[[Graph theory MOC]]
# Category of quivers

The **category of quivers** $\cat{Quiv}$ is a [[category]] where
an object is a [[quiver]]
and a morphism is a [[quiver homomorphism]]. #m/def/cat 
It is therefore identical to the [[Functor category]] $\Set^{\cat Q}$,
where $\cat Q$ is [[the walking]] quiver.

## Important functors

- There is a natural way $\cat{Quiv} \to \cat{Graph}$ to associate every quiver to an “underlying” [[Graph|general graph]], see [[Equivalence of quivers and general graphs]].
- This factorizes the forgetful functor $\opn V$ into the underlying vertex set.
- See [[Underlying quiver]] and [[Free category]]

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