Subject: categories: question on "model functor"

From: Lutz Schroeder <[email-masked]>
Date: Fri, 15 Sep 2000 15:13:52 +0200

The following question looks so natural that somebody's
bound to have looked into it:

Does the functor

Cat^op ---> CAT

A  |-->   [A,Set]

reflect isomorphisms (more generally: limits)?

related questions (i.e. ones that would solve the problem or
at least help solving it) are

-- Are epimorphisms in Cat those functors F:A-->B for which
   F[A] generates B by composites and inverses of isomorphisms (in B)?
-- is the source of all covariant hom-functors an extremal mono-source
   in CAT?

The answers are probably in the literature - but where?

Thanks a lot,

Lutz Schroeder
-- 
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Lutz Schroeder                  Phone +49-421-218-4683
Dept. of Computer Science       Fax +49-421-218-3550
University of Bremen           
[email-masked]           
P.O.Box 330440, D-28334 Bremen 
http://www.informatik.uni-bremen.de/~lschrode
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From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Fri, 15 Sep 2000 18:08:14 +0100 (BST)

> reflect isomorphisms (more generally: limits)?

The question is not very well posed: since Cat and CAT are 2-categories,
one ought to be asking about whether it reflects equivalences. For this,
the answer is negative (and well known): a functor A --> B induces an
equivalence [B,Set] --> [A,Set] iff it induces an equivalence between the
idempotent-completions of A and B. So the inclusion of any non-idempotent-
complete category in its idempotent-completion provides a counterexample.

However, if you insist on asking about isomorphisms rather than 
equivalences, the answer is yes. It's easy to see that if F: A --> B
fails to be surjective (resp. injective) on objects then the induced
functor [F,Set] fails to be injective (resp. surjective); so if [F,Set]
is an isomorphism then F must be bijective on objects, and this combined
with inducing an equivalence of idempotent-completions is enough to make it
an isomorphism.

But this is not a very meaningful result. Provided you assume a sufficiently
powerful form of the axiom of choice, [A,Set] and {B,Set] will be isomorphic
whenever they are equivalent (since each has a proper class of objects in
each isomorphism class, except for the initial object which is unique in
its isomorphism class). The isomorphism will not, of course, be induced by
a functor from A to B; but it will be naturally isomorphic to a functor that
is (at least provided B is idempotent-complete).

Peter Johnstone