From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9992 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: only_marketing_? Date: Tue, 20 Aug 2019 09:55:04 +0100 Message-ID: References: Reply-To: Steve Vickers Mime-Version: 1.0 (1.0) Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable Injection-Info: blaine.gmane.org; posting-host="blaine.gmane.org:195.159.176.226"; logging-data="201783"; mail-complaints-to="usenet@blaine.gmane.org" To: John Baez , categories Original-X-From: majordomo@mlist.mta.ca Tue Aug 20 19:55:41 2019 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.55]) by blaine.gmane.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.89) (envelope-from ) id 1i08M9-000qLM-8v for gsmc-categories@m.gmane.org; Tue, 20 Aug 2019 19:55:41 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39429) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1i08Kp-0004DB-JY; Tue, 20 Aug 2019 14:54:19 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1i08K9-0003z6-FP for categories-list@mlist.mta.ca; Tue, 20 Aug 2019 14:53:37 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9992 Archived-At: Dear John, Those are rather pertinent examples, as the dagger closed and hypergraph cat= egories show up a weakness in my question. I asked about seeking objects, morphisms, identities and associative composi= tion, which seems very natural because it's the basic definition of category= . Everything has a domain and a codomain, an input and an output, and compos= ition is malformed unless it's domain with codomain. This leads many of our c= ategory theoretic intuitions to be based on thinking of objects and morphism= s as being, at some level of abstraction, like sets and functions. Once you have set up the structure of what is input and what is output, it t= akes some effort to forget it. Dagger closed and the associated string diagr= ams provide a mechanism for doing that. A good example is Rel. A morphism from X1 x ... x Xm to Y1 x ... x Yn is jus= t a subset of X1 x ... x Xm x Y1 x ... x Yn, in the light of which it is per= haps perverse to impose domain and codomain structure - unless, perhaps you w= ant to carry on to say which relations are functional. As you propose, this certainly looks like a good way to analyse networks, an= d open systems where there is an interface between internal structure and ex= ternal behaviour, an interface along which we must compose components. I've heard Jamie Vicary and others use the word "compositionality" as someth= ing not quite the same as category theory. Is this what they mean, letting g= o of the strict domain-codomain discipline? All the best, Steve. > On 17 Aug 2019, at 04:44, John Baez wrote: >=20 > Hi - >=20 > Steve wrote: >=20 >> So, to return to John Baez's interview, how might we look for category > theory >> helping to understand the world's problems? We must first look for > objects and >> morphisms,with identities and associative composition, so what are the > real-world >> prototypes of what we are trying to do there? What is the first step > beyond the >> vague aspirations? >=20 > The interviewer didn't give me a chance to say much. Personally I've been > trying to understand the various kind of "networks" that come up in > electrical engineering: >=20 > https://arxiv.org/abs/1504.05625 >=20 > control theory: >=20 > https://arxiv.org/abs/1405.6881 >=20 > chemistry: >=20 > https://arxiv.org/abs/1704.02051 >=20 > and the study of Markov processes: >=20 > https://arxiv.org/abs/1508.06448 >=20 > Researchers in these and many other subjects use diagrams to describe > the networks they're working with. These diagrams are morphisms in > various symmetric monoidal categories. So there are already plenty of > symmetric monoidal categories being put to work in applied math. >=20 > But which ones, exactly? That's what my papers are about. These > categories > turn out to be beautiful and not always familiar; trying to understand the= m > is > making my students and me come up with new ideas. So, right now, I'd say= > researchers in these subjects have more to teach category theorists than > vice > versa. >=20 > Best, > jb >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]