[[Quiver representation theory MOC]]
# Acyclic quiver

Let $\Gamma$ be a quiver. 
An **oriented cycle** is a nontrivial path $p$ with $\dom p = \cod p$,
i.e. a non-identity [[endomorphism]] in the [[free category]] $\underline \Gamma$.
The quiver $\Gamma$ is called **acyclic** iff it has no oriented cycles.

## Properties

- $\Gamma$ is acyclic iff $\Mor \underline \Gamma$ is finite, or equivalently the [[path algebra]] is finite-dimensional.

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#state/tidy | #lang/en | #SemBr