From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/239 Path: news.gmane.org!not-for-mail From: Zinovy Diskin Newsgroups: gmane.science.mathematics.categories Subject: Re: Where does the term monad come from? Date: Tue, 7 Apr 2009 12:50:21 -0400 Message-ID: Reply-To: Zinovy Diskin NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1239147398 13944 80.91.229.12 (7 Apr 2009 23:36:38 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 7 Apr 2009 23:36:38 +0000 (UTC) To: Steve Lack , categories@mta.ca Original-X-From: categories@mta.ca Wed Apr 08 01:37:57 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1LrKra-0005Wg-3p for gsmc-categories@m.gmane.org; Wed, 08 Apr 2009 01:37:54 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LrKK8-0003Fs-RZ for categories-list@mta.ca; Tue, 07 Apr 2009 20:03:20 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:239 Archived-At: On Fri, Apr 3, 2009 at 12:28 AM, Steve Lack wrote: > > Finitary monads can also be considered on other base categories than Set, > especially on locally finitely presentable ones. > > It is true that vector spaces are the algebras for a finitary monad on Set. > There is no need to restrict to finite-dimensional vector spaces; in fact it > is not true that there is a monad on Set whose algebras are the > finite-dimensional vector spaces. > there is something similar in algebraic logic. The class of locally finite cylindric/polyadic algebras is not a variety and the forgetful functor to Set is not monadic (l.f. means that all relations are of finite arities). In categorical logic (hyperdoctrines), these algebras are considered in many-sorted signatures, in fact, as algebras over graphs, and their theory becomes equational (= the corresponding forgetful functor to Graph is monadic). Probably, it's a general phenomenon wrt specifying finitary objects: by indexing them with finite sets (contexts, supports,arities), we get equational theories over graph-like structures. In a wider (and partly speculative) setting, the shift from classical algebraic to categorical logic is a shift from simple signatures and complex theories to complex signatures and simple theories. In a sense, this is what category theory does wherever it applies to classical problems: it greatly simplifies the logic (and the internal structure), but pays for this by a complex vocabulary (the external structure, interface). A typical example is classical vs. categorical set theories. Thus, a categorical model is a device with a structurally complex interface and simple internal logic. An average user prefers, of course, simple-looking interfaces of classical theories (and eventually has to pay for this choice but it happens later on...). So, for marketing categorical models, it's important to provide good manuals for their complicated interfaces -- what Vaughan just did for monads. Zinovy