Re: Is the category of group actions LCCC

[pjf <pjf@seas.upenn.edu>]

On page	-15 (yes, a negative page number) of the TAC reprinting of
Abelian Categories (in the 2003 Forward) I wrote:

    The very large category [described] in Exercise 6-A -- with a few
    variations -- has been a great source of counterexamples over the
    years....In its category of abelian-group objects  Ext(A,B)  is a
    proper class iff there???s a non-zero group homomorphism from  A to  B
    (it needn???t respect the actions) hence the only injective object is
    the zero object (which settled a once-open problem about whether
    there are enough injectives in the category of abelian groups in
    every elementary topos with natural-numbers object.
      http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf


On 2014-09-01 05:12, Timothy Revell wrote:
> Dear All,
>
> I'm wondering whether the category of ALL group actions is locally
> Cartesian closed. This is NOT the functor category [G,Set] for some
> category G with one object, since we allow G to vary. To be more
> specific the category is as follows.
>
>   - The objects are pairs (G,X), where G is a group and X is a G-Set.
>
>   - A morphism (G,X) -> (G', X') is given by a pair (h,f), where
> h:G->G'
> is a group homomorphism and f: X -> X' is a function (a morphism in
> Set)
> such that for all  g in G, x in X
>
>     h(g) * f(x) = f(g * x)
>
> where * on the left denotes the group action of G' on X' and * on the
> right denotes the group action of G on X.
>
>
> All the best,
> Tim


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