18 Dec
2007
18 Dec
'07
2:27 p.m.
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