From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4018 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: week257 Date: Mon, 15 Oct 2007 17:07:53 -0700 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: ger.gmane.org 1241019668 11352 80.91.229.2 (29 Apr 2009 15:41:08 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:41:08 +0000 (UTC) To: categories Original-X-From: rrosebru@mta.ca Mon Oct 15 21:44:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Oct 2007 21:44:10 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhaLb-0005zj-Qc for categories-list@mta.ca; Mon, 15 Oct 2007 21:31:47 -0300 Content-Disposition: inline User-Agent: Mutt/1.4.2.1i Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 75 Original-Lines: 69 Xref: news.gmane.org gmane.science.mathematics.categories:4018 Archived-At: Dear Categorists - I made some mistakes in my account of Cheng's work, saying "monad on a category" at some points where I should have said "monad in a 2-category". Here's a fixed version: Street noted that we can talk about monads, not just in the 2-category of categories, but in any 2-category. I actually explained monads at this level of generality back in "week89". Indeed, for any 2-category C, there's a 2-category Mnd(C) of monads in C. And, he noted that a monad in Mnd(C) is a pair of monads in C related by a distributive law! That's already mindbogglingly beautiful. According to Eugenia, it's practically the last sentence of Street's paper. But in her new work: 18) Eugenia Cheng, Iterated distributive laws, available as arXiv:0710.1120. she goes a bit further: she considers monads in Mnd(Mnd(C)), and so on. Here's the punchline, at least for today: she shows that a monad in Mnd(Mnd(C)) is a triple of monads F, G, H related by distributive laws satisfying the Yang-Baxter equation: \F G/ |H F| G\ /H \ / | | \ / / | | / / \ | | / \ / \ | \ / \ | \ / \ / | | / = / | | / \ / \ | | / \ / \ | \ / | | \ / \ / | | \ / / | | / / \ | | / \ /H \G |F H| G/ \F This is also just what you need to make the composite FGH into a monad! By the way, the pathetic piece of ASCII art above is lifted from "week1", where I first explained the Yang-Baxter equation. That was back in 1993. So, it's only taken me 14 years to learn that you can derive this equation from considering monads in the category of monads in the category of monads in a 2-category. Also, I should have given a reference to earlier work on Gelfand duality in a topos: Bernhard Banachewski and Christopher J. Mulvey, A globalisation of the Gelfand duality theorem, Ann. Pure Appl. Logic 137 (2006), 62-103. Also available at http://www.maths.sussex.ac.uk/Staff/CJM/research/pdf/globgelf.pdf They show that any commutative C*-algebra A in a Grothendieck topos is canonically isomorphic to the C*-algebra of continuous complex functions on the compact, completely regular locale that is its maximal spectrum (that is, the space of homomorphisms f: A -> C). Conversely, they show any compact completely regular locale X gives a commutative C*-algebra consisting of continuous complex functions on X.