nLab finite-dimensional vector space

Idea

A vector space is finite-dimensional if it admits a finite basis. The category FinVect of finite-dimensional vector spaces is of course the full subcategory of Vect whose objects are finite-dimensional.

Properties

Compact closure

Proposition

A vector space $V$ is finite-dimensional precisely if the hom-functor $\hom(V, -): Vect \to Set$ preserves filtered colimits.

Proof

Every vector space $W$ is the filtered colimit of the diagram of finite-dimensional subspaces $W' \subseteq W$ and inclusions between them; applying this to $W = V$, the condition that $\hom(V,-)$ preserves filtered colimits implies that the canonical comparison map

$$colim_{fd\; V' \subseteq V} \hom(V, V') \to \hom(V, V)$$

is an isomorphism, so some element $[f]$ in the colimit represented by $f: V \to V'$ gets mapped to $1_V$, i.e., $i \circ f = 1_V$ for some inclusion $i: V' \hookrightarrow V$. This implies $i$ is an isomorphism, so that $V$ is finite-dimensional.

In the converse direction, observe that $\hom(V, -)$ has a right adjoint (and in particular preserves filtered colimits) if $V$ is finite-dimensional.

To see this, first notice that the dual vector space $V^\ast$ of functionals $f: V \to k$ to the ground field is a dual object to $V$ in the monoidal category sense, so that there is a counit $eva \colon V^\ast \otimes V \to k$ taking $f \otimes v \mapsto f(v)$. The unit is uniquely determined from this counit and can be described using any basis $e_i$ of $V$ and dual basis $f_j$ as the map

$$k \to V \otimes V^\ast$$

taking $1 \mapsto \sum_i e_i \otimes f_i$. We thus have an adjunction $(- \otimes_k V) \; \dashv (- \otimes V^\ast)$, which is mated to an adjunction $\hom(V, -) \dashv \hom(V^\ast, -)$ by familiar hom-tensor adjunctions; thus $\hom(V, -)$ has a right adjoint.

This means that

Proposition

Finite-dimensional vector spaces are exactly the compact objects of Vect in the sense of locally presentable categories, but also the compact = dualizable objects in the sense of monoidal category theory. In particular the category FinVect is a compact closed category.

Related concepts

• finite abelian category

• Schur functor

• finite quantum mechanics in terms of dagger-compact categories