Re: Is the category of group actions LCCC?

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

A small correction. As well as:

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 following stronger statement is true (weakening the stability
required of the exponents):

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

In order to capture Ross' example, this stronger form is needed, since
[f,1]:[B,Set] ---> [A,Set] does not in general preserve exponentials,
while [pi_2,1]:[B,Set] ---> [A*B,Set] does so.

Richard


> 
> 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
> 

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