From: Peter Freyd <pjf@saul.cis.upenn.edu>
To: categories@mta.ca
Subject: Real midpoints
Date: Sun, 26 Dec 1999 13:45:08 -0500 (EST) [thread overview]
Message-ID: <199912261845.NAA19441@saul.cis.upenn.edu> (raw)
It could well be that Vaughan and I are defining the midpoint structure
in the same way. Here's how I described it (using the conventions from
my last posting).
Let F:I --> I v I be a final coalgebra. We will denote the top of I
as T and the bottom as B. Construct the "halving" map, h:I --> I,
(on [-1,1] it will send x to x/2) as:
T v F v B F'v F' F'
I --> 1 v I v 1 ------> I v I v I v I ---> I v I --> I
where F' denotes the inverse of F, and, by a little overloading, T
and B denote the maps constantly equal to T and B. The leftmost
map records the fact that the terminator is a unit for the
ordered-wedge functor.
Let g be the endo-function on I x I defined recursively by:
g<x,y> = if dx = T and dy = T then <x,y> else
if dx = T and uy = B then h(g(dx,uy>) else
if ux = B and dy = T then h(g<ux,dy>) else
if ux = B and uy = B then <x,y>.
The values of g lie in the first and third quadrants, that is, those
points such that either dx = dy = T or ux = uy = B. The two maps
g d x d g u x u
I x I --> I x I --> I x I and I x I --> I x I --> I x I
give a coalgebra structure on I x I. The midpoint operation may be
defined as the induced map to the final coalgebra.
next reply other threads:[~1999-12-26 18:45 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
1999-12-26 18:45 Peter Freyd [this message]
1999-12-29 8:03 ` Vaughan Pratt
2000-01-17 1:23 ` Peter Selinger
2000-01-18 4:07 ` Dusko Pavlovic
2000-01-18 16:16 ` Peter Selinger
2000-01-19 20:23 ` Dusko Pavlovic
Reply instructions:
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
https://en.wikipedia.org/wiki/Posting_style#Interleaved_style
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
--in-reply-to=199912261845.NAA19441@saul.cis.upenn.edu \
--to=pjf@saul.cis.upenn.edu \
--cc=categories@mta.ca \
/path/to/YOUR_REPLY
https://kernel.org/pub/software/scm/git/docs/git-send-email.html
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).