In mathematics, especially representation theory, a quiver is another name for a multidigraph; that is, a directed graph where loops and multiple arrows between two vertices are allowed. Quivers 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 category theory, a quiver can be understood to be the underlying structure of a small category, but without composition or a designation of identity morphisms. That is, there is a forgetful functor from Cat (the category of small categories) to Quiv (the category of multidigraphs). Its left adjoint is a free functor which, from a quiver, makes the corresponding free category.
Definition A quiver Γ consists of:
The set V of vertices of Γ The set E of edges of Γ Two functions: s : E → V {\displaystyle s:E\to V} giving the start or source of the edge, and another function, t : E → V {\displaystyle t:E\to V} giving the target of the edge. This definition is identical to that of a multidigraph that has edges with their own distinct identity. A morphism of quivers is a mapping from vertices to vertices which takes directed edges to directed edges. Formally, if Γ = ( V , E , s , t ) {\displaystyle \Gamma =(V,E,s,t)} and Γ ′ = ( V ′ , E ′ , s ′ , t ′ ) {\displaystyle \Gamma '=(V',E',s',t')} are two quivers, then a morphism m = ( m v , m e ) {\displaystyle m=(m_{v},m_{e})} of quivers consists of two functions m v : V → V ′ {\displaystyle m_{v}:V\to V'} and m e : E → E ′ {\displaystyle m_{e}:E\to E'} such that the following diagrams commute:
That is,
m v ∘ s = s ′ ∘ m e {\displaystyle m_{v}\circ s=s'\circ m_{e}}
and
m v ∘ t = t ′ ∘ m e {\displaystyle m_{v}\circ t=t'\circ m_{e}}
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 {\displaystyle s:E\to V} , t : E → V {\displaystyle t:E\to V} , and the identity morphisms i d V : V → V {\displaystyle \mathrm {id} _{V}:V\to V} and i d E : E → E {\displaystyle \mathrm {id} _{E}:E\to E} . That is, the free quiver is the category
E s ⇉ t V {\displaystyle E\;{\begin{matrix}s\\[-6pt]\rightrightarrows \\[-4pt]t\end{matrix}}\;V}
A quiver is then a functor Γ : Q → S e t {\displaystyle \Gamma :Q\to \mathbf {Set} } . (That is to say, Γ {\displaystyle \Gamma } specifies two sets Γ ( V ) {\displaystyle \Gamma (V)} and Γ ( E ) {\displaystyle \Gamma (E)} , and two functions Γ ( s ) , Γ ( t ) : Γ ( E ) ⟶ Γ ( V ) {\displaystyle \Gamma (s),\Gamma (t)\colon \Gamma (E)\longrightarrow \Gamma (V)} ; this is the full extent of what it means to be a functor from Q {\displaystyle Q} to S e t {\displaystyle \mathbf {Set} } .) More generally, a quiver in a category C is a functor Γ : Q → C . {\displaystyle \Gamma :Q\to C.} The category Quiv(C) of quivers in C is the functor category where:
objects are functors Γ : Q → C , {\displaystyle \Gamma :Q\to C,} morphisms are natural transformations between functors. Note that Quiv is the category of presheaves on the opposite category Qop.
… excerpt ends here. Continue reading the full article.
