From: Robin Houston <r.houston@cs.man.ac.uk>
To: Categories list <categories@mta.ca>
Subject: Re: On defining *-autonomous categories
Date: Tue, 18 Dec 2007 14:27:37 +0000 [thread overview]
Message-ID: <E1J4o1H-0002cz-P2@mailserv.mta.ca> (raw)
On Tue, Dec 18, 2007 at 01:08:07PM +0000, Robin Houston wrote:
> Lemma 2. Let C be a category, and F: C -> C an endofunctor. If FF
> is naturally isomorphic to 1_C, then F is an equivalence.
>
> Proof: Certainly F is essentially surjective, since every object
> X in C is naturally isomorphic to FFX. It then follows by Lemma 1
> that F is full and faithful.
>
>
> For the implication 1 => 2, take F = (- -o D) in Lemma 2.
There is a silly mistake here, caused by the fact that the functor
(- -o D) is contravariant. The error is really in the statement of
Lemma 2; of course the proof still works for contravariant F.
Robin
next reply other threads:[~2007-12-18 14:27 UTC|newest]
Thread overview: 5+ messages / expand[flat|nested] mbox.gz Atom feed top
2007-12-18 14:27 Robin Houston [this message]
-- strict thread matches above, loose matches on Subject: below --
2007-12-19 12:04 Prof. Peter Johnstone
2007-12-18 13:08 Robin Houston
2007-12-17 18:29 Peter Selinger
2007-12-17 2:45 Michael Barr
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