nLab Vect(X)

Definition

For $X$ a suitable space of sorts, the category of vector bundles over $X$ is the category denoted $Vect(X)$ whose

1. objects are vector bundles over $X$,

2. morphisms are vector bundle homomorphisms over $X$.

Specifically for $X$ a topological space, there is the category of topological vector bundles over $X$.

Via direct sum of vector bundles and tensor product of vector bundles this becomes a symmetric monoidal category in two compatible ways, making it a distributive monoidal category, in particular a rig category.

For $X$ a compact Hausdorff space then the Grothendieck group of $Vect(X)$ is the topological K-theory group $K(X)$.

Relation to other categories

Vector spaces

For $X = \ast$ the point space, then this is equivalently the category Vect of plain vector spaces:

$$Vect(\ast) \simeq Vect \,.$$

Vector bundles over general bases

See VectBund

Higher vector bundles

An analog in homotopy theory/higher category theory is the (infinity,1)-category of (infinity,1)-module bundles.