Monoidal product functor is strong monoidal, when?

Monoidal product functor is strong monoidal, when?

[David Roberts]

Hi all,

I have half convinced myself (without checking details) that if I have
a braided monoidal category (C,@), then the monoidal product @: C x C
--> C is strong monoidal. Is this true? What's a reference for this I
could point to?

Thanks,
David

David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com


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Re: Monoidal product functor is strong monoidal, when?

[David Roberts]

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:
>
> Hi David,
>
> Your message caught my attention on my spam tray by Gmail's fault.
>
> I'm not sure what proof would be nice to you, but as far as I could see,
> the construction of a symmetric monoidality of the multiplication of a
>> _symmetric_ monoidal category is largely trivial. Let Fin denote the
> category of finite sets. Then, a symmetric monoidal category (C,@) gives
> you a symmetric monoidal functor Fin ---> Cat which associates to a finite
> set S the category C^S. 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. Inspecting this symmetric monoidal functor,
> you further obtain a map of these symmetric monoidal functors which
> associates to S the multiplication functor
> C^S x C^S ---> C^S induced from the codiagonal map S + S ---> S, where
> "+" in the source denotes the coproduct operation in
> Fin. This is the desired structure.
>
> As you see, we have used the symmetric monoidality of the product
> functor Cat x Cat ---> Cat, which you have because the Cartesian product
> is a limit so preserves products. Thus, a reference you are looking for
> may be
>
> Graeme Segal, Categories and cohomology theories, Topology 13 (1974),
>
> which essentially contains a sufficient argument for this (and is indeed
> earlier than Joyal–Street).
>
> To conclude, "commutation with the braiding" comes immediately from
> the naturality of the codiagonal map, to commute with any
> automorphism of a finite set.
>
> Best regards,
> Takuo Matsuoka
>
>

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Re: Monoidal product functor is strong monoidal, when?

[Matsuoka Takuo]

Hi David,

Your message caught my attention on my spam tray by Gmail's fault.

I'm not sure what proof would be nice to you, but as far as I could see,
the construction of a symmetric monoidality of the multiplication of a
> _symmetric_ monoidal category is largely trivial. Let Fin denote the
category of finite sets. Then, a symmetric monoidal category (C,@) gives
you a symmetric monoidal functor Fin ---> Cat which associates to a finite
set S the category C^S. 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. Inspecting this symmetric monoidal functor,
you further obtain a map of these symmetric monoidal functors which
associates to S the multiplication functor
C^S x C^S ---> C^S induced from the codiagonal map S + S ---> S, where
"+" in the source denotes the coproduct operation in
Fin. This is the desired structure.

As you see, we have used the symmetric monoidality of the product
functor Cat x Cat ---> Cat, which you have because the Cartesian product
is a limit so preserves products. Thus, a reference you are looking for
may be

Graeme Segal, Categories and cohomology theories, Topology 13 (1974),

which essentially contains a sufficient argument for this (and is indeed
earlier than Joyal–Street).

To conclude, "commutation with the braiding" comes immediately from
the naturality of the codiagonal map, to commute with any
automorphism of a finite set.

Best regards,
Takuo Matsuoka


2019年12月17日(火) 7:34 David Roberts <droberts.65537@gmail.com>:
>
> Hi all again,
>
> thanks to those who replied off-list. The canonical reference is
> Joyal–Street's Braided monoidal categories. (Someone else also pointed
> out that algebras for the E_2 operad are equivalent to E_1 algebras in
> the category of E_1 algebras.)
>
> However, my *actual* desired result is that the multiplication of a
> _symmetric_ monoidal category is a braided functor (i.e. commutes with
> the braiding=symmetry in this case). I proved this to my own
> satisfaction, but my proof is not very nice, and I'm searching for a
> cleaner verification of the required commuting diagram. Surely this
> was also known! And if so, what's a good reference (I expect it to be
> even earlier than Joyal–Street).
>
> Thanks,
> David
>
> PS this question relating to Lawvere's 2015 invited CT address might
> be of interest to people here:
>
https://mathoverflow.net/questions/348436/the-barr-boole-galois-topos-a-modification-of-sets-to-play-well-with-schemes
>
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts
> Blog: https://thehighergeometer.wordpress.com
>
> On Wed, 11 Dec 2019 at 16:59, David Roberts <droberts.65537@gmail.com>
wrote:
>>
>> Hi all,
>>
>> I have half convinced myself (without checking details) that if I have
>> a braided monoidal category (C,@), then the monoidal product @: C x C
>> --> C is strong monoidal. Is this true? What's a reference for this I
>> could point to?
>>
>> Thanks,
>> David
>>
>> David Roberts
>> Webpage: https://ncatlab.org/nlab/show/David+Roberts
>> Blog: https://thehighergeometer.wordpress.com
>

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Re: Monoidal product functor is strong monoidal, when?

[David Roberts]

Hi all again,

thanks to those who replied off-list. The canonical reference is
Joyal–Street's Braided monoidal categories. (Someone else also pointed
out that algebras for the E_2 operad are equivalent to E_1 algebras in
the category of E_1 algebras.)

However, my *actual* desired result is that the multiplication of a
_symmetric_ monoidal category is a braided functor (i.e. commutes with
the braiding=symmetry in this case). I proved this to my own
satisfaction, but my proof is not very nice, and I'm searching for a
cleaner verification of the required commuting diagram. Surely this
was also known! And if so, what's a good reference (I expect it to be
even earlier than Joyal–Street).

Thanks,
David

PS this question relating to Lawvere's 2015 invited CT address might
be of interest to people here:
https://mathoverflow.net/questions/348436/the-barr-boole-galois-topos-a-modification-of-sets-to-play-well-with-schemes


David Roberts
Webpage: https://ncatlab.org/nlab/show/David+Roberts
Blog: https://thehighergeometer.wordpress.com

On Wed, 11 Dec 2019 at 16:59, David Roberts <droberts.65537@gmail.com> wrote:
>
> Hi all,
>
> I have half convinced myself (without checking details) that if I have
> a braided monoidal category (C,@), then the monoidal product @: C x C
> --> C is strong monoidal. Is this true? What's a reference for this I
> could point to?
>
> Thanks,
> David
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts
> Blog: https://thehighergeometer.wordpress.com


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]