Re: Is the category of group actions LCCC?

[Ross Street <ross.street@mq.edu.au>]

On 1 Sep 2014, at 7:12 pm, Timothy Revell <timothy.revell@strath.ac.uk> wrote:

> I'm wondering whether the category of ALL group actions is locally
> Cartesian closed. 

This is what I answered Timothy:
======
No, it’s not.
Since the category has a terminal object (1,1), being a LCCC would imply it
was cartesian closed. However, that would imply (G,X) \times — preserved
the initial object (1,0), which is false: (G,X)\times (1,0) = (G,0).
======

But it seems there is more to the story. 
The thing stopping the category of actions from
being cartesian closed is that the category Gp of groups is not. However,
the category Gpd of groupoids and the category Cat of categories are.
The (2-)category Cat//’Set’ of all category actions is defined as follows:
objects (A,F) are functors F : A —> Set and morphisms (f,t) : (A,F) —>  (B,G)
are functors f : A —> B with natural transformation t : F ==> G f.
This (2-)category is cartesian closed: the internal hom [(B,G),(C,H)] is
([B,C], K) where [B,C] is the functor category and K(g) = [B,Set](G, H g).

However Cat//’Set’ is not locally cartesian closed basically because Cat
is not. It is not even locally cartesian closed as a bicategory.
The 2-category Gpd is cartesian closed; it is not locally cartesian closed;
it is locally cartesian closed as a bicategory. 

Similarly, Gpd//’Set’ is locally cartesian closed as a bicategory.
Often, in dealing with groups, we find groupoids help.
This case is a good example and I hope helps in the applications
you have in mind, Timothy.

Ross



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