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From: "Dr. P.T. Johnstone"
Newsgroups: gmane.science.mathematics.categories
Subject: Re: Pullback preserving functor
Date: Fri, 29 Jan 1999 15:35:47 +0000 (GMT)
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> 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