Subject: categories: Pullback perserving functor

From: Hongseok Yang <[email-masked]>
Date: Thu, 28 Jan 1999 23:13:03 -0600 (CST)

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?)

Thanks,
Hongseok

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From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Fri, 29 Jan 1999 15:35:47 +0000 (GMT)

> 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

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From: F W Lawvere <[email-masked]>
Date: Fri, 29 Jan 1999 09:51:13 -0500 (EST)

Whether or not these functors form a cartesian-closed category
depends strongly on the nature of the domain category.  For example,
if the domain category is an abelian category as opposed to it
being a pretopos.  Related matters are discussed in the recent
paper by  Borceux and Pedicchio and the papers there cited:
Left-exact presheaves on a small pretopos, Journal of Pure and
Applied Algebra, vol. 135, no. 1, 4 Febr. 1999, pp 9 - 22.

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F. William Lawvere			Mathematics Dept. SUNY 
[email-masked]               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

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