From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9343 Path: news.gmane.org!.POSTED!not-for-mail From: Alexander Kurz Newsgroups: gmane.science.mathematics.categories Subject: the dual category Date: Thu, 14 Sep 2017 15:53:21 +0100 Message-ID: References: Reply-To: Alexander Kurz NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 (Mac OS X Mail 10.3 \(3273\)) Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1505483457 29701 195.159.176.226 (15 Sep 2017 13:50:57 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 15 Sep 2017 13:50:57 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri Sep 15 15:50:53 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1dsr1B-0007Zj-83 for gsmc-categories@m.gmane.org; Fri, 15 Sep 2017 15:50:53 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:40361) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1dsr1v-0000Sy-6X; Fri, 15 Sep 2017 10:51:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1dsr0G-0001NQ-L9 for categories-list@mlist.mta.ca; Fri, 15 Sep 2017 10:49:56 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9343 Archived-At: I would like to add another example to Eduardo=E2=80=99s. In computer science both algebras and coalgebras for an endofunctor on = sets are useful structures and both initial algebras and final = coalgebras play an important role in the semantics of programming = languages. It is now an important feature that algebras and coalgebras over set are = not dual to each other. Only the invention of the dual category reveals = the underlying duality. The ensuing tension between `abstract=E2=80=99 duality and `concrete=E2=80= =99 non-duality is certainly one reason why the study of set-coalgebras = is fascinating.=20 For example, whereas it is well-known that the initial sequence of a = finitary set-endofunctor converges in omega steps, a result by Worrell = shows that the final sequence of a finitary set-endofunctor converges in = omega+omega steps. Best wishes, Alexander > On 12 Sep 2017, at 17:09, Eduardo J. Dubuc wrote: >=20 > On 11/09/17 13:19, Joyal, Andr? wrote: >> Dear John, and category theorists, >>=20 >> The fact that every category has an opposite introduces >> a symmetry in mathematics that would not be there otherwise. >> >> The category of sets is not self dual, but a disjoint union of sets >> is a coproduct, dual to a product. >>=20 >> Thurston does not show esteem for logic. >> Most mathematicians are taking logic for granted; they just use it >> as a part of their natural language. >> It is obvious that human understanding depends on the >> the laws of thought, on logic. >> In a sense, category theory is a branch of mathematical logic, >> since it greatly improves mathematical thinking in general. >> A category theorist might say (not too loudly) that mathematical = logic >> is a branch of category theory. >>=20 >> Best, >> andr? >>=20 >=20 > The opposite category (*) may look a senseless obscurity and make some > people nauseous, but it seems to me it made an important contribution = to > the understanding of mathematics. It took a long time to form part of > mathematical thinking (and still is). For example, Bourbaki treatment = of > limits (of sets say) define and develops basic properties of = projective > limits, including the universal property. Later does the same for > inductive limits, and includes a proof of the dual statements !!. He = had > to do so since it had not incorporated categories and the opposite > category. He states what a universal property is, but can not state = that > the respective universal properties (for limits and colimits) are one > the dual of the other. >=20 > (*) The axioms of a category are self dual. Another examples are = abelian > categories, and a very subtle one, namely, Quillen's model categories. >=20 > Many categories are not self dual, and this is underneath the duality > between algebra and geometry. >=20 > best e.d. >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]