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The notion of essential fiber of a functor is an enhancement of the naive notion of fiber which, for functors, would violate the principle of equivalence. It is a category-theoretic version of a homotopy fiber.
Let be a functor and an object. The essential fiber of over is the following category:
The essential fiber can be identified with the pseudopullback of along the functor from the terminal category which picks out the object . It can also be identified with a homotopy fiber in the canonical model structure on Cat. When groupoids are identified with homotopy 1-types, the essential fiber actually coincides with the classical homotopy fiber (up to equivalence).
If is an isofibration, then any of its essential fibers is equivalent to the corresponding strict fiber. This includes the case when is a Grothendieck fibration.
On the other hand, when is a Street fibration (the version of Grothendieck fibration which respects the principle of equivalence), then essential fibers do not coincide with strict fibers, and essential fibers are the more useful notion. In particular, the correspondence between fibrations and pseudofunctors only goes through for Street fibrations if one defines the pseudofunctor using essential fibers.
Last revised on October 4, 2021 at 09:25:35. See the history of this page for a list of all contributions to it.