Re: Functor algebras

Re: Functor algebras

[categories]

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

Re: Functor algebras

[categories]

Date: Mon, 12 Jan 1998 16:16:14 +0000
From: J Robin B Cockett <J.R.B.Cockett@dpmms.cam.ac.uk>

> 
> 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

Re: Functor algebras

[categories]

Date: Thu, 08 Jan 1998 11:52:54 MEZ
From: Martin Hofmann <mh@mathematik.tu-darmstadt.de>

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

Functor algebras

[categories]

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