Re: Functor algebras

[categories <cat-dist@mta.ca>]

Date: Wed, 7 Jan 1998 23:06:24 -0500 (EST)
From: Ernie Manes <manes@math.umass.edu>

> 
> 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