A vector bundle is a morphism that looks locally on the target like a product of the target with a vector space.

We will call the target space the base and the space of definition the total space. The preimage of a point of the base is called the fiber.

Is that the correct mathematical definition? It doesn't mention what kind of spaces we look at, what kind of morphism I'm talking about, what the product is, locally in which sense, vector space over which field, do we allow infinite dimension, ... so it's not a mathematical definition in the pedantic sense. I will give you pedantic definitions in this article, just to satisfy my need to write down what I consider to be a good terminology.

Some concepts related to vector bundles

There are some notions close to vector spaces that I want to discuss as well: complex vector bundles, holomorphic vector bundles, fiber bundles, G-bundles and Hurewicz fibrations on the one side, locally free sheaves, quasi-coherent sheaves, sheaves of abelian groups on the other side.
I'll give you quick definitions in the non-rigorous spirit of the first definition above:

• A complex vector bundle is a vector bundle whose fibers are vector spaces over the complex numbers.
• A holomorphic vector bundle is a complex vector bundle whose transition functions are biholomorphic (I will mention transition functions below).
• A fiber bundle is a morphism that looks locally on the target like a product of the target with the fiber over some point.
• A G-bundle, for G some group, is a fiber bundle that has a G-action on the total space that preserves the fibers and is free and transitive.
• A Hurewicz fibration is a map that allows the following: Whenever you fix a map to the total space, and a homotopy of the composition with the Hurwicz fibration, this homotopy can be lifted to a homotopy of the map to the total space.
• A locally free sheaf is a sheaf of modules under the sheaf of rings of regular functions that locally looks like a free module.
• A quasi-coherent sheaf is a sheaf of modules under the sheaf of rings of regular functions that locally looks like a cokernel of a morphism of locally free sheaves.
• A sheaf of abelian groups is just a sheaf whose sections happen to be abelian groups. And also the sections over the empty set are required to be the trivial group, not something else.

The connection between the bundle world and the sheaf world is, of course, the sheaf of sections of a bundle. Such a sheaf of sections of a vector bundle happens to be a locally free sheaf, and to every locally free sheaf one can associate a vector bundle that turns this construction into an equivalence of categories.

A good definition is ...

Before I talk about transition functions, I want to discuss some aspects of a good mathematical definition (of course there are exceptions):

• It has to be specific. You don't want to talk about general something when in fact, what you have in mind is something quite concrete. You can generalize later.
• It has to capture only the aspects that are important, nothing else. You don't want to study aspects of implementation of a concept instead of the concept. Example: You can define natural numbers as certain sets in many different ways, while most of the time it doesn't matter which kind of set representation you choose.
• It should relate to familiar useful notions in mathematics instead of re-inventing the wheel. Also, the names should not be too different from historical usage.
• It should not use too much highfalutin abstract language if it can be said in a down-to-earth elementary way just as short or mildly longer anyway.
• It should be invariant under transformations of the structures and properties that matter. Example: A property of vector spaces is defined poorly if it isn't invariant under vector space isomorphisms - since it is then not really a property of vector spaces but a property of a certain set which happens to carry a vector space structure. This is also called "evil" in some circles.
• It should separate stuff from structure, in other words, data from properties. This is not only a matter of exposition, but it is most important in exposition.

A bad definition of vector bundles

A bad definition of vector bundles would look like that:

A continuous map $p : E \to B$ together with a cover $U_i$ of $B$ is called vector bundle if $p^{-1}(U_i) \simeq B \times \mathbb{R}^n$. A morphism of vector bundles from $p : E \to B$ to $p' : E' \to B'$ is a commutative diagram with $p,p'$ and continuous maps $E \to E'$ and $B \to B'$ in it, such that the map $E \to E'$, when restricted to any fiber, is a linear map.

I don't want to go into detail what's wrong with this definition (that I just made up), but instead give you what I consider a good definition (and I expect there to be disagreement).

Good definitions of various things

Fix a category of spaces, for example topological spaces, manifolds, algebraic varieties or schemes. Let $k$ be a field. If we want to consider a vector space of dimension $n$ as a space, we write $A^n_k$ (for affine space).

Then, to any given space $B$ we can associate the category of spaces over $B$, whose objects are morphisms $E \to B$ and a morphism is a morphism of spaces $E \to E'$ such that the commutative triangle made from this and the morphisms $E \to B$ and $E' \to B$ commutes.

From this it is clear that an isomorphism in the category of spaces over $B$ is an isomorphism of total spaces that preserves the fibers, which means it commutes with the morphisms to $B$.

We also remind you that for any morphism $p : E \to B$ we call a morphism $s : B \to E$ a section of $p$ if $p \circ s = id_B$.

• A trivialized fiber bundle over a space $B$ with fiber $F$ and trivialization $\psi$ is a morphism $E \to B$ together with an isomorphism $\psi$ to a projection morphism $B \times F \to B$.
• A trivial fiber bundle over a space $B$ with fiber $F$ is a morphism $E \to B$ such that there exists a trivialization $\psi$ that turns it into a trivialized fiber bundle.
• A locally trivialized fiber bundle over an open cover $\{U_i\}_{i \in I}$ of $B$ with fiber $F$ is a morphism $p : E \to B$ together with morphisms $\{\psi_i\}_{i \in I}$ called local trivialization, such that each morphism $p|_{p^{-1}(U_i)} : p^{-1}(U_i) \to U_i$ is a trivialized fiber bundle over the space $U_i$ with fiber $F$ and trivialization $\psi_i$.
• A fiber bundle over a space $B$ with fiber $F$ is a morphism $E \to B$ such that there exists an open cover $\{U_i\}_{i \in I}$ and a local trivialization $\{\psi_i\}_{i \in I}$ such that these together form a locally trivialized fiber bundle.
• A locally trivialized $k$-vector bundle of rank $n$ over an open cover $\{U_i\}_{i \in I}$ of a space $B$ is a trivialized fiber bundle $p : E \to B$ with fiber $A^n_k$, fixed $k$-vector space structure on each fiber $p^{-1}(x)$ for $x \in B$ and local trivialization $\{\psi_i\}$ such that for all $i \in I$ and each point $x \in U_i$ the morphism $\psi_i|_x : p^{-1}(x) \to A^n_k$ is an isomorphism of vector spaces as well.
• Let $G$ be a group object in spaces. A $G$-valued 2-cocycle subordinate to an open cover $\{U_i\}_{i \in I}$ is a family of morphisms $\{\phi_{ij} : U_i \cap U_j \to G\}_{(i,j) \in I \times I}$ that satisfy the cocycle conditions: $\phi_{ii} = id_{U_i}$ and $\phi_{ki} = \phi_{kj}\cdot\phi_{ji}$, where $\cdot$ is pointwise multiplication in $G$.
• A $k$-vector space object over a space $B$ is a morphism $p : E \to B$ together with a section $z : B \to E$ called the zero section, a morphism $a : E \times_B E \to E$ over $B$ called addition and a morphism $m : k \times E \to E$ over $B$ called multiplication, such that for each point $x \in B$ the fiber $p^{-1}(x)$ becomes a $k$-vector space with zero $z(x)$, vector addition $a|_{p^{-1}(x)\times p^{-1}(x)}$ and scalar multiplication $m|_{k \times p^{-1}(x)}$.
• A $k$-vector bundle of rank $n$ over a space $B$ is a $k$-vector space object $p : E \to B$ such that there exists an open cover $\{U_i\}_{i \in I}$ and a local trivialization $\{\psi_i\}_{i \in I}$ such that these, together with the induces $k$-vector space structure on the fibers of $p$ form a trivialized $k$-vector bundle of rank $n$.

I want to remark that a vector bundle is not just a fiber bundle with fiber $F$ a vector space! The vector space structure has to vary continuously along the base, and it needs to be fixed once and for all. Also, it might puzzle you that the definition of a vector bundle is so different from "a morphism that is locally a vector bundle". This is because we have to fix the vector space structure on the fibers, not only an isomorphism class of vector spaces. The consequence of using different (seemingly easier) definitions would be to get different, non-equivalent categories.

To each locally trivialized vector bundle we can associate a 2-cocycle $\{\phi_{ij}\}$ by taking the local trivialization $\{\psi_i\}$ and looking at the morphism $\psi_{i} \circ \psi_{j}^{-1} : (U_i \cap U_j) \times A^n_k \to (U_i \cap U_j) \times A^n_k$ which is, by definition, a morphism over $U_i \cap U_j$, so it is reasonable to compose with the projection to $A^n_k$ and we get an adjoint morphism, called transition function $\phi_{ij} : U_i \cap U_j \to Hom(A^n_k,A^n_k) = GL_n(k)$.

There is no way to get back the original locally trivialized vector bundle from the cocycle, but we can glue together trivial vector bundles over the $U_i$ along the morphisms $(U_i \cap U_j) \times A^n_k \to (U_i \cap U_j) \times A^n_k$ that can be built from the cocycle, to get a locally trivialized vector bundle which is isomorphic, as locally trivialized vector bundle, to the original one.
The notion of isomorphism of vector bundles translates directly into the notion of cohomologous 2-cocycles.

Conclusion

So now we have somehow three different, yet somehow equivalent approaches to introduce vector bundles: one via fiber bundles, one via sheaves and one via cocycles. Which one is "right"?

For geometric intuition, one might argue for each of the three approaches. Traditionally, topologists tend to use fiber bundle language and algebraists tend to use sheaf language. Sheaf language is convenient to compute cohomology, either abstractly or via cocycles. To this purpose, one needs the category of quasi-coherent sheaves or all abelian sheaves, since they have better homological properties (being abelian and having enough injectives respectively). While this makes the sheaf language seem superior, the fiber bundle approach has the advantage of exhibiting a vector bundle as something that might be a Hurewicz fibration, which it is, in the topological setting. The systematic use of fibrations and cofibrations isn't so much found in the mainstream literature (although the term "fibration" is used frequently) but there is no reason to assume that won't change in future.

I would advise to introduce vector bundles by introducing locally trivialized vector bundles first. Most people tend to work with trivialized things, but we can try hard to make sure our constructions and definitions don't depend on the trivialization.

This process happens often in mathematics: one has some abstract concept to think about, and then there are various equivalent presentations of it, which are often useful to prove theorems or imagine geometry. I think it is equally important to separate the concepts from the presentations as it is to separate data from properties in a definition.

Whew, that was a rant!
I'm sure I butchered some definition here. Please tell me where. Oh, and I deliberately left out some definitions of morphisms here, because I think they should be obvious. Are they?