Quiver (mathematics)

In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. 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 category theory, a quiver can be understood to be an underlying structure of a category, but without identity morphisms and composition. 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 free category.

A quiver Γ consists of:

This definition is identical to that of a multidigraph.

A morphism of quivers is defined as follows. If ${\displaystyle \Gamma =(V,E,s,t)}$ and ${\displaystyle \Gamma '=(V',E',s',t')}$ are two quivers, then a morphism ${\displaystyle m=(m_{v},m_{e})}$ of quivers consist of two functions ${\displaystyle m_{v}:V\to V'}$ and ${\displaystyle m_{e}:E\to E'}$ such that following diagrams commute:

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