From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: Re: Functor algebras
Date: Mon, 12 Jan 1998 14:40:36 -0400 (AST) [thread overview]
Message-ID: <Pine.OSF.3.90.980112144030.29842P-100000@mailserv.mta.ca> (raw)
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
next reply other threads:[~1998-01-12 18:40 UTC|newest]
Thread overview: 4+ messages / expand[flat|nested] mbox.gz Atom feed top
1998-01-12 18:40 categories [this message]
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1998-01-08 20:29 categories
1998-01-08 20:28 categories
1998-01-07 16:59 categories
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