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Eigil Rischel (Mar 24 2020 at 09:15):In a poset-enriched category (or a general bicategory), we can ask for a "Frobenius monoid", i.e a monoid structure on an object which has an adjoint comonoid structure satisfying a certain equation.
Is there any work done on "lax frobenius monoids", i.e where the Frobenius equation holds only up to a (not necessarily invertible) 2-cell?
Specifically, I'd like this inequality:
(And the analogous one for the other equation, clearly)
Jules Hedges (Mar 24 2020 at 11:18):I think it looks faintly familiar, I might have seen it in the context of diagram rewriting. (My first thought was cartesian bicategories, but I don't think it's right, you get similar looking equations but not the same ones)
Eigil Rischel (Mar 24 2020 at 13:54):My question was actually inspired by seeing this preprint (https://arxiv.org/abs/2003.09453) on Cartesian bicategories today and noticing a possible connection to my own work - but I only get that inequality, not an equality, in the Frobenius axiom (it's to do with causal statistical models, the problem is that information can't travel backwards along the wire on the LHS)
Nicolas Blanco (Mar 24 2020 at 15:33):You may be interested in this paper https://www.irif.fr/~mellies/tensorial-logic/9-dialogue-categories-and-frobenius-algebras.pdf
Also Mike Shulman mentioned lax Frobenius monoid in this post https://golem.ph.utexas.edu/category/2017/11/starautonomous_categories_are.html
Nicolas Blanco (Mar 24 2020 at 15:59):However, it is important to see that Mike Shulman is working with polycategories so the definition you gave cannot be expressed in this setting.
In particular the special Frobenius law cannot be stated in general for polycategory.