category theory

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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

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