*-Autonomous Functor Categories, revision

*-Autonomous Functor Categories, revision

[Peter Freyd]

  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.