Some properties of Fib as a fibred 2-category

AbstractWe consider some basic properties of the 2-category Fib of fibrations over arbitrary bases, exploiting the fact that it is fibred over Cat. We show a factorisation property for adjunctions in Fib, which has direct consequences for fibrations, e.g. a characterisation of limits and colimits for them. We also consider oplax colimits in Fib, with the construction of Kleisli objects as a particular example. All our constructions are based on an elementary characterisation of Fib as a 2-fibration