Some thoughts: * The result about composition of fibrations holds in any 2-category with comma objects and 2-pullbacks, not just Cat. (Think of the Chevalley criterion for fibrations.) * By duality on 2-cells it thus also applies to opfibrations, and hence to bifibrations. * It is bifibration structure that gives you the left adjoints you ask for. * For the right adjoints, look at the dual 2-category, where your fibrations become bifibrations. Hence it seems to me that your conjectures are all true, and even generalize widely. Steve.
On 6 Jun 2014, at 10:47, Neil Ghani <neil.ghani@strath.ac.uk> wrote:
Dear All
We know that if p and q are fibrations, then their composition p.q is a fibration.
But what about quantification … that is if reindexing along every morphism has a right/left adjoint in p and q, then does reindexing along every morphism in p.q have a right/left adjoint? Under some circumstances?
Thanks for any thoughts Neil
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