Factorization systems for vector bundles

Question

Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?

Answer 1

Normally, we consider vector bundles over some base space $X$. The simplest case is if $X$ is a point. Then the category of vector bundles over $X$ is just the category of vector spaces (over whatever your base field $k$ is). This category has exactly four weak factorization systems: (isomorphisms, all), (injections, surjections), (surjections, injections), (all, isomorphisms). As an exercise, I encourage you to write down the weak factorization systems on other spaces, like two-point spaces. There are plenty!

If, rather than $Vect(X)$, you don't want to fix a base space, then you're taking the Grothendieck construction with respect to the functor that takes a space $X$ to $Vect(X)$. In this setting, there is a general technique for piecing together factorization systems from the base $Top$ and the fibers $Vect(X)$ to factorization systems on the total space. For more, check out The Grothendieck construction for model categories and On bifibrations of model categories. So, the answer is that there are plenty of factorization systems running around, both on the categories $Vect(X)$ and on $Top$, and on the Grothendieck construction.