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:EV, t:EV, and the identity morphisms idV:VV and idE:EE. That is, the free quiver is

E
\;\begin{matrix} s \\ \rightrightarrows \\ t \end{matrix}\; V

A quiver is then a functor Γ:QSet.

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

  • objects are functors Γ:QC,
  • morphisms are natural transformations between functors.

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

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