From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9989 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Alexander Kurz Newsgroups: gmane.science.mathematics.categories Subject: Re: only_marketing_? Date: Fri, 16 Aug 2019 09:30:07 -0700 Message-ID: References: Reply-To: Alexander Kurz Mime-Version: 1.0 (Mac OS X Mail 11.5 \(3445.9.1\)) 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="255620"; mail-complaints-to="usenet@blaine.gmane.org" Cc: categories@mta.ca To: Steve Vickers Original-X-From: majordomo@mlist.mta.ca Sun Aug 18 17:59:27 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 1hzNaY-0014NR-OK for gsmc-categories@m.gmane.org; Sun, 18 Aug 2019 17:59:26 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39151) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1hzNZV-0003dv-KW; Sun, 18 Aug 2019 12:58:21 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1hzNYo-0002Oe-B6 for categories-list@mlist.mta.ca; Sun, 18 Aug 2019 12:57:38 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9989 Archived-At: > 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? Wrt real-world prototypes of morphism, John Baez in his recent SYCO 4 = talk [1] suggested that the idea of a morphism as a component of a = network (and composition as "wiring up'' components) could have = applications in many areas outside of computer science, including = engineering, biology and ecology. This is still vague, so I am not pretending to answer Steve's question, = but I think it points in a direction that is promising. All the best, Alexander [1] http://events.cs.bham.ac.uk/syco/4/slides/Baez.pdf Btw, slides of (almost) all talks of SYCO 4 are linked from the schedule = at=20 http://events.cs.bham.ac.uk/syco/4/ >=20 > Steve Vickers >=20 >> On 12 Aug 2019, at 01:03, Vaughan Pratt = wrote: >>=20 >> Formulating the *future *with category theory? My understanding is = that >> the *present *is already formulated less with set theory than with = category >> theory, at least its morphisms under composition, and has been for a = long >> time. >>=20 >> In 1969 Jack Schwartz introduced the programming language SETL based = on set >> theory. However programmers found it much easier to write programs = as >> functions composed from simpler functions than to implement them with = sets >> based on membership, and SETL never caught on. >>=20 >> There have also been sporadic attempts to introduce set theory into = K-12 >> mathematics, under such rubrics as "New maths", starting with the >> operations of union and intersection, but these have not caught on = either. >>=20 >> Category theory is an abstract formulation of functions taking = composition >> as the primitive operation. Functions are in wide use in both = mathematics >> and software. Whenever a function calls another function from within = it, >> that's composition. >>=20 >> In recent years I've been promoting a viewpoint of algebra that = emphasizes >> the associativity of composition as the root of not just algebra but >> bialgebra in the sense of typed Chu spaces. The defining properties = of >> both homomorphisms and Chu transforms follow from associativity. My = most >> recent talk on this was at FMCS in June, the "ten" slides are here >> . (They're less cryptic when = the >> speaker is there to explain them, especially with an audience of only = one >> or two and with no time pressure.) >>=20 >> One might call this "category light" by virtue of working in the = *class* >> CAT, i.e. just categories, no functors etc. By Yoneda (unintended = pun >> there, it's actually "biYoneda") the functors are there, they're = just >> "under the hood". Just as you only need to know what sort of engine = is in >> the car you're driving if you have to service it, you only need to = know >> about functors, natural transformations, etc. when you become a = "category >> mechanic" so to speak. >>=20 >> Sets are automatic because they arise wherever morphisms gather = together in >> those spaces we call homsets. >>=20 >> Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]