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