From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/586 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: Functor algebras Date: Mon, 12 Jan 1998 14:40:36 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241017056 26377 80.91.229.2 (29 Apr 2009 14:57:36 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:57:36 +0000 (UTC) To: categories Original-X-From: cat-dist Mon Jan 12 14:40:39 1998 Original-Received: (from cat-dist@localhost) by mailserv.mta.ca (8.8.8/8.8.8) id OAA12734; Mon, 12 Jan 1998 14:40:36 -0400 (AST) Original-Lines: 48 Xref: news.gmane.org gmane.science.mathematics.categories:586 Archived-At: Date: Mon, 12 Jan 1998 16:16:14 +0000 From: J Robin B Cockett > > Date: Tue, 06 Jan 1998 17:26:17 -0600 > From: Uday S Reddy > > Happy New Year, everyone. > > I have been wondering about a little question. Category theory texts > talk about "algebras" for an endofunctor, which are arrows of type FA -> > A, and dually coalgebras A -> GA. I am interested in the symmetric > case, arrows of type FA -> GA for endofunctors F and G. > Have such structures been studied? > > This is only scratching the surface. One can ask for a family of such > arrows for an algebra. One can consider functors F,G: C -> D between > different categories leading to algebras of the form GA> where > A is an object of C, and f an arrow in D, and so on. I am also > interested in the "diagonal" case, arrows of type FAA -> GAA where F and > G are functors C^op x C -> C. (Note that all these structures have a > "natural" notion of homomorphisms.) > > I would appreciate any pointers to the literature. > > Uday Reddy The category with objects GA> and evident maps is sometimes called an inserter. It is a weighted limit - a sort of "lax equalizer" of the two functors F and G: it may be written as F//G to distinguish it from the comma category (which is written F/G). It is used in the construction of datatypes (Hagino's thesis - as mentioned earlier - see also Dwight Spencer and my paper "Strong categorical datatypes II" TCS 139 (1995) 69-113 and its predecessor). Furthermore, one can express the parametricity properties of combinators and modules using these categories (see Peter Vesely's MSc thesis on the Charity site (http:/www.cpsc.ucalgary.ca/projects/charity/home.html) and Maarten Fokkinga's thesis - and paper in a recent MSCS issue - where I believe he uses the term "transformer" rather than combinator). I recently gave a working presentation to IFIP 2.1 entitled a "A reminder on inserters" ... this because I felt the connection to datatypes and the software structuring and parametricity ramifications of this seemingly innocuous limit had still not been sufficiently recognized or exploited. Robin Cockett