nLab essential fiber

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 $p:E\to B$ over an object $b \in B$ can be thought of as the category of ways that $b$ can arise from applying $p$ to some object in $E$.

Definition

Definition

For $p \colon E\to B$ be a functor and $b\in B$ an object the essential fiber of $p$ over $b$ is the category whose:

• objects are pairs $(e,\phi)$ where $e\in E$ is an object and $\phi\colon p(e)\cong b$ is an isomorphism,

• morphisms$(e,\phi)\to (e',\phi')$ are morphisms $f\colon e\to e'$ in $E$ such that $\phi' \circ p(f) = \phi$,

• composition operation is the evident one.

Remark

The notion of essential fiber in Def. can be identified with that of the pseudopullback or “isocomma object” of $p$ along the functor $b\colon 1\to B$ from the terminal category which picks out the object $b$:

$$\array{ E \times_B 1 &\longrightarrow& E \\ \Big\downarrow && \Big\downarrow \mathrlap{{}_p} \\ 1 \!\!\! &\underset {b} {\longrightarrow} & B }$$

Moreover, the notion can also be identified with a homotopy fiber in the canonical model structure on Cat.

Finally, with groupoids identified as homotopy 1-types, the essential fiber of a functor between groupoids and thought of as $\infty$-groupoids actually coincides with its homotopy fiber in the classical sense of homotopy theory (well-defined up to weak homotopy equivalence).

Relationship to fibrations

If $p$ is an isofibration, then any of its essential fibers (Def. ) 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 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 $f \colon A \to B$ is conservative if and only if all its essential fibers are groupoids.

Proposition

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

and thus:

Proposition

If the functor $p \colon E \to B$ is faithful and conservative then all its essential fibers are equivalent to discrete categories.

Example

There typically is a nontrivial preorder of ways that a set can arise as the underlying set of a topological space, because the forgetful functor from $\mathrm{Top}$ to $\mathrm{Set}$ is faithful but not conservative.

On the other hand, there is a mere set (i.e., a discrete category) of ways that a set can arise as the underlying set of a group, because the forgetful functor from $\mathrm{Grp}$ to $\mathrm{Set}$, being monadic, is both faithful and conservative.

Remark

The automorphism group of $b \in B$ always acts on the essential fiber of $b$.

For example, on objects, $\alpha \in \mathrm{Aut}_B(b)$ acts to send $(e,\phi)$ to $(e, \alpha \circ \phi)$.

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 $p \colon E \to B$ is faithful and conservative, the essential fiber over $b \in B$ is equivalent to the discrete category on the set

$$\coprod_e \big( \mathrm{Aut}_B (b) / \mathrm{Aut}_E (e) \big) \,$$

where $e \in E$ ranges over one representative of each isomorphism class in $E$ whose image is the isomorphism class of $b$.

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

Proposition

If the functor $p \colon E \to B$ is faithful, conservative and it is injective on isomorphism classes, then for any $e \in E$, the essential fiber over $b = f(e)$ is equivalent to the discrete category on the set

$$\mathrm{Aut}_B (b) / \mathrm{Aut}_E (e) \,.$$

Thus $\mathrm{Aut}_B (b)$ acts transitively on this set.

Example

The forgetful functor from complex vector spaces to real vector spaces is faithful, conservative and injective on isomorphism classes. The essential fiber over a given real vector space is the set of complex structures on this vector space, and if this vector space is $\mathbb{R}^{2n}$, proposition implies that this set of complex structures is isomorphic to (the underlying set of) the coset space

$$\mathrm{GL}(2n,\mathbb{R})/\mathrm{GL}(n,\mathbb{C}) \,.$$

Related entries

• fiber

• homotopy fiber

• fibered category

References

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

• Category Theory Community Server: Homotopy fibers (2023)