category theory

# Contents

## Definition

A bifibration of categories is a functor

$E \to B$

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

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

## Examples

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.

Revised on September 23, 2014 23:39:31 by Hew Wolff? (4.35.166.120)