nLab
Vect(X)

Contents

Context

Category theory

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Definition

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

  1. objects are vector bundles over XX,

  2. morphisms are vector bundle homomorphisms over XX.

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

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

Relation to other categories

Vector spaces

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

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

Higher vector bundles

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

Last revised on May 26, 2017 at 02:06:31. See the history of this page for a list of all contributions to it.