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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