Contents

bundles

cohomology

# Contents

## Idea

A fibre bundle or fiber bundle is a bundle in which every fibre is isomorphic, in some coherent way, to a standard fibre or typical fiber. Usually one also requires that it be locally trivial, hence locally of the form of a Cartesian product with that typical fiber.

## Definitions

In the most general sense, a bundle over an object $B$ in a category $C$ is a morphism $p: E \to B$ in $C$.

In appropriate contexts, a fibre bundle over $B$ with standard fibre $F$ may be defined as a bundle over $B$ such that, given any global element $x: 1 \to B$, the pullback of $E$ along $x$ is isomorphic to $F$. Certainly this definition is appropriate whenever $C$ has a terminal object $1$ which is a separator, as in a well-pointed category; even then, however, one often wants the more restrictive notion below.

One often writes a typical fibre bundle in shorthand as $F \to E \to B$ or

$\array { F & \rightarrow & E \\ & & \downarrow \\ & & B }$

even though there is not a single morphism $F \to E$ but instead one for each global element $x$ (and none at all if $B$ has no global elements!).

If $C$ is a site, then a locally trivial fibre bundle over $B$ with typical fibre $F$ is a bundle over $B$ with a cover $(j_\alpha: U_\alpha \to B)_\alpha$ such that, for each index $\alpha$, the pullback $E_\alpha$ of $E$ along $j_\alpha$ is isomorphic in the slice category $C/{U_\alpha}$ to the trivial bundle $U_\alpha \times F$ (a local trivialization).

One can also drop $F$ and define a slightly more general notion of locally trivial bundle over $B$ as a bundle over $B$ with a cover $(j_\alpha: U_\alpha \to B)_\alpha$ such that, for each index $\alpha$, there is a fibre $F_\alpha$ and an isomorphism in $C/{U_\alpha}$ between the pullback $E_\alpha$ and the trivial bundle $U_\alpha \times F_\alpha$. Every locally trivial fibre bundle is obviously a locally trivial bundle; the converse holds if $B$ is connected.

Now suppose that $E$ is a fibre bundle over $B$ with typical fibre $F$, locally trivialised over an (open) cover $(U_\alpha)_\alpha$ via isomorphisms $(t_\alpha \colon E_\alpha \simeq U_\alpha\times F)_\alpha$. Given an index $\alpha$ and an index $\beta$, let $U_{\alpha,\beta}$ be the fibred product (pullback) of $U_\alpha$ and $U_\beta$. Note that we can pull back the commutative triangle formed by $t_\alpha$ along $j_\alpha^*j_\beta$ to get a new isomorphism $t_{\alpha,\beta} \colon E_{\alpha,\beta} \simeq U_{\alpha,\beta} \times F$:

We can do the same on the other side, leading to the following situation:

Then we have an automorphism $g_{\alpha,\beta}$ of $U_{\alpha,\beta} \times F$ in $C/{U_{\alpha,\beta}}$ given by $g_{\alpha,\beta} = t_{\beta,\alpha} \circ t_{\alpha,\beta}^{-1}$. The $g_{\alpha,\beta}$ are the transition morphisms of the locally trivial fibre bundle $E$.

If $E$ in fact admits a global trivialisation $t \colon E \simeq B \times F$, then one can see that the two isomorphisms $t_{\alpha,\beta}$ and $t_{\beta,\alpha}$ are equal to the pullback of (the commutative triangle formed by) $t$ along $j_{\alpha,\beta} : U_{\alpha,\beta} \to B$, so that the transition morphisms $g_{\alpha,\beta}$ are all identities.

Often one considers special kinds of bundles, by requiring structure on the standard fibre $F$ and/or conditions on the transition morphisms $g_{\alpha,\beta}$. For example:

## Special cases

### $G$-bundle

If $G$ is a group object in $C$ that acts on $F$, then a $G$-bundle (or bundle with structure group $G$) over $B$ with standard fibre $F$ is a locally trivial fibre bundle over $B$ with standard fibre $F$ together with morphisms $U_{\alpha,\beta} \to G$ that, relative to the action of $G$ on $F$, give the transition maps $g_{\alpha,\beta}$. (The morphism $U_{\alpha,\beta} \to G$ is also written $g_{\alpha,\beta}$, conflating action with application.)

### $G$-principal bundles

More specifically, a (right or left) principal $G$-bundle over $B$ is a $G$-bundle over $B$ with standard fibre $G$, associated with the action of $G$ on itself by (right or left) multiplication.

### Structure-preserving bundles

If $F$ is an object of a concrete category over $C$, then we can consider locally trivial fibre bundles with standard fibre $F$ such that the transition morphisms are structure-preserving morphisms. If the automorphism group $Aut(F)$ can be internalised in $C$, then this the same as an $Aut(F)$-bundle, but the concept makes sense in any case.

### Vector bundles

As a fairly specific example, if $F$ is a topological vector space (and $C$ is a category with structure to support this, such as Top or Diff), then a vector bundle over $B$ with standard fibre $F$ is a $GL(F)$-bundle over $B$ with standard fibre $F$, where $GL(F)$ is the general linear group with its defining action on $F$.

### Associated bundles

Given a right principal $G$-bundle $\pi: P\to X$ and a left $G$-space $F$, all in a sufficiently strong category $C$ (such as Top), one can form the quotient object $P\times_G F = (P\times F)/{\sim}$, where $P \times F$ is a product and $\sim$ is the smallest congruence such that (using generalized elements) $(p g,f)\sim (p,g f)$; there is a canonical projection $P\times_G F\to X$ where the class of $(p,f)$ is mapped to $\pi(p)\in X$, hence making $P\times_G F\to X$ into a fibre bundle with typical fiber $F$, and the transition functions belonging to the action of $G$ on $F$. We say that $P\times_G F\to X$ is the associated bundle to $P\to X$ with fiber $F$.

## Generalizations

### In higher category theory

In higher category theory the notion of fiber bundle generalizes. See

### In commutative algebra

Under the interpretation of modules as generalized vector bundles, locally trivial fiber bundles correspond to locally free modules. See there for more.

### In noncommutative geometry

In noncommutative geometry both principal and associated bundles have analogues. The principal bundles over noncommutative spaces typically have structure group replaced by a Hopf algebra; the most well-known class whose base is described by a single algebra are Hopf–Galois extensions; the global sections of the associated bundle are formed using cotensor product. Transition functions can be to some extent emulated using noncommutative localizations, which yield nonaffine generalizations of Hopf–Galois extensions. Another generalization is when Hopf–Galois extensions in the sense of comodule algebras are replaced by entwining structures with analogous Galois condition.

## Properties

### Relation to fibrations

###### Proposition

Every locally trivial topological fiber bundle projection is a Serre fibration, and a Hurewicz fibration if it is also a numerable bundle.

###### Proof

It is clear that the projection out of a Cartesian product is a Serre fibration. With this, the statement follows from local triviality and the local recognition of Serre fibrations (this Prop) and of Hurewicz fibrations (this Prop.), respectively.

###### Remark

Beware that the converse statement is far from being true.

Fiber bundles (then spelled fibre-bundles) were originally defined in:

Other sources:

With an eye towards application in mathematical physics:

Discussion of fiber bundles internal to finitely complete categories: