$E$ 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)

Last revised on June 29, 2018 at 14:16:06.
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