V_a$. In other words, the following diagram commutes: $$\xymatrix{ V_{s(a)}\ar[r]^{f_{s(a)}}\ar[d]^{V_a} & W_{s(a)}\ar[d]^{W_a} \\
                      V_{t(a)}\ar[r]^{f_{t(a)}} & W_{t(a)}.
           }$$

Composition of morphisms is defined vertex-wise in the obvious way.

Representations of a quiver $Q$, together with their morphisms, form a category ${\mbox{{\rm Rep \!\!}}}(Q)$. We denote by ${\mbox{{\rm rep \!\!}}}(Q)$ its full subcategory whose objects are finite-dimensional representations. These categories are abelian; we can see this by showing that they are equivalent to module categories (see Proposition \[prop::Morita\]).

Let $Q$ be a quiver. The *path algebra* of $Q$ is the associative $k$-algebra $kQ$ defined as follows.

-   For all non-negative integers $\ell$, let $(kQ)_\ell$ be the $k$-vector with basis the set of paths of length $\ell$ in $Q$. Then the underlying vector space of $kQ$ is $\bigoplus_{\ell=0}^{\infty} (kQ)_\ell$.

-   Multiplication is defined on paths by $$w_2\cdot w_1 = \begin{cases} w_2w_1 & \textrm{if } s(w_2)=t(w_1) \\ 0 & \textrm{else,} \end{cases}$$ and extended to all of $kQ$ by linearity.

We denote by ${\mathfrak{m}}$ the two-sided ideal of $kQ$ generated by the arrows of $Q$. In other words, ${\mathfrak{m}}= \bigoplus_{\ell=1}^{\infty} (kQ)_\ell$.

If $I$ is any two-sided ideal of $kQ$, then we denote by ${\mbox{{\rm Rep \!\!}}}(Q,I)$ the full subcategory of ${\mbox{{\rm Rep \!\!}}}(Q)$ whose objects 