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

• For $C$ any category with pullbacks, the codomain fibration $cod : [I,C] \to C$ is a bifibration.

• Dually, for $C$ any category with pushouts, the domain opfibration $dom : [I,C] \to C$ is a bifibration.

• The canonical functor Mod $\to$ CRing is a bifibration.