essential fiber


Category theory

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Basic facts





The essential fiber of a functor is a non-evil replacement for the fiber. It is a category-theoretic version of a homotopy fiber.


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

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

The essential fiber can be identified with the pseudopullback of pp along the functor b:1Bb\colon 1\to B from the terminal category which picks out the object bb. 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 pp is an isofibration, then any of its essential fibers is equivalent to the corresponding strict fiber. This includes the case when pp is a Grothendieck fibration.

On the other hand, when pp is a Street fibration (the non-evil version of a Grothendieck fibration), 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.

Revised on September 16, 2010 16:59:48 by Urs Schreiber (