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Eduardo Hoefel (Jun 10 2025 at 13:31):Dear all,
Does anyone know a reference for a proof of the following statement? It seems to be correct.
Consider a monoidal category in which every object admits exactly one comonoid structure. Then the monoidal structure in the category is necessarily cartesian.
This statement appears to hold without additional assumptions, but I am open to adding some if needed. For instance, I don’t mind assuming that the category is braided, even though that doesn’t seem strictly necessary.
Best regards,
Eduardo
Paolo Perrone (Jun 10 2025 at 13:37):A similar result to this is Fox' theorem.
It also needs the additional assumption that every morphism is a homomorphism of comonoids.
Paolo Perrone (Jun 10 2025 at 13:42):In case you are familiar with Markov categories, you can interpret (and prove) the idea in the following way:
1) Morphisms of comonoids are exactly the deterministic maps.
2) Deterministic maps always make their outputs conditionally independent given their inputs.
3) Therefore, their outputs are completely determined by their projections.
4) Therefore, the universal property of the product holds.
Eduardo Hoefel (Jun 10 2025 at 13:48):Thank you Paolo ! I didn't know about Markov Categories, but it sounds like a quite interesting solution to this question.
Paolo Perrone said:
A similar result to this is Fox' theorem.
It also needs the additional assumption that every morphism is a homomorphism of comonoids.
Paolo Perrone (Jun 10 2025 at 13:50):A nice thing about Markov categories is that they allow to transfer ideas from probability to other areas of math. The idea of independence (and conditional independence) is particularly useful, also outside of probability.
Tobias Fritz (Jun 10 2025 at 14:01):Eduardo Hoefel said:
Consider a monoidal category in which every object admits exactly one comonoid structure. Then the monoidal structure in the category is necessarily cartesian.
That is actually not true, and many Markov categories are counterexamples to this! See Remark 11.29 here, which implies for example that [[FinStoch]] is a monoidal category in which every object has a unique comonoid structure.
The fact that FinStoch is not cartesian is in a sense what probability is all about, namely the fact that a joint distribution of two random variables is not uniquely determined by its two marginals.
Tobias Fritz (Jun 10 2025 at 14:19):For people less familiar with probability theory, a simpler counterexample is SetMulti, by which we mean the category of sets and multivalued functions, or in other words the subcategory of Rel consisting of the total relations. This is also a positive Markov category and hence my previous statement applies.
It seems plausible that even (Rel,x) itself, which is not quite a Markov category, has unique comonoids. But I haven't checked the details of that.
Eduardo Hoefel (Jun 10 2025 at 14:25):Thank you Tobias ! I will check your counter examples and read more about Markov Categories.
Notification Bot (Jun 10 2025 at 18:15):Eduardo Hoefel has marked this topic as resolved.
Matteo Capucci (he/him) (Jun 12 2025 at 10:01):You also need to assume they are commutative, as far as I remember.