nLab dual fibration

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

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

where () op:CatCat(-)^{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 EE, and whose morphisms are equivalence classes of spans LvXhRL \stackrel{v}{\longleftarrow} X \stackrel{h}{\longrightarrow} R where vv is vertical (i.e., p(v)=idp(v) = id) and hh is horizontal (i.e., pp-cartesian). (See Pavlovic and Kock references.)

References

Last revised on March 29, 2023 at 13:08:21. See the history of this page for a list of all contributions to it.