Reference. quiver [nlab-quiver]

November 1, 2025
nLab authors
https://ncatlab.org/nlab/show/quiver

Backlinks

Definition. Quiver [002l]

November 27, 2025

A quiver is a collection of edges which may stretch between (ordered) pairs of points called vertices. Thus making a quiver a kind of graph. In the graph theoretical sense a quiver is a directed pseudo-multigraph so breaking that down [0]:

It's directed meaning each arrow has a direction associated with it going from a source vertex to a target vertex
It's a pseudograph meaning each pair of vertices has some number of edges (including zero) between them, and loops (edges that connect a vertex to itself) are allowed
It's a multigraph which means we constrain the kinds of edges to disallow loops (an edge from a vertex to itself.)

Lean representation

November 27, 2025

The lean representation of this construct is given as follows:


    universe v v₁ v₂ u u₁ u₂

    class Quiver (V : Type u) where
      Hom : V → V → Sort v