geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
A quiver is a collection of edges which may stretch between (ordered) pairs of “points”, called vertices.
A quiver is like a category with units and composition forgotten. Indeed, a category is a quiver with extra structure. To formalize this idea, we say there is a forgetful functor
where Quiv is the category of quivers and Cat is the category of (small strict) categories. Moreover, this forgetful functor has a left adjoint
sending each quiver to the free category on that quiver.
A quiver is a kind of graph and is often called a directed graph (or digraph) by category theorists. However, in the context of graph theory, the term “directed graph” is often taken to mean that there is at most one edge from one vertex to another. See directed graph.
The walking quiver^{1} $X$ is the category with
one object $X_0$, called the object of vertices;
one object $X_1$, called the object of edges;
two morphisms $s, t\colon X_1 \to X_0$, called the source and target;
together with identity morphisms.
A quiver is a functor $G\colon X \to$ Set.
More generally, a quiver in a category $C$ is a functor $G\colon X \to C$.
The category of quivers in $C$, Quiv$(C)$, is the functor category $C^{X}$, where:
objects are functors $G\colon X \to C$,
morphisms are natural transformations between such functors.
In the basic case where $C$ is Set, the category Quiv(Set) is equivalent to the category of presheaves on $X^{op}$. So: the category of quivers, Quiv, is the category of presheaves on the category $X^{op}$.
A quiver $G$ consists of two sets $E$ (the set of edges of $G$), $V$ (the set of vertices of $G$) and two functions
(the source and target functions). More generally, a quiver internal to a category (more simply, in a category) $C$ consists of two objects $E$, $V$ and two morphisms $s, t\colon E \rightrightarrows V$.
If $G = (E, V, s, t)$ and $G' = (E', V', s', t')$ are two quivers in a category $C$, a morphism $g\colon G \to G'$ is a pair of morphisms $g_0\colon V \to V'$, $g_1\colon E \to E'$ such that $s' \circ g_1 = g_0 \circ s$ and $t' \circ g_1 = g_0 \circ t$.
In graph theory, a quiver is often (cf. p. 4 or Figure 2-2) called a directed pseudograph (or some variation on that theme), but category theorists often just call them directed graphs.
Let $G_0 = G(X_0)$ and $G_1 = G(X_1)$.
A quiver in $C$ is a presheaf on $X^{op}$ with values in $C$.
A quiver is a globular set which is concentrated in the first two degrees.
A quiver can have distinct edges $e,e'\in G_1$ such that $s(e) = s(e')$ and $t(e) = t(e')$. A quiver can also have loops, namely, edges with $s(e) = t(e)$.
A quiver is complete? if for any pair of vertices $v,v'\in G_0$, there exists a unique directed edge $e\in G_1$ with $s(e) = v, t(e) = v'$.
Saying quiver instead of directed (multi)graph indicates focus on a certain set of operation intended on that graph. Notably there is the notion of a quiver representation.
Now, one sees that a representation of a graph $G$ in the sense of quiver representation is nothing but a functor $\rho\colon Q := F(G) \to Vect$ from the free category $F(G)$ on the quiver $G$:
Given a graph $G$ with collection of vertices $G_0$ and collection of edges $G_1$, there is the free category $F(G)$ on the graph whose collection of objects coincides with the collection of vertices, and whose collection of morphisms consists of finite sequences of edges in $G$ that fit together head-to-tail (also known as paths). The composition operation in this free category is the concatenation of sequences of edges.
Here we are taking advantage of the adjunction between Cat (the category of small categories) and Quiv (the category of directed graphs). Namely, any category has an underlying directed graph:
and the left adjoint of this functor gives the free category on a directed graph:
Since this is the central operation on quivers that justifies their distinction from the plain concept of directed graph, we may adopt here the point of view that quiver is synonymous with free category.
So a representation of a quiver $Q = F(G)$ is a functor
Concretely, such a thing assigns a vector space to each vertex of the graph $G$, and a linear operator to each edge. Representations of quivers are fascinating things, with connections to ADE theory, quantum groups, string theory, and more.
It may be handy to identify a quiver with its free category. This can be justified in the sense that the functor $F\colon Quiv \to Cat$ is an embedding ($k$-surjective for all $k \gt 0$) on the cores. In other words, isomorphisms between quivers may be identified with equivalences between free categories with no ambiguity.
However, at the level of noninvertible morphisms, this doesn't work; while $U$ is faithful, it is not full. In other words, there are many functors between free categories that are not morphisms of quivers.
Nevertheless, if we fix a quiver $G$ and a category $D$, then a representation of $G$ in $D$ is precisely a functor from $F(G)$ to $D$ (or equivalently a quiver morphism from $G$ to $U(D)$), and we may well want to think of this as being a morphism (a heteromorphism) from $G$ to $D$. As long as $D$ is not itself a free category, this is unlikely to cause confusion.
For $Q$ a quiver, write $k Q$ for the path algebra of $Q$ over a ground field $k$. That is, $k Q$ is an algebra with $k$-basis given by finite composable sequences of arrows in $Q$, including a “lazy path” of length zero at each vertex. The product of two paths composable paths is their composite, and the product of non-composable paths is zero.
A module over $k Q$ is the same thing as a representation of $Q$, so the theory of representations of quivers can be viewed within the broader context of representation theory of (associative) algebras.
If $Q$ is acyclic, then $k Q$ is finite-dimensional as a vector space, so in studying representations of $Q$, we are really studying representations of a finite dimensional algebra, for which many interesting tools exist (Auslander-Reiten theory, tilting, etc.).
Gabriel's theorem says that connected quivers of finite type are precisely those whose underlying undirected graph is a Dynkin diagram in the ADE series, and that the indecomposable quiver representations are in bijection with the positive roots in the root system of the Dynkin diagram. (Gabriel 72).
Some general-purpose references include
Harm Derksen, Jerzy Weyman, Quiver representations AMS Notices, 2005.
William Crawley-Boevy, Lectures on quiver representations (pdf).
Alistair Savage, Finite-dimensional algebras and quivers (arXiv:math/0505082), Encyclopedia of Mathematical Physics, eds. J.-P. Françoise, G.L. Naber and Tsou S.T., Oxford, Elsevier, 2006, volume 2, pp. 313-320.
Classification results for quivers appeared in
Quivers (referred to as directed pseudographs) were a tool in parts of the work of Ringel and Youngs in the second half the twentieth century to prove Heawood’s formula for every finite genus, cf. e.g. Fig. 2.3 the monograph
Beware that, strictly speaking, for Ringel, “quiver” means “embedded quiver” (into a given surface); in particular the author distinguishes between the two possible orientations of an embedded loop.
Quivers embedded in surfaces are studied in:
A special kind of quiver (finite, no loops, no parallel arcs) is treated in
Also called “the elementary ”parallel process“ ” by William Lawvere in p. 272. ↩