A bifibration of categories is a functor
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 $f^* : E_{b_2} \to E_{b_1}$.
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 forgetful functor Top $\to$ Set is a bifibration. See also topological concrete category.
The forgetful functor Grpd $\to$ Set is a bifibration.
The forgetful functor Cat $\to$ Set is a bifibration.
Ordinary Grothendieck fibrations correspond to pseudofunctors to Cat, by the Grothendieck construction and hence to prestacks. For these one may consider descent.
If the fibration is even a bifibration, there is a particularly elegant algebraic way to encode its descent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.
Fibrations and opfibrations on a category $C$ (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads. If $C$ has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the bifibrations satisfying the Beck-Chevalley condition; see (von Glehn).
In (CagneMelliès) the authors develop a notion of Quillen bifibration which combines the two concepts of Grothendieck bifibration and Quillen model structure, establishing conditions for when in a bifibration model structures on all the fibers and on the base combine to give one on the total category.
The basic theory of 2-fibrations, fibrations of bicategories, is developed in (Buckley). One (unpublished) proposal by Buckley and Shulman, for a “2-bifibration”, a bifibration of bicategories, is a functor of bicategories that is both a 2-fibration and a 2-opfibration. This should correspond to a pseudofunctor into $2 Cat_{adj}$, just as for 1-categories. There is a difference from this case, however, in that although a 2-fibration (corresponding to a “doubly contravariant” pseudofunctor $B^{coop} \to 2Cat$) has both cartesian lifts of 2-cells and 1-cells, a 2-opfibration (corresponding to a totally covariant pseudofunctor $B\to 2Cat$) has opcartesian lifts of 1-cells but still cartesian lifts of 2-cells.
A bifibration $F:E\to B$ such that the transition functors also have right adjoints is sometimes called a trifibration (cf. Pavlović 1990, p.315) or $\ast$-bifibration.
Paul-André Melliès, Noam Zeilberger, Type refinement and monoidal closed bifibrations, (arXiv.1310.0263).
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.
Tamara von Glehn, Polynomials and models of type theory, PhD thesis
Mitchell Buckley?, Fibred 2-categories and bicategories, (arXiv:1212.6283)
Pierre Cagne, Paul-André Melliès, On bifibrations of model categories, (arXiv:1709.10484)
Last revised on June 27, 2018 at 09:51:37. See the history of this page for a list of all contributions to it.