Re: Is the category of group actions LCCC?

[Claudio Hermida <claudio.hermida@gmail.com>]

On 2014-09-04, 10:05 PM, Richard Garner wrote:
>  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 is indeed the case and it appears as Corollary 4.12 in

Claudio Hermida, Some properties of Fib as a fibred 2-category, Journal of Pure 
and Applied Algebra, Volume 134, Issue 1, 5 January 1999, Pages 83-109, ISSN 
0022-4049, http://dx.doi.org/10.1016/S0022-4049(97)00129-1.
(http://www.sciencedirect.com/science/article/pii/S0022404997001291)


>
>  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.

That is also correct, but Cat is not lccc. To get this to work, one must
restrict Cat to the broad subcategory whose morphisms satisfy the
Conduche condition (which is the same as exponentiability in Cat), as
exposed in the nLab page

http://nlab.mathforge.org/nlab/show/Conduche+functor

Claudio



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