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/ ]