Quiver (mathematics) - Category-theoretic Definition

Category-theoretic Definition

The above definition is based in set theory; the category-theoretic definition generalizes this into a functor from the free quiver to the category of sets.

The free quiver (also called the walking quiver, Kronecker quiver, 2-Kronecker quiver or Kronecker category) Q is a category with two objects, and four morphisms: The objects are V and E. The four morphisms are s:E→V, t:E→V, and the identity morphisms id_V:V→V and id_E:E→E. That is, the free quiver is

A quiver is then a functor Γ:Q→Set.

More generally, a quiver in a category C is a functor Γ:Q→C. The category of quivers in C, Quiv(C), is the functor category where:

• objects are functors Γ:Q→C,
• morphisms are natural transformations between functors.

Note that Quiv is the category of presheaves on the opposite category QOp.