Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: wiring diagrams for compact closed categories

Antonin Delpeuch (Jan 11 2022 at 18:53):

Wiring diagrams are a combinatorial version of string diagrams for symmetric monoidal categories (for instance see https://arxiv.org/abs/2101.12046). Has anyone already defined somewhere a variant of this notion for compact closed categories? Maybe @Evan Patterson , @David Spivak @Dmitry V. know?

Evan Patterson (Jan 11 2022 at 19:34):

For self-dual compact closed categories (hypergraph categories), there is the operad of undirected wiring diagrams. UWDs are actually much simpler than directed wiring diagrams, and we use them all the time in AlgebraicJulia as an unbiased syntax for composing relations, spans, structured cospans, and so on.

I am not sure about general compact closed categories. I figure you could handle them by adding some extra data to a UWD but I don't remember seeing that written down.

John Baez (Jan 11 2022 at 19:40):

Just by the way, every hypergraph category is self-dual compact closed, but 'hypergraph' means more. (I know Evan knows this, this remark is just for the newbies out there....)

Evan Patterson (Jan 11 2022 at 20:50):

Thanks for clarifying that. I should have said that UWDs are a syntax for hypergraph categories. For a self-dual compact closed category that is not a hypergraph category, not every UWD would give a valid way of composing morphisms.

Evan Patterson (Jan 11 2022 at 21:05):

I remembered a more relevant reference: String diagrams for traced and compact categories are oriented 1-cobordisms by Spivak, Schultz, and Rupel.

Evan Patterson (Jan 11 2022 at 21:12):

Also, last year at the Topos Colloquium, Marcy Robertson talked about the combinatorics of modular operads:

https://topos.site/topos-colloquium/slides/2021-07-29.pdf
https://www.youtube.com/watch?v=a88dtP1CGT8

I think that this is related to giving a combinatorial description of composition in compact closed categories, but I haven't understood the details.

John Baez (Jan 11 2022 at 21:14):

I thought modular operads described ways of gluing together Riemann surfaces (which are 2-dimensional gadgets, a kind of souped-up version of string diagrams in a way).

John Baez (Jan 11 2022 at 21:17):

Oh, never mind - the word 'modular' is connected to Riemann surface theory, but that's probably just one application of modular operads. Getzler and Kapranov started this subject with a paper Modular operads, and their abstract says

Modular operads are a special type of operad: in fact, they bear the same relationship to operads that graphs do to trees (i.e. simply connected graphs). One of the basic examples of a modular operad is the collection of Deligne-Mumford-Knudsen moduli spaces $M_{g,n}$ of stable pointed algebraic curves; hence the word "modular.''

John Baez (Jan 11 2022 at 21:18):

So they were interested in the Riemann surface ($\sim$ algebraic curve) example, but the basic idea is that "modular operads are to graphs as operads are to trees".

Evan Patterson (Jan 11 2022 at 21:22):

I was just going on slides 8-10 in that talk, where the description of a modular operad looks a lot like what you can do in a compact closed category: there is an involution on objects, you can compose morphisms, and there are "contraction" operations that look like the cap/cup.

John Baez (Jan 11 2022 at 21:49):

Okay, maybe the idea is that once you can build up trees (as in an ordinary operad) and you've got caps and cups, you can build up graphs.

Antonin Delpeuch (Jan 12 2022 at 06:57):

Thanks Evan! Ok then I will write up a definition, indeed I do not think it is particularly difficult, I just wanted to cite any relevant work that already got it right :)

Evan Patterson (Jan 12 2022 at 07:52):

Cool, those are the references I know. Looking forward to seeing your definitions.

Tobias Fritz (Jan 12 2022 at 10:22):

In Rewriting modulo symmetric monoidal structure, Bonchi et al have characterized string diagrams in symmetric monoidal categories as cospans of acyclic hypergraphs in which the legs satisfy a suitable monogamy condition. This matches up nicely with the structured cospan approach to networks in general.

I bet that their characterization can be adapted to the compact closed case relatively easily by relaxing the conditions on the cospans involved a bit. My guess would be that one can do it by dropping the acyclicity entirely and replacing their monogamy condition, which applies to each of the two legs separately (Defn 3.6), by a "joint monogamybigamy" condition: $f$ and $g$ still need to be injective, and every node in $G$ must be the source or target of an edge plus being in the image of $f$ or $g$ in exactly two ways overall. This would formalize the idea that every wire connects exactly two ports, and each port is either incident to a box or an overall input or output interface.

Tobias Fritz (Jan 12 2022 at 10:29):

Actually this is not quite right yet: my proposed definition does not allow for two input interfaces to be wired together by a cap, or two output interfaces to be wired together by a cup. But something along these lines must be possible.

David Spivak (Jan 14 2022 at 01:27):

Sorry for the delay. The wiring diagrams for traced and for compact closed categories are basically the same: they're oriented 1-cobordisms. I wrote a paper with Schultz and Rupel on it String diagrams for traced and compact categories are oriented 1-cobordisms

Antonin Delpeuch (Jan 20 2022 at 11:02):

Thanks! In the end just went for a simple combinatorial description, it did not feel like a lot of overhead. I needed it to construct the free compact closed category generated by a symmetric monoidal category: https://arxiv.org/abs/2201.07527