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:
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)