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Stream: learning: questions

Topic: monoidal category cartesian 'from an object'

Matteo Capucci (he/him) (Mar 04 2025 at 17:57):

Say a monoidal category $(\cal C, I, \otimes)$ is cartesian from $H$ if $H$ is a commutative comonoid in $\cal C$ which induces a strong monoidal representable copresheaf ${\cal C}(H,-):\cal C \to \bf Set$.
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 $0$ is cartesian from $0$ since

$${\cal C}(0,A \otimes B) \cong {\cal C}(0,A) \times {\cal C}(0,B) \cong 1.$$

Matteo Capucci (he/him) (Mar 04 2025 at 17:59):

(I'm reasoning by analogy with another situation, in which the concept is not trivial. So I'm interested in understanding this one.)

Vincent Moreau (Mar 04 2025 at 23:33):

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.

Matteo Capucci (he/him) (Mar 05 2025 at 08:38):

True!

Vincent Moreau (Mar 05 2025 at 17:29):

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.

Vincent Moreau (Mar 05 2025 at 17:57):

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!

Vincent Moreau (Mar 05 2025 at 18:13):

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.

Vincent Moreau (Mar 05 2025 at 18:36):

This implies a stability of Hom cartesian comonoids by colimits as well

Matteo Capucci (he/him) (Mar 06 2025 at 07:57):

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 $L \dashv R : \cal C \leftrightarrows D$ of categories where $\cal D$ is cartesian monoidal, $\cal C$ is monoidal, and both $L$ and $R$ are colax. By doctrinal adjunction, this lifts to ad adjunction of colax monoidal functors iff $R$ is strong, in which case:

$${\cal C}(LX, A \otimes B) \cong {\cal D}(X, R(A \otimes B)) \cong {\cal D}(X, RA \times RB) \cong {\cal D}(X, RA) \times {\cal D}(X,RB) \cong {\cal C}(LX,A) \times {\cal C}(LX,B)$$

So neat!

Your example with the tensor product of topological spaces involves $\cal C = \bf Top$, $\cal D= \bf Set$, $R=U$ and $L=\Delta$ (discrete space). My example of monoidal categories with an initial object involves $\cal C$ such category, $\cal D=1$, $R=!$ and $L=0$.

Matteo Capucci (he/him) (Mar 06 2025 at 08:33):

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!