nLab
Vect

Definition

Given a field $k$, the category $Vect_k$ is category whose objects are vector spaces and whose morphisms are linear maps.

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

The study of $Vect$ is called linear algebra.

Related categories

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

This is a compact closed category (see here).

$\Fin Vect$ is where most of ordinary linear algebra lives, although much of it makes sense in all of $Vect$. 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 $Fin$ Hilb.

category: category