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Jules Hedges (Sep 27 2021 at 10:16):(This question arose from a very interesting discussion on twitter involving a whole bunch of people, starting at https://twitter.com/stringdiagram/status/1442058728359043074?s=21)
@_julesh_ Did you mean to say commutative? You can't define it in a monoidal category, and it doesn't seem needed in a proof.
- Dan Marsden (@StringDiagram)It's an old bit of foklore proved by Fox that a symmetric monoidal product is a cartesian product iff it has a unique supply (*) of commutative comonoids, and every morphism is a comonoid homomorphism
(*) that's Brendan's terminology: aka every object has that structure in a way that's compatible with the tensor product
This suggestions a provisional definition: a "noncommutative finite product category" is a (not necessarily symmetric) monoidal category with a unique supply of comonoids, and every morphism is a comonoid homomorphism
I wonder what tricks this thing can do? Are there any interesting examples? Can you equivalently define it with a univeral property? Etc.
Dan Marsden (Sep 27 2021 at 10:46):I think one issue with the weakening to a monoidal category is that the compatibility with the tensor is phrased in terms of the symmetry to get the types to line up.
Jules Hedges (Sep 27 2021 at 11:01):Oh yeah, that's a bit unfortunate
Dan Marsden (Sep 27 2021 at 11:20):The bit I'm most interested in at the moment is how many of the assumptions are needed to show Cartesian monoidal structure. I think \emph{possibly} that symmetric + supply of monoids compatible with tensors + every morphism is a comonoid homomorphism might be enough. I've not checked this yet, so after my previous blunder, I wouldn't assume this correct yet...
Chad Nester (Sep 27 2021 at 11:23):This thing (https://arxiv.org/pdf/2109.09634.pdf) popped up on arxiv recently. It deals with the braided case.
Nathanael Arkor (Sep 27 2021 at 11:23):This isn't quite the same question, but Fox's theorem was recently analysed for braided monoidal categories in Kraehmer–Mahaman's Clones from comonoids.
Chad Nester (Sep 27 2021 at 11:23):ha!
Nathanael Arkor (Sep 27 2021 at 11:23):Wow :laughing:
Dan Marsden (Sep 27 2021 at 12:58):Cartesian-Monoidal-Structure.pdf
Dan Marsden (Sep 27 2021 at 13:02):It seems sufficient that in a symmetric monoidal category with a supply of monoids compatible with the tensor, such that every morphism is a comonoid homomorphism, the monoidal structure is Cartesian. In fact, I don't think you need associativity of the comonoids either, so unital "diagonal morphisms" seems enough. In particular, you don't need the commutativity or unicity as far as I could see in my proofs with the assumptions spelt out.
Dan Marsden (Sep 27 2021 at 13:03):Of course, once you've proved this, you get a lot more for free, including the commutativity, associativity and unicity.