= Path-algebra module equivalence
{title2=$\operatorname{Rep}(Q)\simeq kQ\text{-}\operatorname{Mod}$}
For a <quiver> with finitely many vertices, send a <representation of a quiver> $(X_i,f_\rho)$ to $\bigoplus_iX_i$. Vertex <idempotents> act as projections, and paths act as composites of the arrow maps. Conversely, a unital <module> $M$ gives vertex spaces $e_iM$ and arrow maps $x\mapsto\rho x$. Compatible vertex maps are exactly <module homomorphisms>. These constructions give inverse equivalences of <categories>.
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