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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

There For exist any variants field with better category-theoretic properties. One example is the category ofbornological$k$ , vector the spaces, category which is both complete$Vect_k$ , is complete cocomplete , and closed cocomplete monoidal . See and e.g. Houzel closed 1973 monoidal , with respect to the Meyer tensor 1999 product of vector spaces , .Meyer 2004 for more.

Splitting lemma

Assuming the axiom of choice (and essentially by the basis theorem): \begin{proposition}\label{ShortExactSequencesSplit} In $Vect$ every short exact sequence splits. \end{proposition} (See there.)

\begin{remark} On FinDimVect this is a categorification of the rank-nullity theorem. \end{remark}

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:

On There bornological are vector various spaces: categories oftopological vector spaces, for instance bornological topological vector spaces.