quiver representations and representation morphisms

Let $Q=(Q_0,Q_1,s,t)$ be a quiver, i.e. $Q_0$ is a set of vertices, $Q_1$ is a set of arrows and $s,t:Q_1\to Q_0$ are functions such that $s$ maps each arrow to its source and $t$ maps each arrow to its target.

A representation $𝕍$ of $Q$ over a field $k$ is a family of vector spaces ${\{V_i\}}_{i\in Q_0}$ over $k$ together with a family of $k$-linear maps ${\{f_a:V_{s(a)}\to V_{t(a)}\}}_{a\in Q_1}$.

A morphism $F:𝕍\to 𝕎$ between representations $𝕍=(V_i,g_a)$ and $𝕎=(W_i,h_a)$ is a family of $k$-linear maps ${\{F_i:V_i\to W_i\}}_{i\in Q_0}$ such that for each arrow $a\in Q_1$ the following relation holds:

$$F_{t(a)}\circ g_a=h_a\circ F_{s(a)}.$$

Obviously we can compose morphisms of representations and in this the case class of all representations and representation morphisms together with the standard composition is a category. This category is abelian.

It can be shown that for each finite quiver $Q$ (i.e. with $Q_0$ finite) and field $k$ there exists an algebra $A$ over $k$ such that the category of representations of $Q$ is equivalent to the category of modules over $A$.

A representation $𝕍$ of $Q$ is called trivial iff $V_i=0$ for each vertex $i\in Q_0$.

A representation $𝕍$ of $Q$ is called locally finite-dimensional iff for each vertex $i\in Q_0$ and finite-dimensional iff $𝕍$ is locally finite-dimensional and $V_i=0$ for almost all vertices $i\in Q_0$.