[[Linear algebra MOC]]
# Category of vector spaces
The **category of vector spaces** $\Vect_{\mathbb{K}}$ over a field $\mathbb{K}$ is an example of a [[Glossary of categories#Concrete categories|concrete category]],
that is to say its objects are sets with additional structure
and its morphisms are mappings that preserve that structure.
In this case, each object is a [[Vector space]]
and each of its morphisms is a [[Linear map]] 
— a mapping which preserves scalar multiplication and vector addition.
It is identical to [[Category of left modules|$\lMod{\mathbb K}$]], the different name is just for emphasis.

[[Matrix multiplication algebra as a category]].

## Universal constructions
- The [[trivial vector space]] $0$ of dimension zero is both the initial and terminal object
  i.e. the [[Initial and terminal objects|zero object]].
  See [[Initial and terminal objects#In the Category of vector spaces]].

## Skeleton

The canonical [[skeleton category]] $\opn{Sk}(\Vect_{\mathbb{K}})$ is the restriction to objects of the form $\mathbb{K}^{(\alpha)}$ for some [[Cardinal]] $\alpha$.
This of course assumes [[Axiom of Choice|AC]].

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