From: "Stephen Lack" <S.Lack@uws.edu.au>
To: <categories@mta.ca>
Subject: RE: An autonomous category
Date: Wed, 15 Mar 2006 11:58:36 +1100 [thread overview]
Message-ID: <039A7CE5BC8F554C81732F2505D511720290EF4B@BONHAM.AD.UWS.EDU.AU> (raw)
Dear Marco,
This has been considered by Brian Day. He spoke about it in a talk
*-autonomous convolution
in the Australian Category Seminar on 5 March 1999,
You can also transform this via the log/exponential functions to an
additive tensor product on the extended (positive and negative) reals.
Regards,
Steve Lack.
-----Original Message-----
From: cat-dist@mta.ca on behalf of Marco Grandis
Sent: Tue 14/03/2006 12:45 AM
To: categories@mta.ca
Subject: categories: An autonomous category
The Lawvere category of extended positive real numbers has also an
autonomous structure, with a multiplicative tensor product (instead
of the original additive one). Has this been considered somewhere?
To be more explicit:
The well-known article of Lawvere on "Metric spaces..." (Rend. Milano
1974, republished in TAC Reprints n. 1) introduced the category of
extended positive real numbers, from 0 to oo (infinity included),
with arrows x \geq y, equipped with a strict symmetric monoidal
closed structure: the tensor product is the sum, the internal hom is
truncated difference (with oo - oo = 0).
Now, the same category can be equipped with a multiplicative tensor
product, x.y.
Provided we define 0.oo = oo (so that tensoring by any element
preserves the initial object oo), this is again a strict symmetric
monoidal closed structure, with hom(y, z) = z/y. Now, the
'undetermined forms' 0/0 and oo/oo are defined to be 0.
The new multiplicative structure is even *-autonomous, with
involution x* = 1/x (and 'nearly' compact).
(Note that this choice of values of the undetermined forms comes from
privileging the direction x \geq y, which is necessary if we want
to view metric spaces, normed categories etc. as enriched categories).
Marco Grandis
next reply other threads:[~2006-03-15 0:58 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2006-03-15 0:58 Stephen Lack [this message]
-- strict thread matches above, loose matches on Subject: below --
2006-03-13 13:45 Marco Grandis
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=039A7CE5BC8F554C81732F2505D511720290EF4B@BONHAM.AD.UWS.EDU.AU \
--to=s.lack@uws.edu.au \
--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).