nLab trifibration

Idea

$E$ is a trifibration if both $E \to B$ and $E^{d} \to B$ are bifibrations, where the category $E^{d}$, fibered over $B$, is the dual fibration obtained by changing the direction of all the vertical arrows in $E$. The arrows of $E^{d}$ are the equivalence classes of spans with a vertical arrow pointing at the source, and a cartesian arrow pointing at the target (Pavlovic90, p. 315).

A fibration $E \to B$ is a bifibration if and only if for every $t: I \to J$ in $B$, the inverse image functor $t^{\ast}: E_{J} \to E_{I}$ has a left adjoint $t_{!}: E_{I} \to E_{J}$. It is a trifibration if and only if there is also a right adjoint $t_{\ast}: E_{I} \to E_{J}$.

Since a trifibration is not a fibration in three ways as its name suggests, alternative terminology has been used, including $\ast$-bifibration in (FBMF).

References

• Duško Pavlović, Categorical Interpolation: Descent and the Beck-Chevalley Condition without Direct Images , pp.306-325 in Category theory Como 1990, LNM 1488 Springer Heidelberg 1991. (pdf)

• Mike Shulman, Framed bicategories and monoidal fibrations, Theory and Applications of Categories, Vol. 20, No. 18, 2008, pp. 650–738, (tac)