Quiver (mathematics)

In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, in other words a multidigraph. They are commonly used in representation theory: a representation V of a quiver assigns a vector space V(x) to each vertex x of the quiver and a linear map V(a) to each arrow a. In , a quiver can be understood to be the underlying structure of a , but without composition or a designation of identity morphisms. That is, there is a forgetful functor from Cat to Quiv. Its left adjoint is a free functor which, from a quiver, makes the corresponding . A quiver Γ consists of: The set V of vertices of Γ The set E of edges of Γ Two functions: s:E \to V giving the start or source of the edge, and another function, t:E \to V giving the target of the edge. This definition is identical to that of a multidigraph. A morphism of quivers is defined as follows. If and are two quivers, then a morphism of quivers consists of two functions and such that the following diagrams commute: That is, and The above definition is based in set theory; the category-theoretic definition generalizes this into a functor from the free quiver to the . 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 \to V, t: E \to V, and the identity morphisms \mathrm{id}_V: V \to V and \mathrm{id}_E: E \to E. That is, the free quiver is A quiver is then a functor \Gamma:Q \to \mathbf{Set}. More generally, a quiver in a category C is a functor \Gamma: Q \to C. The category Quiv(C) of quivers in C is the where: objects are functors \Gamma:Q \to C, morphisms are natural transformations between functors. Note that Quiv is the on the Qop. If Γ is a quiver, then a path in Γ is a sequence of arrows such that the head of ai+1 is the tail of ai for i = 1, ..., n−1, using the convention of concatenating paths from right to left.