Contents

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 $f^* : E_{b_2} \to E_{b_1}$.

## 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.

## Relation to distributive laws

Fibrations and opfibrations on a category $C$ (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads. If $C$ has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the bifibrations satisfying the Beck-Chevalley condition; see (von Glehn).

## Bifibration of model categories

In (CagneMelliès) the authors develop a notion of Quillen bifibration which combines the two concepts of Grothendieck bifibration and Quillen model structure, establishing conditions for when in a bifibration model structures on all the fibers and on the base combine to give one on the total category.

## Bifibration of bicategories

The basic theory of 2-fibrations, fibrations of bicategories, is developed in (Buckley). One (unpublished) proposal by Buckley and Shulman, for a “2-bifibration”, a bifibration of bicategories, is a functor of bicategories that is both a 2-fibration and a 2-opfibration. This should correspond to a pseudofunctor into $2 Cat_{adj}$, just as for 1-categories. There is a difference from this case, however, in that although a 2-fibration (corresponding to a “doubly contravariant” pseudofunctor $B^{coop} \to 2Cat$) has both cartesian lifts of 2-cells and 1-cells, a 2-opfibration (corresponding to a totally covariant pseudofunctor $B\to 2Cat$) has opcartesian lifts of 1-cells but still cartesian lifts of 2-cells.