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Stream: theory: category theory

Topic: free cartesian structure over a monoidal category

Matteo Capucci (he/him) (Dec 08 2022 at 15:08):

(crossposting from the nForum)
Fox's theorem famously states that given a monoidal category, the cofree way to make it into a cartesian one is to take the category of comonoids therein. This gives a right adjoint to the forgetful functor $\bf CartMonCat \to MonCat$. What about the left adjoint?

Matteo Capucci (he/him) (Dec 08 2022 at 15:09):

Neil Ghani suggested, very plausibly, that it's the functor freely throwing in diagonals/terminal morphisms, and quotienting hom-sets so that both are natural

Morgan Rogers (he/him) (Dec 08 2022 at 15:19):

That does sound plausible... is there a good reason to believe that there should be a left adjoint in the first place?

Morgan Rogers (he/him) (Dec 08 2022 at 15:22):

(also, this is for monoidal categories and strong monoidal functors, right?)

Matteo Capucci (he/him) (Dec 08 2022 at 16:19):

Well, Fox says so first of all. There's also a paper by Lack (IIRC) about property-like structure which treats these kind of situations.

Matteo Capucci (he/him) (Dec 08 2022 at 16:19):

Morgan Rogers (he/him) said:

(also, this is for monoidal categories and strong monoidal functors, right?)

Fox says strict. I guess weakening to strong wouldn't break the result.

Morgan Rogers (he/him) (Dec 08 2022 at 16:20):

I see, this is a situation where we know the adjoint exists and you're asking if there is a nice presentation of it. Cool!

Matteo Capucci (he/him) (Dec 08 2022 at 16:22):

Exactly :)

Nathanael Arkor (Dec 08 2022 at 16:43):

Matteo Capucci (he/him) said:

Neil Ghani suggested, very plausibly, that it's the functor freely throwing in diagonals/terminal morphisms, and quotienting hom-sets so that both are natural

This sounds right. It's a general fact that any morphism of algebraic theories $T \to T'$ induces a left-adjoint $\mathrm{Alg}(T) \to \mathrm{Alg}(T')$ which, informally, freely adds the $T'$-structure to a $T$-algebra. This fact carries over to 2-algebraic theories. The left adjoint will add the structure which is "missing": which is the natural comonoid structure on each object.

Claudio Pisani (Dec 13 2022 at 11:03):

This is related to the monoidal closed structure on colored operads (that is, symmetric multicategories),
where $A\otimes B$ is the Boardman-Vogt tensor product and $[A,B]$ is the operad of functors $A\to B$
(see my "sequential multicategories"paper for details).
Indeed, if $1$ is the terminal operad, $1\otimes A$ and $[1,A]$ give respectively the reflection
and the coreflection of $A$ in sequential operads (those whose operations are sequences of unary operations).
If we restrict to the "representable" or "monoidal" operads, the sequential ones are the cocartesian monoidal categories, so that we get the reflection and the coreflection of monoidal categories in the cocartesian ones.
While $[1,A]$ is the operad of monoids in $A$, $1\otimes A$ is obtained, as you said, by adding a commutative monoid structure on each object of $A$ and quotienting to make each operation in $A$ to commute with these monoid structures.

Since the above result holds in the context of operads, when we restrict to the representable operads
the relevant morphismsare lax functors of monoidal categories.
This puzzles me since, as you mentioned, Fox's result is stated for strong functors.
That is, it seems to me somewhat strange that both the lax form of Fox's theorem
(corollary 3.19 in the cited paper) and the original strong form can be both true.

Matteo Capucci (he/him) (Dec 13 2022 at 11:58):

It works for lax morphisms too? That's great!

Matteo Capucci (he/him) (Dec 13 2022 at 11:58):

Is 'the cited paper' your 'sequential multicategories'?

Claudio Pisani (Dec 13 2022 at 14:05):

yes