nLab
Vect
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
Given a field $k$ , the category of $k$ -vector spaces $Vect_k$ is the category whose

objects are vector spaces ,

morphisms are linear maps .

If the field $k$ is understood, one often just writes $Vect$ .

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

The study of $Vect$ is called linear algebra .

Properties
For a field $k$ , the category $Vect_k$ is cocomplete .

Splitting lemma
Assuming the axiom of choice (and essentially by the basis theorem ):

(See

there .)

Finite-dimensional vector spaces
The full subcategory of Vect consisting of finite-dimensional vector spaces may be denoted FinDimVect .

This is a compact closed category (see here ).

$FinDimVect$ is where most of ordinary linear algebra lives, although much of it makes sense in all of $Vect$ . See also at quantum information theory in terms of dagger-compact categories .

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

Modules
More generally, for $R$ any ring (not necessarily a field ) then the analog of $Vect$ is the category $R$ Mod of $R$ -modules and module homomorphisms between them.

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

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

Last revised on August 19, 2023 at 13:14:42.
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