EE is a trifibration if both EBE \to B and E dBE^{d} \to B are bifibrations, where the category E dE^{d}, fibered over BB, is the dual fibration obtained by changing the direction of all the vertical arrows in EE. The arrows of E dE^{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 EBE \to B is a bifibration if and only if for every t:IJt: I \to J in BB, the inverse image functor t *:E JE It^{\ast}: E_{J} \to E_{I} has a left adjoint t !:E IE Jt_{!}: E_{I} \to E_{J}. It is a trifibration if and only if there is also a right adjoint t *:E IE Jt_{\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).


  • 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)

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