Trivial fiber bundle vs product space

[cianfa72]

Not following all the arguments that closely but I think I agree with Martin. Once you learn some category theory, products are defined as something satisfying a specific universal property and are unique up to isomorphism. It's fairly easy to construct "different" products that satisfy the dame universal property.

Let me try to recap my understanding about this. In the context of category theory, given two objects $X_1$ and $X_2$ of a given category (say topological spaces), one can define/construct different products. A product consists of two pieces of information: an object $X$ (of the same category) and a pair of projections $\pi_1: X \to X_1$, $\pi_2: X \to X_2$ which satisfy the universal property. Such a product (i.e. object plus projections named canonical projections) is unique up to canonical isomorphism.

What is such canonical isomorphism ?

Say we have an object $X'$ plus two projections $\pi'_1: X' \to X_1$, $\pi'_2: X' \to X_2$ satisfying the universal property, i.e. it is a product. Then there exist an unique (canonical) isomorphism $\varphi: X \to X'$ such that $\pi'_1 = \pi_1 \circ \varphi$ and $\pi'_2 = \pi_2 \circ \varphi$.

 

[lavinia]

Here is a definition from Milnor's Characterisitc Classes , one of the most important texts in 20th century topology.

"
DEFINITION. A real vector bundle § over B consists of the following:
1) a topological space E= E(§) called the total space,
2) a (continuous) map p: E →B called the projection map, and
3) for each be B the structure of a vector space* over the real numbers in the set p^(-1)( b ) .

These must satisfy the following restriction:

Condition of local triviality. For each point b of B there should
exist a neighborhood UC B, an integer n>0, and a homeomorphism :

h UX R^n →p^(-1)(U)

so that, for each b in U, the correspondence x →h(b, x) defines an isomorphism between the vector space R^n and the vector space p^(-1) (b).

Such a pair (U,h) will be called a local coordinate system for § about b If it is possible to choose U equal to the entire base space, then § will be called a trivial bundle." pg 13 (italics added)

In the category of vector bundles and vector bundle morphisms, triviality is a structural feature. This is why Milnor says, "If it is possible to choose U equal to the entire base space, then § will be called a trivial bundle."