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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{quiver} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{graph_theory}{}\paragraph*{{Graph theory}}\label{graph_theory} [[!include graph theory - contents]] \hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory} [[!include representation theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak \noindent\hyperlink{slick_definition}{Slick definition}\dotfill \pageref*{slick_definition} \linebreak \noindent\hyperlink{nutsandbolts_definitions}{Nuts-and-bolts definitions}\dotfill \pageref*{nutsandbolts_definitions} \linebreak \noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak \noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_free_categories}{Relation to free categories}\dotfill \pageref*{relation_to_free_categories} \linebreak \noindent\hyperlink{relation_to_representation_theory_of_algebras}{Relation to representation theory of algebras}\dotfill \pageref*{relation_to_representation_theory_of_algebras} \linebreak \noindent\hyperlink{Classification}{Classification}\dotfill \pageref*{Classification} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} A \emph{quiver} (\hyperlink{Gabriel72}{Gabriel 72}) is a collection of [[edges]] which may stretch between (ordered) [[pairs]] of ``points'', called \emph{[[vertices]]}. Hence a quiver is a kind of [[graph]], often called a [[directed graph]] (or [[digraph]]) by [[category theory|category theorists]]. The term ``quiver'' (German: \emph{K\"o{}cher}) for a \emph{[[directed graph]]} is due to \hyperlink{Gabriel72}{Gabriel 72}, where it says on the first page: \begin{quote}% For such a 4-tuple $\big[$ $V \underoverset{\underset{t}{\longrightarrow}}{\overset{s}{\longrightarrow}}{} E$ $\big]$ we propose the term \emph{quiver}, and not \emph{graph}, since there are already too many notions attached to the latter word. \end{quote} While therefore, as concepts in themselves, \emph{quiver} and \emph{directed graph} are just the same, using the term \emph{quiver} serves to indicate certain constructions that one is interested in (a \emph{[[concept with an attitude]]}), notably the corresponding \emph{[[quiver representations]]} which much of the theory revolves around. The key result here is \emph{[[Gabriel's theorem]]} from the same article \hyperlink{Gabriel72}{Gabriel 72} that introduced the terminology \emph{quiver}. On the other hand, beware that in [[graph theory]], the term ``[[directed graph]]'' is often taken to mean that there is at most one edge from one vertex to another. To emphasize that a quiver allows more edges between vertices one may speak of \emph{directed [[pseudographs]]}. See at \emph{[[directed graph]]} for more. From yet another perspective, quivers/directed pseudographs are also like [[small categories]] with [[identity morphisms]] and [[composition]] forgotten. Conversely, a [[small category]] may be regarded as a quiver/directed pseudograph equipped with [[extra structure]]. Formally this is witnessed by a [[forgetful functor]] \begin{displaymath} U\colon Cat \to Quiv \end{displaymath} where [[Quiv]] is the category of quivers and [[Cat]] is the category of ([[small category|small]] [[strict category|strict]]) categories. Moreover, this forgetful functor has a [[left adjoint]] \begin{displaymath} F\colon Quiv \to Cat \end{displaymath} sending each quiver to the [[free category]] on that quiver. \hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions} \hypertarget{slick_definition}{}\subsubsection*{{Slick definition}}\label{slick_definition} The \textbf{[[walking structure|walking]] quiver}\footnote{Also called ``the elementary ''parallel process`` '' by [[William Lawvere|Lawvere]] in \hyperlink{GeneralizedGraphs}{p. 272}, ``a category that has no compositions in it'' by [[William Lawvere|Lawvere]], in his lecture at [[Como]].} $X$ is the [[category]] with \begin{itemize}% \item one [[object]] $X_0$, called the object of \emph{vertices}; \item one object $X_1$, called the object of \emph{edges}; \item two [[morphisms]] $s, t\colon X_1 \to X_0$, called the \emph{source} and \emph{target}; \item together with [[identity morphisms]]. \end{itemize} A \textbf{quiver} is a [[functor]] $G\colon X \to$ [[Set]]. More generally, a \textbf{quiver [[internalization|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: \begin{itemize}% \item objects are functors $G\colon X \to C$, \item morphisms are [[natural transformation|natural transformations]] between such functors. \end{itemize} 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 $X^{op}$. \hypertarget{nutsandbolts_definitions}{}\subsubsection*{{Nuts-and-bolts definitions}}\label{nutsandbolts_definitions} A \textbf{quiver} $G$ consists of two sets $E$ (the set of \emph{edges} of $G$), $V$ (the set of \emph{vertices} of $G$) and two functions \begin{displaymath} s, t\colon E \rightrightarrows V \end{displaymath} (the source and target functions). More generally, a \textbf{quiver internal to a category} (more simply, \emph{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 \textbf{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. \hyperlink{DG2nd}{p. 4} or \hyperlink{White84}{Figure 2-2}) called a \textbf{directed pseudograph} (or some variation on that theme), but category theorists often just call them \textbf{directed graphs}. \hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks} Let $G_0 = G(X_0)$ and $G_1 = G(X_1)$. \begin{itemize}% \item A quiver in $C$ is a [[presheaf]] on $X^{op}$ with values in $C$. \item A quiver is a [[globular set]] which is concentrated in the first two degrees. \item 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 $s(e) = t(e)$. \item A quiver is \textbf{[[complete graph|complete]]} if for any pair of vertices $v,v'\in G_0$, there exists a unique directed edge $e\in G_1$ with $s(e) = v, t(e) = v'$. \end{itemize} \hypertarget{terminology}{}\subsection*{{Terminology}}\label{terminology} Saying \emph{quiver} instead of \emph{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 \emph{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 \emph{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 [[sequence]]s of edges in $G$ that fit together head-to-tail (also known as \emph{[[path]]s}). The composition operation in this free category is the concatenation of sequences of edges. Here we are taking advantage of the [[adjoint functor|adjunction]] between [[Cat]] (the category of small categories) and [[Quiv]] (the category of directed graphs). Namely, any category has an underlying directed graph: \begin{displaymath} U\colon Cat \to Quiv \end{displaymath} and the left adjoint of this functor gives the free category on a directed graph: \begin{displaymath} F\colon Quiv\to Cat \end{displaymath} 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 \emph{quiver} is synonymous with \emph{free category}. So a [[representation]] of a quiver $Q = F(G)$ is a functor \begin{displaymath} R\colon Q \to Vect \end{displaymath} 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. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{relation_to_free_categories}{}\subsubsection*{{Relation to free categories}}\label{relation_to_free_categories} It may be handy to \emph{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 ($k$-[[k-surjectivity|surjective]] for all $k \gt 0$) on the [[cores]]. In other words, [[isomorphisms]] between quivers may be identified with [[equivalence of categories|equivalences]] between free categories with no ambiguity. However, at the level of noninvertible morphisms, this doesn't work; while $U$ is [[faithful functor|faithful]], it is \emph{not} [[full functor|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 [[quiver representation|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. \hypertarget{relation_to_representation_theory_of_algebras}{}\subsubsection*{{Relation to representation theory of algebras}}\label{relation_to_representation_theory_of_algebras} For $Q$ a quiver, write $k Q$ for the \emph{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.). \hypertarget{Classification}{}\subsubsection*{{Classification}}\label{Classification} [[Gabriel's theorem]] (\hyperlink{Gabriel72}{Gabriel 72}) says that connected quivers with a [[finite number]] of [[indecomposable object|indecomposable]] [[quiver representations]] over an [[algebraically closed field]]are precisely the \emph{[[Dynkin quivers]]}: those whose underlying [[undirected graph]] is a [[Dynkin diagram]] in the [[ADE classification|ADE series]], and that the [[indecomposable object|indecomposable]] [[quiver representations]] are in [[bijection]] with the positive [[root (in representation theory)|roots]] in the [[root system]] of the Dynkin diagram. (\hyperlink{Gabriel72}{Gabriel 72}). \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[McKay quiver]], [[McKay correspondence]] \item [[quiver gauge theory]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The concept of quiver in the context of [[quiver representation]] (and their classification in [[Gabriel's theorem]]) originates with \begin{itemize}% \item [[Peter Gabriel]], \emph{Unzerlegbare Darstellungen. I}, Manuscripta Mathematica 6: 71--103, (1972) (\href{https://link.springer.com/article/10.1007/BF01298413}{doi:10.1007/BF01298413}, \href{http://www.ams.org/mathscinet-getitem?mr=332887}{MR332887} \href{http://dx.doi.org/10.1007/BF01298413}{doi}) \end{itemize} Some general-purpose references include \begin{itemize}% \item Harm Derksen, Jerzy Weyman, \href{http://www.ams.org/notices/200502/fea-weyman.pdf}{Quiver representations} \emph{AMS Notices}, 2005. \item William Crawley-Boevy, \emph{Lectures on quiver representations} (\href{http://www.amsta.leeds.ac.uk/~pmtwc/quivlecs.pdf}{pdf}). \item Alistair Savage, \emph{Finite-dimensional algebras and quivers} (\href{http://www.arxiv.org/abs/math/0505082}{arXiv:math/0505082}), \emph{Encyclopedia of Mathematical Physics}, eds. J.-P. Fran\c{c}oise, G.L. Naber and Tsou S.T., Oxford, Elsevier, 2006, volume 2, pp. 313-320. \end{itemize} Quivers (referred to as \emph{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 \begin{itemize}% \item Gerhard Ringel: \emph{Map Color Theorem}. Springer. Grundlehren Band 209. 1974 \end{itemize} 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: \begin{itemize}% \item Arthur T. White: \emph{Graphs, Groups and Surfaces}. North Holland. Completely revised and enlarged edition (1985) \end{itemize} A special kind of quiver (finite, no loops, no parallel arcs) is treated in \begin{itemize}% \item Gregory Gutin, Jørgen Bang-Jensen: \emph{Digraphs: Theory, Algorithms and Applications}. Springer Monographs in Mathematics. Second Edition (2009) \item [[William Lawvere]]: \emph{Qualitative Distinctions Between Some Toposes of Generalized Graphs}, Contemporary Mathematics 92 (1989) \end{itemize} [[!redirects quiver]] [[!redirects quivers]] [[!redirects directed pseudograph]] [[!redirects directed pseudographs]] \end{document}