nLab dual fibration

Idea

The dual fibration of a (Grothendieck) fibration is a fibration over the same base but with all of the fibers dualized.

Definition

Let $p : E \to B$ be a fibration, and let $F : B^{op} \to Cat$ be the corresponding indexed category. The dual fibration $p^d : E^d \to B$ can be defined (Borceux) as the fibration associated (via the Grothendieck construction) to the composite

$$B^{op} \stackrel{F}{\longrightarrow} Cat \stackrel{(-)^{op}}{\longrightarrow} Cat$$

where $(-)^{op} : Cat \to Cat$ is the operation sending any category to its opposite (note this operation preserves the direction of the 1-cells, although it reverses the direction of the 2-cells in Cat). (See Borceux reference.) Alternatively, the dual fibration may be defined in more elementary terms as a category with the same objects as $E$, and whose morphisms are equivalence classes of spans $L \stackrel{v}{\longleftarrow} X \stackrel{h}{\longrightarrow} R$ where $v$ is vertical (i.e., $p(v) = id$) and $h$ is horizontal (i.e., $p$-cartesian). (See Pavlovic and Kock references.)

Related concepts

• hyperdoctrine

• trifibration

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)

• Francis Borceux, Handbook of Categorical Algebra (see Volume 2, 8.3).

• Anders Kock, The dual fibration in elementary terms, arXiv:1501.01947.