A bifibration of categories is a functor

EB E \to B

that is both a Grothendieck fibration as well as an opfibration.

Therefore every morphism f:b 1b 2f : b_1 \to b_2 in a bifibration has both a push-forward f *:E b 1E b 2f_* : E_{b_1} \to E_{b_2} as well as a pullback f *:E b 2E b 1f^* : E_{b_2} \to E_{b_1}.


Relation to monadic descent

Ordinary Grothendieck fibrations correspond to pseudofunctors to Cat, by the Grothendieck construction and hence to prestacks. For these one may consider descent.

If the fibration is even a bifibration, there is a particularly elegant algebraic way to encode its descent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.


A bifibration F:EBF:E\to B such that F op:E opBF^{op}:E^{op}\to B is a bifibration as well is called a trifibration (cf. Pavlović 1990, p.315).


Revised on May 20, 2015 09:12:56 by Urs Schreiber (