Category of vector spaces

The category of vector spaces $𝖵 𝖾 𝖼 𝗍 𝕂$ over a field $𝕂$ is an example of a 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 $𝕂 𝖬 𝗈 𝖽$ , 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 zero object. See In the Category of vector spaces.

Skeleton

The canonical skeleton category $S k (𝖵 𝖾 𝖼 𝗍 𝕂)$ is the restriction to objects of the form $𝕂 (𝛼)$ for some Cardinal $𝛼$ . This of course assumes AC.

#state/develop | #SemBr