Re: Composition of Fibrations and Quantification

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

Adding to Steve's most appropriate comments.
One also wants to have the so-called Beck-Chevalley conditions.

Recall that a fibration has internal sums iff it is a bifibration
where cocartesian arrows are stable under pullbacks along cartesian arrows.
This property is easily seen to be preserved by composition.

But P has internal products iff P^op has internal sums. However, we
don't have (P \circ Q)^op = P^op \circ Q^op in particular because
since the right composite doesn't exist. (If P is a fibration over BB
then P^op still is a fibration over BB and not over BB^op).
Still I guess one can check directly that composition preserves the
property of having small products. Maybe it's even in Bart Jacob's book?

Thomas



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