Re: Re: Monoidal product functor is strong monoidal, when?

Re: Re: Monoidal product functor is strong monoidal, when?

[Matsuoka Takuo]

Hi David,

Sure. Once you establish

>(1) CAlg(V) admits finite coproducts, and the forgetful functor CAlg(V) --> V canonically enhances to a symmetric monoidal functor (CAlg(V),∐) --> (V,⊠).

in some way, the Eckmann–Hilton argument shows that
the multiplication map A⊠A --> A is canonically equivalent to the
underlying map of the codiagonal map A∐A --> A in CAlg(V), hence

>(2) Using the notation (1'), for any A,B∈CAlg(V), there is a canonical enhancement of the multiplication map A⊠A --> A from a morphism in V to a morphism in CAlg(V).

Thanks for sharing this.

Incidentally, when I previously wrote

> The symmetric monoidal structure of C x C gives
> you a symmetric monoidal structure on the functor Fin ---> Cat
> associating C^S x C^S to S.

I of course only meant that there will be such a symmetric monoidal
functor (where the association S ---> C^S x C^S then does not say
more than that the one point set goes to C x C), but NOT that there
is a functor S ---> C^S x C^S before having a symmetric monoidal
structure on C x C. I'm sorry about the imprecision.

Best regards,
Takuo


2020年1月20日(月) 15:03 David Roberts <droberts.65537@gmail.com>:
>
> Hi Takuo,
>
> thanks for that observation, it's rather nice way to put it.
>
> Aaron Mazel-Gee had another way to show it, which he shared with me
> privately, and gave me permission to pass on to the list, copied
> below.
>
> It turns out that there are two diagrams that prove the result about
> symmetric monoidal functors (if one takes an elementary approach, and
> not using strictification, as Joyal–Street do in the published
> 'Braided tensor categories'), which are two halves of the generalised
> resultoassociahedron on the middle of page 39 of
> http://web.science.mq.edu.au/~street/BatanAustMSMq.pdf, originally
> appearing in work of Bar-Natan in 1993 (or so, it's a little hard to
> recognise). If one categorified this result, then one could have a
> 3-arrow that filled this polyhedral diagram of 2-arrows.
>
> =====
> On Tue, 17 Dec 2019 at 08:46, Aaron Mazel-Gee wrote:
>>Hi David,
>
>>It sounds like this is (once again) the opposite what you're looking for, but I would say that this is a special instance of a more general fact.
>
>>Let (V,⊠) be a symmetric monoidal ∞-category, and write CAlg(V) for its ∞-category of commutative algebras (a.k.a. E_∞-algebras).
>
>>(1) CAlg(V) admits finite coproducts, and the forgetful functor CAlg(V) --> V canonically enhances to a symmetric monoidal functor (CAlg(V),∐) --> (V,⊠).
>
>>(1') In particular, for any pair of objects A,B∈CAlg(V), one might write A⊠B∈CAlg(V) for their coproduct.
>
>>(2) Using the notation (1'), for any A,B∈CAlg(V), there is a canonical enhancement of the multiplication map A⊠A --> A from a morphism in V to a morphism in CAlg(V).
>
>>(3) For any ∞-operad O, note the existence of a forgetful functor CAlg(V) --> Alg_O(V).
>
>
>
>>Now, take V to be the (2,1)-category Cat, equipped with the cartesian symmetric monoidal structure.  Then, an E_∞-algebra in V is nothing more or less than a symmetric monoidal category C.  By (2), the symmetric monoidal product C x C --> C admits a canonical enhancement to a symmetric monoidal functor.  And if you like, you can apply (3) with O=E_2.
>
> ======
>
> Thanks,
> David
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts
> Blog: https://thehighergeometer.wordpress.com
>
> On Sun, 12 Jan 2020 at 01:15, Matsuoka Takuo <motogeomtop@gmail.com> wrote:

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