* Pullback perserving functor
@ 1999-01-29 5:13 Hongseok Yang
1999-01-29 14:51 ` F W Lawvere
1999-01-29 15:35 ` Pullback preserving functor Dr. P.T. Johnstone
0 siblings, 2 replies; 3+ messages in thread
From: Hongseok Yang @ 1999-01-29 5:13 UTC (permalink / raw)
To: categories
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
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: Pullback perserving functor
1999-01-29 5:13 Pullback perserving functor Hongseok Yang
@ 1999-01-29 14:51 ` F W Lawvere
1999-01-29 15:35 ` Pullback preserving functor Dr. P.T. Johnstone
1 sibling, 0 replies; 3+ messages in thread
From: F W Lawvere @ 1999-01-29 14:51 UTC (permalink / raw)
To: categories
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.
*******************************************************************************
F. William Lawvere Mathematics Dept. SUNY
wlawvere@acsu.buffalo.edu 106 Diefendorf Hall
716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA
*******************************************************************************
On Thu, 28 Jan 1999, Hongseok Yang wrote:
>
> 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
>
>
>
^ permalink raw reply [flat|nested] 3+ messages in thread
* Re: Pullback preserving functor
1999-01-29 5:13 Pullback perserving functor Hongseok Yang
1999-01-29 14:51 ` F W Lawvere
@ 1999-01-29 15:35 ` Dr. P.T. Johnstone
1 sibling, 0 replies; 3+ messages in thread
From: Dr. P.T. Johnstone @ 1999-01-29 15:35 UTC (permalink / raw)
To: categories
> 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
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