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essential fiber in nLab

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homotopy theory, (∞,1)-category theory, homotopy type theory

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Contents

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.

The essential fiber of a functor p:E→Bp:E\to B over an object b∈Bb \in B can be thought of as the category of ways that bb can arise from applying pp to some object in EE.

Definition

Definition

For p:E→Bp \colon E\to B be a functor and b∈Bb\in B an object the essential fiber of pp over bb is the category whose:

  • objects are pairs (e,ϕ)(e,\phi) where e∈Ee\in E is an object and ϕ:p(e)≅b\phi\colon p(e)\cong b is an isomorphism,

  • morphisms(e,ϕ)→(e′,ϕ′)(e,\phi)\to (e',\phi') are morphisms f:e→e′f\colon e\to e' in EE such that ϕ′∘p(f)=ϕ\phi' \circ p(f) = \phi,

  • composition operation is the evident one.

Relationship to fibrations

If pp is an isofibration, then any of its essential fibers (Def. ) 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 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 Grothendieck fibrations and pseudofunctors only goes through for Street fibrations if one defines the pseudofunctor using essential fibers.

Properties

Some properties of a functor are reflected in properties of its essential fibers (Def. ). A good intuition is that the more a functor resembles an injective function, the simpler its essential fibers are.

Proposition

A functor f:A→Bf \colon A \to B is conservative if and only if all its essential fibers are groupoids.

Proposition

If the functor p:E→Bp \colon E \to B is faithful, all its essential fibers are preorders. (The converse is not true.)

and thus:

When the essential fiber is essentially a set as in proposition , this allows us to describe the essential fiber as a union of orbits:

Proposition

If the functor p:E→Bp \colon E \to B is faithful and conservative, the essential fiber over b∈Bb \in B is equivalent to the discrete category on the set

∐ e(Aut B(b)/Aut E(e)) \coprod_e \big( \mathrm{Aut}_B (b) / \mathrm{Aut}_E (e) \big) \,

where e∈Ee \in E ranges over one representative of each isomorphism class in EE whose image is the isomorphism class of bb.

When p:E→Bp \colon E \to B is not only faithful and conservative but also injective on isomorphism classes, there is at most one isomorphism class in EE whose image is the isomorphism class of bb. Thus the coproduct in proposition has at most one summand, and the automorphism group of bb acts transitively on the relevant set:

References

The above results on essential fibers were proved in this discussion:

Last revised on January 30, 2024 at 08:28:59. See the history of this page for a list of all contributions to it.