From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9993 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Bob Coecke Newsgroups: gmane.science.mathematics.categories Subject: Re: only_marketing_? Date: Wed, 21 Aug 2019 00:32:20 +0100 Message-ID: References: Reply-To: Bob Coecke Mime-Version: 1.0 (Mac OS X Mail 9.3 \(3124\)) Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable Injection-Info: blaine.gmane.org; posting-host="blaine.gmane.org:195.159.176.226"; logging-data="37526"; mail-complaints-to="usenet@blaine.gmane.org" To: Steve Vickers , categories , John Baez Original-X-From: majordomo@mlist.mta.ca Wed Aug 21 11:54:27 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 1i0NJz-0009cz-Gk for gsmc-categories@m.gmane.org; Wed, 21 Aug 2019 11:54:27 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39517) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1i0NJf-0005x3-6H; Wed, 21 Aug 2019 06:54:07 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1i0NIy-0003Al-Ru for categories-list@mlist.mta.ca; Wed, 21 Aug 2019 06:53:24 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9993 Archived-At: Hi Steve, John and everyone,=20 =46rom the perspective of quantum theory there of course is still a = clear input-output thing going on, as these morphisms are subject to the = causality of the world in which we live, captured by the symmetric = monoidal structure that caries that causal structure. Then the key = thing is not to be Cartesian i.e. the whole is not the sum of the parts. = Parts may even be pure noise while the whole is a sharp state. In that = context, compact closure is a beautifiul thing given that, just like = being Cartesian, it is more a property than it is a structure, due to = being `maximally wholistic=E2=80=99. Compactness seems to appear everywhere when you look around in the real = world, for example in language which is also compact closed if one = adopts Lambek=E2=80=99s pregroups for the grammar, and then attaching = meaning to things gives one a thick (non-symmetric) compact closed = category. Here, inputs and outputs truly become meaningless it seems. =20= =E2=80=94 As a pure category theory problem, there is still something interesting = to be done to better characterise compactness as a property. We made = some (admittedly very modest) baby steps here: https://arxiv.org/abs/1803.00708 https://arxiv.org/abs/1805.12088 I believe that compact closure deserves the same attention as the = Cartesian structures has gotten, if not more. The reason to say more is = that this case is further away from how me (are made to) think, so = having some maths in place is particularly useful. =E2=80=94 On a related note, in 2001 I was about to leave Montreal for Oxford, but = then 9/11 happened, and the planes didn=E2=80=99t go for a while. A = number of other people turned out to be in Montreal too those days, = including Bill Lawvere, and there were some talks at UDM (can=E2=80=99t = remember if this was because of being stuck, or planned). Bill Lawvere = explained then very nicely the `real world motivation=E2=80=99 for many = of the decissions he made for developing certain parts of Category = theory, mostly drawn from how to organise society . That was an = eye-opener. =20 One typically thinks of computer science as the main domain to drive = parts of the development of category theory from an applied perspective, = but some of the fathers of the field clearly also had other real world = motivations beyond pure mathematics, which makes perfect sense. = Category theory deserves to be about the real world, and hence play an = important role beyond pure maths, just like many other developments in = math like geometry and analysis. That=E2=80=99s why some of us started = with the dedicated ACT conference + school series. The thing to do now is not to just get category theory about the real = world, but also into the real world, and that=E2=80=99s not going to = happen by just writing papers. The investment hunger seems to be there = now, so let=E2=80=99s do it! Personally, I am going to give up part of = my academic position to do just that. Cheers, Bob. > On 20 Aug 2019, at 09:55, Steve Vickers = wrote: >=20 > Dear John, >=20 > Those are rather pertinent examples, as the dagger closed and = hypergraph categories show up a weakness in my question. >=20 > I asked about seeking objects, morphisms, identities and associative = composition, 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 composition is malformed unless it's domain with codomain. = This leads many of our category theoretic intuitions to be based on = thinking of objects and morphisms as being, at some level of = abstraction, like sets and functions. >=20 > Once you have set up the structure of what is input and what is = output, it takes some effort to forget it. Dagger closed and the = associated string diagrams provide a mechanism for doing that. >=20 > A good example is Rel. A morphism from X1 x ... x Xm to Y1 x ... x Yn = is just a subset of X1 x ... x Xm x Y1 x ... x Yn, in the light of which = it is perhaps perverse to impose domain and codomain structure - unless, = perhaps you want to carry on to say which relations are functional. >=20 > As you propose, this certainly looks like a good way to analyse = networks, and open systems where there is an interface between internal = structure and external behaviour, an interface along which we must = compose components. >=20 > I've heard Jamie Vicary and others use the word "compositionality" as = something not quite the same as category theory. Is this what they mean, = letting go of the strict domain-codomain discipline? >=20 > All the best, >=20 > Steve. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]