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 <A, f:FA->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: Tue, 06 Jan 1998 17:26:17 -0600 From: Uday S Reddy <reddy@cs.uiuc.edu>
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 <A, f:FA->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
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: Tue, 06 Jan 1998 17:26:17 -0600 From: Uday S Reddy <reddy@cs.uiuc.edu>
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 <A, f:FA->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 <A, f:FA->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
participants (4)
-
Ernie Manes -
J Robin B Cockett -
Martin Hofmann -
Uday S Reddy