* Re: Functor algebras
@ 1998-01-08 20:29 categories
0 siblings, 0 replies; 4+ messages in thread
From: categories @ 1998-01-08 20:29 UTC (permalink / raw)
To: 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
^ permalink raw reply [flat|nested] 4+ messages in thread
* Re: Functor algebras
@ 1998-01-12 18:40 categories
0 siblings, 0 replies; 4+ messages in thread
From: categories @ 1998-01-12 18:40 UTC (permalink / raw)
To: 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
^ permalink raw reply [flat|nested] 4+ messages in thread
* Re: Functor algebras
@ 1998-01-08 20:28 categories
0 siblings, 0 replies; 4+ messages in thread
From: categories @ 1998-01-08 20:28 UTC (permalink / raw)
To: 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
^ permalink raw reply [flat|nested] 4+ messages in thread
* Functor algebras
@ 1998-01-07 16:59 categories
0 siblings, 0 replies; 4+ messages in thread
From: categories @ 1998-01-07 16:59 UTC (permalink / raw)
To: 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
^ permalink raw reply [flat|nested] 4+ messages in thread
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