nLab
essential fiber

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Idea

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.

Definition

Let $p:E\to B$ be a functor and $b\in B$ an object. The essential fiber of $p$ over $b$ is the following category:

• its objects are pairs $(e,\phi)$ where $e\in E$ is an object and $\phi\colon p(e)\cong b$ is an isomorphism.
• its morphisms $(e,\phi)\to (e',\phi')$ are morphisms $f\colon e\to e'$ in $E$ such that $\phi' \circ p(f) = \phi$.

The essential fiber can be identified with the pseudopullback of $p$ along the functor $b\colon 1\to B$ from the terminal category which picks out the object $b$. 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).

Relationship to fibrations

If $p$ is an isofibration, then any of its essential fibers is equivalent to the corresponding strict fiber. This includes the case when $p$ is a Grothendieck fibration.

On the other hand, when $p$ 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.