nLab
Vect

$\mathrm{Vect}$ is the category whose objects are vector spaces and whose morphisms are linear operators. This definition implicitly presumes that we are considering vector spaces over a fixed field $k$. To make the dependence on the field explicit, we may write ${\mathrm{Vect}}_k$ or $kVect$.

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

$Fin\mathrm{Vect}$ is where most of ordinary linear algebra lives, although much of it makes sense in all of $\mathrm{Vect}$. On the other hand, anything involving transposes or inner products really takes place in $\mathrm{Fin}$ Hilb.

category: category