Re: Monoidal product functor is strong monoidal, when?

[Matsuoka Takuo <motogeomtop@gmail.com>]

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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