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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!

Vincent Moreau (Nov 07 2025 at 10:52):

I recently came accros the paper Classifying strict discrete opfibrations with lax morphisms, congrats! and thanks for mentioning the example of this chat :)

Vincent Moreau (Nov 07 2025 at 10:57):

In the past months I fell deeply in love with Chu spaces, so let me give another example that is similar to the one coming from topological spaces. Indeed, there is a fully faithful functor from $\mathbf{Top}$ to $\mathbf{Chu}(\mathbf{Set}, 2)$, sending a topological space on the Chu space made of the points and the opens of the original topological space. It is well known that this functor is fully faithful, and what I understood very recently was this this functor was also strong monoidal, when $\mathbf{Top}$ is equipped with its SMCC structure, and $\mathbf{Chu}(\mathbf{Set}, 2)$ with its usual *-autonomous structure!

Vincent Moreau (Nov 07 2025 at 11:04):

Now, for any SMCC $\mathbf{C}$ with appropriate pullbacks and $\bot$ any object of $\mathbf{C}$, we have a *-autonomous category $\mathbf{Chu}(\mathbf{C}, \bot)$ whose objects are triples $X = (X_+, X_-, \Vdash_{X} : X_+ \otimes X_- \to \bot)$. Moreover, there is an adjunction whose left adjoint $\mathbf{C} \to \mathbf{Chu}(\mathbf{C}, \bot)$ sends an object $A$ on the Chu space $!A$ defined as

$$(!A)_+ \ :=\ A \qquad (!A)_- \ :=\ A \multimap \bot \qquad \Vdash_{!A} \ :=\ \operatorname{ev}_{A, \bot}$$

In the case where $\mathbf{C}$ is $\mathbf{Set}$ and $\bot$ is $2$, we get back the functor that sends a set on its "discrete" Chu space, so this is very similar to the situation with $\mathbf{Top}$.

fosco (Nov 07 2025 at 11:06):

I also love that result; it appears in the Coimbra notes by Vaughan Pratt

Vincent Moreau (Nov 07 2025 at 11:07):

Now, the right adjoint $\mathbf{Chu}(\mathbf{C}, \bot) \to \mathbf{C}$, which sends $X$ on $X_+$, is strong monoidal for the usual tensor product of Chu spaces, so we can apply all that we said in the chat! In particular, if $\mathbf{C}$ is monoidal cartesian, then any $!A$ sees $\mathbf{Chu}(\mathbf{C}, \bot)$ as a cartesian monoidal category :)

Matteo Capucci (he/him) (Nov 10 2025 at 09:47):

That's really nice, Vincent! It makes me realize that the notion of cartesian object should be relevant for categorical semantics of linear logic. It seems plausible to me that one could start from a single SMCC $C$ of linear types and ask the exponential modality $!$ to land in the 'cartesian part' of $C$...