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 $\pi'_2 = \pi_2 \circ \varphi$.

 

[martinbn]

I am starting to think that the question of the thread is not about fibler bundles, but whether isomorphic objects are to be considered different or the same. If you have two different constructions, say of "projective space", which are isomorphic then do we have one object called "the projective space" or do you have two distinct objects and when we say "let X be the projective space" we mean any of the the two.