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 idV:V→V and idE: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.
Read more about this topic: Quiver (mathematics)
Famous quotes containing the word definition:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)