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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