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Fabio Zanasi (Oct 15 2024 at 17:34):Together with @Gabriele Lobbia , @Wojciech Rozowski and @Ralph Sarkis we just put this on ArXiv:
Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams
https://arxiv.org/abs/2410.09229
It is our attempt at answering a question that I've been wondering since I came to know the work of Mardare, Plotkin and Panangaden on Quantitative Algebraic Reasoning: what is the monoidal counterpart? How do we define a 'quantitative' version of monoidal theories, where we can have axioms of the form 'string diagram 1 is at distance at most epsilon from string diagram 2'? What it means for such a theory to be a presentation of an (enriched) category? It turns out even just defining all that requires quite a bit of machinery, which is what we develop in this work.