nLab
Vect

Context

Category theory

Linear algebra

Contents

Definition

Given a field kk, the category of kk-vector spaces Vect kVect_k is the category whose

  1. objects are vector space,

  2. morphisms are linear maps.

If the field kk is understood, one often just writes VectVect.

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

The study of VectVect is called linear algebra.

Finite-dimensional vector spaces

The full subcategory of Vect consisting of finite-dimensional vector spaces is denoted FinVect\Fin Vect.

This is a compact closed category (see here).

FinVect\Fin Vect is where most of ordinary linear algebra lives, although much of it makes sense in all of VectVect. See also at finite quantum mechanics in terms of dagger-compact categories.

On the other hand, anything involving transposes or inner products really takes place in FinFin Hilb.

Modules

More generally, for RR any ring (not necessarily a field) then the analog of VectVect is the category RRMod of RR-modules and module homomorphisms between them.

Vector bundles

For XX a suitable space of sorts, there is the category Vect(X) of vector bundles over XX. Specifically for XX a topological space, there is the category of topological vector bundles over XX. For X=*X = \ast the point space, then this is equivalently the category of plain vector spaces:

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

category: category

Revised on May 26, 2017 01:46:15 by Urs Schreiber (92.218.150.85)