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Alexander Gietelink Oldenziel (Jul 21 2020 at 12:19):Hello Friends, I hope I am doing this right.
I have a simple question: monoidal categories are approx. monoids in Cat, what about the dual notion comonoids in Cat [or monoids in Cat^{op} if you will]? Have these comonoidal categories ever been considered, and if so where can I learn about them? Thanks in advance for any help!
Jules Hedges (Jul 21 2020 at 12:52):I believe they are trivial, at least if you define them this way, for the same reason that comonoids in Set are trivial
Jules Hedges (Jul 21 2020 at 12:53):If is a monoidal category whose monoidal product is the categorical product (ie. a cartesian monoidal category) then every object of is a commutative comonoid in exactly one way, namely where the comultiplication is copy and the counit is delete
Jules Hedges (Jul 21 2020 at 12:57):The symmetry breaks because monoidal categories are monoids in Cat considered with its cartesian product specifically, not anything else. (Dually, if your monoidal product is a coproduct then monoids would become trivial)
Alexander Gietelink Oldenziel (Jul 21 2020 at 13:47):Thanks Jules Hedges! That's very clear.
John Baez (Jul 21 2020 at 20:31):Every category is a comonoid in (Cat, ) in a unique way, just like every set is a comonoid in (Set, ) in a unique way.
It's a lot of fun to prove the second fact, and once you do you'll see that it only use the fact that (Set, ) is cartesian, so the same proof works for (Cat, ).
John Baez (Jul 21 2020 at 20:32):I'm writing the here because to speak of a "comonoid in a category X", we need X to be a monoidal category - so we should specify the monoidal structure.
Emily (Jul 24 2020 at 00:09):Does this question become non-trivial when phrased for enriched categories, where the product can be non-Cartesian?
That is, if we take a monoidal category V that isn't Cartesian, then the induced product of V-categories also won't be Cartesian. Are there non-trivial "comonoidal V-categories", meaning pseudomonoids in Cats_V^op?
(Maybe related to this is the following nLab page: monoidal categories with diagonals)
By the way, we could also dualise things by working with Cats_V^co instead of Cats_V^op, or even with Cats_V^coop. That is, by reversing the direction of the 2-morphisms of Cats_V, rather than of the 1-morphisms. Since the associators and unitors of a monoidal category are invertible, this ends up making no difference.
OTOH, passing from pseudomonoids to lax monoids, we'd get lax notions of monoidal categories, like a lax monoidal category. In Cats_V, this would give us:
Reid Barton (Jul 24 2020 at 00:12):Sure, for example, you could just take the case of one object and then you get a comonoid in V, if I'm not mistaken.
Emily (Jul 25 2020 at 04:59):I was looking through my notes and I found a forgotten memo about precisely this topic: it seems Day considered this notion already in Section 2.4 of his PhD thesis.
Link: http://unsworks.unsw.edu.au/fapi/datastream/unsworks:58748/SOURCE01?view=true.
(The definition is given on page 62, meaning page 67 of the PDF. See also Section 3.3 there)
John Baez (Jul 25 2020 at 05:06):Multiplicative Group Scheme said:
Does this question become non-trivial when phrased for enriched categories, where the product can be non-Cartesian?
That is, if we take a monoidal category V that isn't Cartesian, then the induced product of V-categories also won't be Cartesian. Are there non-trivial "comonoidal V-categories", meaning pseudomonoids in Cats_V^op?
Yeah, that's a lot better.