Re: Is the category of group actions LCCC?

["Fred E.J. Linton" <fejlinton@usa.net>]

On Tue, 02 Sep 2014 08:28:21 PM EDT, Timothy Revell
<timothy.revell@strath.ac.uk> asked:
  
> I'm wondering whether the category of ALL group actions is locally
> Cartesian closed. ... 

Borrowing ideas from several earlier followups to this question, I might 
suggest the following as a quick and easy argument for a negative answer:

1) an initial object in that "category of ALL group actions" is given
by the trivial group's action on the empty set, ({e}, 0);

2) the slice of that category over ({e}, 0) "is" just the category Grp
of groups and group homomorphisms (well, really, it's the full subcategory
of ALL group actions on the empty set :-) but that's essentially just Grp);

3) the category of group actions is no more a LCCC than Grp is a CCC.

Or have I overlooked some obvious fly in my proposed ointment :-) ?

Cheers, -- Fred Linton




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