Representation of a quiver

Definition

Let $k$ be a field and let $Q = (Q_0, Q_1, s, t)$ be a quiver. A representation $V$ of $Q$ over $k$ consists of:

1. a vector space $V_i$ over $k$ for each vertex $i \in Q_0$;
2. a linear map $V_a \colon V_{s(a)} \to V_{t(a)}$ for each arrow $a \in Q_1$.

A morphism $\varphi \colon V \to W$ of representations is a family of linear maps $\varphi_i \colon V_i \to W_i$, $i \in Q_0$, such that $\varphi_{t(a)} \circ V_a = W_a \circ \varphi_{s(a)}$ for every arrow $a$. If $Q_0$ is finite and each $V_i$ is finite-dimensional, $V$ is finite-dimensional, and its dimension vector is $(\dim V_i)_{i \in Q_0}$.