Re: Is the category of group actions LCCC?

[Richard Garner <richard.garner@mq.edu.au>]

It seems that the following is in fact true:

Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is
cartesian closed with exponents stable under pullback, and Pi's exist
along product projections (and satisfy BCC), then E is cartesian closed.

The product of (a, phi) with (b, psi) in E is of course (a x b,
pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the
product projections in B.

The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi),
ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and 
ev: [b,c] x b ----> c in B.

This in particular applies to Cat//’Set’ as in Ross' message, seen as a
fibration over Cat with reindexing along f:A--->B given by
[f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to
pullbacks, but they don't satisfy BCC; however, right adjoints to
pullback along product projections are given just by (conical) limit
functors, and these do satisfy BCC. So the preceding construction
applies (and a bit of fiddling about shows that this does indeed agree
with Ross' prescription).

As for local cartesian closure: if B is lccc, each fibre is lccc with
fibrewise Pi's stable under pullback, and E--->B has all products, then
it seems that each slice fibration p/A: E/A--->B/pA will satisfy the
conditions in the second paragraph, whence E is also lccc.

Richard

On Thu, Sep 4, 2014, at 10:19 AM, Ross Street wrote:
> 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
> 

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]