Re: Is the category of group actions LCCC?

[Steve Lack <steve.lack@mq.edu.au>]

Dear Tim,

Write Act for the category you mention. Then Act/(G,0) is clearly equivalent to Grp/G.
Since Grp/G is not cartesian closed, neither is Act/(G,0).

Regards,

Steve Lack.

On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk> 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
> 
> 
> -- 
> Timothy Revell,
> Department of Computer and Information Sciences,
> University of Strathclyde.
> The University of Strathclyde is a charitable body, registered in
> Scotland, with registration number SC015263.
> 


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