From: <laurent.mehats@gmail.com>
To: categories@mta.ca
Subject: Re: Conditions for adjoints
Date: Sun, 25 Oct 2009 10:48:39 +0100 [thread overview]
Message-ID: <E1N24Wa-0004Lt-N8@mailserv.mta.ca> (raw)
robin@ucalgary.ca a écrit :
> (Apologies to those who received the earlier type mashed version ...)
>
> Jeremy Dawson and I were discusing whether one can express the conditions
> for an adjoint without requiring functors ... this is what we came up
> with:
>
>
> There is an adjoint between two categories if and only if
> there are object functions F and G (not functors) and
> for each X in \X and Y in \Y there are functions:
>
> #: \X(X,G(Y)) -> \Y(F(X),Y) ---- sharp
> @: \Y(F(X),Y) -> \X(X,G(Y)) ---- flat
>
> between the homsets such that
> (1) @(#(1)) = 1 and dually #(@(1)) = 1 (inverse on identities)
> (2) @(1) @(#(1) #(f)) = f and dually #(@(g) @(1)) #(1) = g
> (3) @(#(f @(1)) h k) = f @(h) @(#(1) k)
> and dually
> #(x y @(#(1) z)) = #(x @(1)) #(y) z.
>
> I find it hard to believe that such conditions have not been recorded.
> Does anyone have a reference or similar conditions which do not require
> functors?
>
> -robin
Hello,
Such conditions are discussed in detail in:
Kosta Došen, Cut Elimination in Categories, Trends in Logic 6, Kluwer, 1999.
Those you mention already appear on p. 258 of:
Kosta Došen, Deductive Completeness, Bull. Symbolic Logic Volume 2, Number
3 (1996), 243-283.
(http://www.math.ucla.edu/~asl/bsl/0203/0203-001.ps).
Regards,
Laurent Méhats
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
next reply other threads:[~2009-10-25 9:48 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
2009-10-25 9:48 laurent.mehats [this message]
-- strict thread matches above, loose matches on Subject: below --
2009-10-24 19:51 robin
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=E1N24Wa-0004Lt-N8@mailserv.mta.ca \
--to=laurent.mehats@gmail.com \
--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).