Re: Pullback preserving functor

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

> Would someone let me know the answer and the proof or counter example of
> the following question?
>
> Suppose the category C has a pullback for every pair of morphism
> (f : X -> Y, g : W -> Y). Let K be the full subcategory of the functor
> category Func(C,Set) whose objects are pullback perserving functors.
> Is K ccc? (If so, how I can show this?)


The answer is no. First note that K is closed under products in the
functor category. Also, it contains all the representable functors; 
so, if it were cartesian closed, the exponential G^F would have to be 
given by 

G^F(c) \cong nat((c,-),G^F) \cong nat((c,-)\times F,G)

i.e. K would have to be closed under exponentials in [C,Set]. However,
it isn't in general. For a simple counterexample, let C be the category
with five objects a,b,c,d,e and six non-identity morphisms

a --> b, a --> c, b --> d, c --> d, a --> d, a --> e ;

note that C has just one nontrivial pullback square

            a -----> b
            |        |
            |        |
            v        v
            c -----> d

Let F be the functor given by F(a) = F(b) = F(c) = F(d) = \emptyset,
F(e) = {*}, and let G be F + F. Then (taking the above definition of G^F)
G^F(a) has two elements, but G^F(b), G^F(c) and G^F(d) are singletons.

Peter Johnstone