative squares of "good" embeddings (if not even weakly commutative?)

Amar Hadzihasanovic (Sep 02 2022 at 20:02):

@Tom Hirschowitz Yes I'm aware that there are good nerves of monoidal categories, in fact "better" ones than this would be (it would be faithful but not fully faithful), but right now I am interested in getting this one right :)