I am doing Category Theory and part of my project is to understand the category of vector bundles. Is there any reference(s) for vector bundles from a categorical perspective?
I also aim to understand things such as limits (in particular, pullbacks). Also cartesian and cocartesian liftings if there is a functor say $F\colon Vect(X) \to \text{Set}$. Basically I want to understand that given such a functor, does it admit cartesian and cocartesian liftings. $Vect(X)$ denotes the category of vector bundles over a topological space $X$.
You might like Segal’s paper classifying spaces and spectral sequences(see the section 'categories and classifying spaces'). There he views the category of vector bundles on a space $X$ to be the functor category $Fun(Open_*(X), (Vect, iso))$ of functors from the category of pointed open sets in $X$ to vector spaces/isomorphisms.
Using this point of view you get the classifying map of a vector bundle for free. The classifying space of vector bundles can be taken to be the space of all vector spaces/isomorphisms(i.e. the realization of the category of vector spaces). The classifying map is just given by the functor defining the vector bundle.
This point of view is nice bc it’s categorical (just like you wanted) and it’s exactly how things go for bundles of any sort (not just vector spaces). It motivates for instance the classification theorem for grothendieck fibrations, whose fibers are certain types of categories(e.g. categories, groupoids, etc). These kinds of fibrations will be classified, using the same reasoning as above, by the space of all categories of type you're interested in(categories, groupoids, etc).