From: Peter Freyd <pjf@saul.cis.upenn.edu>
To: categories@mta.ca
Subject: *-Autonomous Functor Categories, revision
Date: Sat, 26 Nov 2005 07:50:40 -0500 (EST) [thread overview]
Message-ID: <200511261250.jAQCoetu005155@saul.cis.upenn.edu> (raw)
Mike Barr has pointed out that the proof in my last posting of
LEMMA: The object I = H^R is injective in *F*.
doesn't work. (It was actually the fourth proof I had come up with.
I wondered why it was so much simpler). So here's one that does work
(and is just about as simple).
Let
O
|
H^R
|
H^A --> H^B --> T --> O
be exact (all vertical arrows point down). We seek a retraction for
H^R --> T. Since H^R is projective (as is any representable) we may
choose a map H^R --> H^B to yield a commutative triangle. The full
subcategory of representables is closed under finite limits, so let
H^C --> H^R
| |
H^A --> H^B
be a pullback in *F* and let
B --> A
| |
R --> C
be the corresponding pushout in the category of f.p R-modules. The
map from H^C to T is the zero map and we use the hypothesis that
H^R --> T is monic to infer that H^C --> H^R, hence R --> C, are
zero maps. Let O --> K --> B --> A be exact. It is an exercise in
abelian categories that R --> C = 0 implies K --> B --> R is
epi. Now (finally using the projectivity of R) choose a retraction
R --> K. The map H^A --> H^B --> H^K --> H^R is of course, a zero
map and we may factor H^B --> H^K --> H^R as H^B --> T --> H^R.
The map T --> H^R is easily checked to be the retraction we seek.
reply other threads:[~2005-11-26 12:50 UTC|newest]
Thread overview: [no followups] expand[flat|nested] mbox.gz Atom feed
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=200511261250.jAQCoetu005155@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).