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Say a monoidal category is cartesian from if is a commutative comonoid in which induces a strong monoidal representable copresheaf .
Is anything known about comonoids like this?
I could come up with only one example, namely any semicocartesian monoidal category monoidal category with an initial object is cartesian from since
(I'm reasoning by analogy with another situation, in which the concept is not trivial. So I'm interested in understanding this one.)
Regarding your example, I think it works even if the initial object is not the monoidal unit, in which case it still has a unique (commutative, if the ambiant monoidal category is symmetric) comonoid strucutre.
True!
One example is given by any discrete space in Top equipped with the "separated continuity" tensor product, see e.g. Borceux volume II end of section 7.1. More generally, by doctrinal adjunction this holds for any adjunction of oplax functors where the codomain of the right adjoint is monoidal cartesian! This looks like but is not quite a linear-non-linear adjunction, which is an adjunction of lax functors.
Oh, and now that I'm thinking about it, this subsumes the example of the initial object, which we get back in case the monoidal cartesian category is the terminal one!
Actually, I think that this adjunction boils down to the two following observations about your "Hom cartesian" comonoids: (i) all the objects of a cartesian category, with their unique structure of comonoids, are Hom cartesian, and (ii) Hom cartesian comonoids are preserved by left adjoints of adjunctions of oplax functors.
This implies a stability of Hom cartesian comonoids by colimits as well
Vincent Moreau said:
by doctrinal adjunction this holds for any adjunction of oplax functors where the codomain of the right adjoint is monoidal cartesian!
Wow that is cool! :star_struck:
Let me unpack it. You have an adjunction of categories where is cartesian monoidal, is monoidal, and both and are colax. By doctrinal adjunction, this lifts to ad adjunction of colax monoidal functors iff is strong, in which case:
So neat!
Your example with the tensor product of topological spaces involves , , and (discrete space). My example of monoidal categories with an initial object involves such category, , and .
It's so weird how it's somehow dual to a LNL adjunction... like take D be commutative comonoids in C, you have a strong monoidal functor U back to C. Suppose I had a right adjoint K, ie C admits the cofree commutative comonoid construction. Then K is automatically lax by doctrinal adjunction but it needn't be strong in general, as far as I understand. So this situation would be completely disjoint from the one pictured above!