Date: Wed, 7 Jan 1998 12:59:50 -0400 (AST) Subject: Functor algebras 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 Date: Thu, 8 Jan 1998 16:29:05 -0400 (AST) Subject: Re: Functor algebras Date: Wed, 7 Jan 1998 23:06:24 -0500 (EST) From: Ernie Manes > > 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 > > Algebras of form FA -> GA were considered in some detail by the Prague school in the 1970s. Email Jiri Adamek for precise references. egm Date: Thu, 8 Jan 1998 16:29:46 -0400 (AST) Subject: Re: Functor algebras ---------- Forwarded message ---------- Date: Thu, 08 Jan 1998 11:52:54 MEZ From: Martin Hofmann To: cat-dist@mta.ca Subject: Re: Functor algebras Dear Uday, There has been an Edinburgh PhD thesis by Tatsuya Hagino on the subject of the se dialgebras. He defines a strongly normalising lambda calculus based on initial terminal dialgebras and also does some general theory. Hope this helps, Martin -- Martin Hofmann AG Logik und mathemat. Grundl. der Informatik Fachbereich Mathematik Technische Hochschule Darmstadt Schlossgartenstrasse 7 D-64289 Darmstadt Germany Tel. : x49-6151-16-3615 FAX : x49-6151-16-4011 e-mail: mh@mathematik.th-darmstadt.de WWW : http://www.mathematik.th-darmstadt.de/ags/ag14/mitglieder/hofmann-e.html Date: Mon, 12 Jan 1998 14:40:36 -0400 (AST) Subject: Re: Functor algebras 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