Let $(Q,I)$ be a bound quiver over a field $k$.

Let $\mathbb{V}$ be a representation of $Q$ over $k$ composed by $\{f(q)\}_{{q\in Q_{1}}}$ a family of linear maps. If

is a path in $Q$, then we have the evaluation map

For stationary paths we define $f_{{e_{x}}}:V_{x}\to V_{x}$ by $f_{{e_{x}}}=0$. Also, note that if $\rho$ is a relation in $Q$, then

where all $w_{i}$’s have the same source and target. Thus it makes sense to talk about evaluation in $\rho$, i.e.

In particular

is a linear map.

Recall that the ideal $I$ is generated by relations (see this entry) $\{\rho_{1},\ldots,\rho_{n}\}$.

Definition. A representation $\mathbb{V}$ of $Q$ over $k$ with linear mappings $\{f(q)\}_{{q\in Q_{1}}}$ is said to be bound by $I$ if

for every $i=1,\ldots,n$.

It can be easily checked, that this definition does not depend on the choice of (relation) generators of $I$.

The full subcategory of the category of all representations which is composed of all representations bound by $I$ is denoted by $\mathrm{REP}(Q,I)$. It can be easily seen, that it is abelian.