nLab Vect

Definition

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

1. objects are vector spaces,

2. 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):

Proposition

In $Vect$ every short exact sequence splits.

(See there.)

Remark

On FinDimVect this is a categorification of the rank-nullity theorem.

Related categories

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:

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

category: category