# The category $\bf{FinVect}$ of finite vector spaces is rigid.

I am following Pavel Etingof et al's book on tensor categories.

They give FinVect as an example of a rigid monoidal category, with evaluation map given by $$\text {ev}_V(\epsilon\otimes v)=\epsilon(v)$$ and coevaluation map $$\text{coev}_V:\mathbb K \rightarrow V\otimes V^*$$ "the usual embedding".

My question is embarassingly simple: what is this "usual embedding?" I was trying to see what it had to be using the definition of evaluation and coevaluation, but got nowhere.

## 1 Answer

What Etingof et al call the "usual embedding" is given by linear extension of the map$$1_K\mapsto\sum_{i = 1}^{n}v_i\otimes v_i^*$$

where the $$v_i$$ are a basis of V and $$v_i^*$$ denoted the basis dual to the $$v_i$$'s.

This makes perfect sense if you want $$ev \circ f\otimes id\circ coev = Trace(f)$$ for any $$f:V\longrightarrow V$$.

Try $$f=id$$ and you'll find that $$ev\circ coev = dim(V)$$

• To rephrase things slightly, if $V$ is finite dimensional, there is an isomorphism with $\hom(V,V)$ and $V\otimes V^{*}$, and the usual embedding is the inclusion of scalar maps into all linear maps. – Aaron Dec 4 '18 at 11:52