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

References

    Reference. quiver [nlab-quiver]

    Backlinks

      Definition. Category [002m]

      • November 27, 2025

      There are a few ways you can define a category. In the most basic intuitive sense a category consists of a collection of things called objects and binary relationships (or transitions) between those objects called morphisms (or arrows). We can combine these relationships by composing them, and for each object there is an identity morphism that acts as a neutral element for composition. (1)

      In the context of quivers a category can be defined as a quiver with a rule saying for how we can compose two edges that fit together to get a new edge. Furthermore, each vertex (object) has an edge starting and ending at that vertex (the identity morphism). The classical definition is something like this:

      The data

      • November 27, 2025

      A category \(C\) consists of:

      • A collection (or class (2)) of objects, denoted as \(\text {Ob}(C)\) or \(C_0\).
      • A collection (or set (2)) of morphisms (or arrows), denoted \(C_1\) or \(C(x, y)\) for \(x, y \in \text {Ob}(C)\).
        • For every morphism \(f \in C(x, y)\), there are two associated objects: the source (or domain) \(x\) and the target (or co-domain) \(y\). In standard function notation, we write \(f: x \to y\) where \(x = \text {dom}(f)\) and \(y = \text {cod}(f)\). NLab has a nice convention where it denotes the source \(s\) of a morphism as \(s(f)\) and the target \(t\) as \(t(f)\).
        • For every pair of morphisms \(f \in C(x, y)\) and \(g \in C(y, z)\) (s.t. \(t(f) = s(g)\) i.e. the morphisms type check), there is a composition morphism \(g \circ f \in C(x, z)\). Written out we can denote this as: \[ C(x, y) \times C(y, z) \to C(x, z) \] in diagrammatic order this is often written as \(f; g\) we can equivalently use a more graphical notation:
        • For every object \(x \in \text {Ob}\) there is an identity morphism: \[ (\text {id}_x : x \to x) \in C(x, x) \]
        Note: Some additional notations for morphisms include \(\text {hom}(x, y)\), \(\text {hom}_C(x, y)\) or \(C_1(x, y)\). Additionally, people use the notation \(\text {Mor(C)}\) to denote the following disjoint union \[ \text {Mor}(C) = \bigsqcup _{x, y \in \text {Ob}(C)} C(x, y) \] Which just expresses the idea that the collection of all morphisms in a category is made up of the morphisms between each pair of objects.

      The axioms

      • November 27, 2025

      The above are often called data of a category. In addition to this data, a category must satisfy the following axioms or (conditions):

      • Morphisms need to be associative which means that for every triple of morphisms \(f \in C(w, x)\), \(g \in C(x, y)\), and \(h \in C(y, z)\) the following holds: \[ h \circ (g \circ f) = (h \circ g) \circ f \]
      • For each morphism \(f \in C(x, y)\) the identity morphisms act as neutral elements for composition: \[ \text {id}_y \circ f = f = f \circ \text {id}_x \] This is also known as the left and right unit laws or just unity in general.

      Remarks

      • November 27, 2025
      • A category such as the one described above is often also called a 1-category to distinguish it from higher categories such as 2-categories, n-categories.