From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9994 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: Scott Morrison Newsgroups: gmane.science.mathematics.categories Subject: Re: only_marketing_? Date: Wed, 21 Aug 2019 13:30:22 +1000 Message-ID: References: Reply-To: Scott Morrison Mime-Version: 1.0 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="48014"; mail-complaints-to="usenet@blaine.gmane.org" Cc: John Baez , categories To: Steve Vickers Original-X-From: majordomo@mlist.mta.ca Wed Aug 21 11:56:58 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 1i0NMP-000CKv-QR for gsmc-categories@m.gmane.org; Wed, 21 Aug 2019 11:56:57 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:39537) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1i0NM6-0006HL-Kd; Wed, 21 Aug 2019 06:56:38 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1i0NLM-0003EY-1n for categories-list@mlist.mta.ca; Wed, 21 Aug 2019 06:55:52 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9994 Archived-At: (On the subject of "letting go of the strict domain-codomain discipline"...= ) In my work with Kevin Walker (e.g. https://arxiv.org/abs/1009.5025, particularly the disklike n-categories of \S 6) we talk about n-categories for which notions of input and output are solely "in the eye of the beholder", and argue that this is a convenient and natural language for the role of categories in TFT. As an application of this, in a very recent paper with Kevin and Paul Wedrich, https://arxiv.org/abs/1907.12194, we give what is arguably the "first interesting example" of a 4-category, built out of Khovanov homology. Formalising this gadget as a disklike 4-category, we can say enough to produce invariants of oriented 4-manifolds. Attempting instead to formalise this gadget as a "conventional" "domain-codomain" 4-category, we were much less satisfied --- we can check the axioms for a braided monoidal 2-category, but after that it's not particularly clear which duality properties one would need to check (in Lurie's language, perhaps this is working out what an SO(4)-fixed point structure actually is?) in order to continue on to building 4-manifold invariants. best regards, Scott On Wed, Aug 21, 2019 at 3:56 AM Steve Vickers w= rote: > > Dear John, > > Those are rather pertinent examples, as the dagger closed and hypergraph = categories show up a weakness in my question. > > I asked about seeking objects, morphisms, identities and associative comp= osition, which seems very natural because it's the basic definition of cate= gory. Everything has a domain and a codomain, an input and an output, and c= omposition is malformed unless it's domain with codomain. This leads many o= f our category theoretic intuitions to be based on thinking of objects and = morphisms 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, i= t takes some effort to forget it. Dagger closed and the associated string d= iagrams provide a mechanism for doing that. > > 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. > > As you propose, this certainly looks like a good way to analyse networks,= and open systems where there is an interface between internal structure an= d external behaviour, an interface along which we must compose components. > > I've heard Jamie Vicary and others use the word "compositionality" as som= ething not quite the same as category theory. Is this what they mean, letti= ng go of the strict domain-codomain discipline? > > All the best, > > Steve. > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]