From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10107 Path: news.gmane.io!.POSTED.ciao.gmane.io!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: Monoidal product functor is strong monoidal, when? Date: Mon, 20 Jan 2020 14:32:57 +1030 Message-ID: References: Reply-To: David Roberts Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Injection-Info: ciao.gmane.io; posting-host="ciao.gmane.io:159.69.161.202"; logging-data="112569"; mail-complaints-to="usenet@ciao.gmane.io" Cc: "categories@mta.ca list" To: Matsuoka Takuo Original-X-From: majordomo@rr.mta.ca Mon Jan 20 20:51:21 2020 Return-path: Envelope-to: gsmc-categories@m.gmane-mx.org Original-Received: from smtp2.mta.ca ([198.164.44.55]) by ciao.gmane.io with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.92) (envelope-from ) id 1itd4z-000T5Y-Ir for gsmc-categories@m.gmane-mx.org; Mon, 20 Jan 2020 20:51:21 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:41434) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1itd3b-0005d4-Dh; Mon, 20 Jan 2020 15:49:55 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1itd3S-0008DJ-UH for categories-list@rr.mta.ca; Mon, 20 Jan 2020 15:49:46 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.io gmane.science.mathematics.categories:10107 Archived-At: 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=E2=80=93Street 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. =3D=3D=3D=3D=3D 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,=E2=8A=A0) be a symmetric monoidal =E2=88=9E-category, and write CA= lg(V) for its =E2=88=9E-category of commutative algebras (a.k.a. E_=E2=88= =9E-algebras). >(1) CAlg(V) admits finite coproducts, and the forgetful functor CAlg(V) --= > V canonically enhances to a symmetric monoidal functor (CAlg(V),=E2=88=90= ) --> (V,=E2=8A=A0). >(1') In particular, for any pair of objects A,B=E2=88=88CAlg(V), one might= write A=E2=8A=A0B=E2=88=88CAlg(V) for their coproduct. >(2) Using the notation (1'), for any A,B=E2=88=88CAlg(V), there is a canon= ical enhancement of the multiplication map A=E2=8A=A0A --> A from a morphis= m in V to a morphism in CAlg(V). >(3) For any =E2=88=9E-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 symm= etric monoidal structure. Then, an E_=E2=88=9E-algebra in V is nothing mor= e or less than a symmetric monoidal category C. By (2), the symmetric mono= idal product C x C --> C admits a canonical enhancement to a symmetric mono= idal functor. And if you like, you can apply (3) with O=3DE_2. =3D=3D=3D=3D=3D=3D 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 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 finit= e > 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=E2=80=93Street). > > 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 > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]