[FOM] 827: Tangible Incompleteness Restarted/1

[Timothy Y. Chow]

On Sun, 29 Sep 2019, José Manuel Rodríguez Caballero wrote:
> There is a group of mathematicians (school of Gabriel) who prefer to 
> substitute the word GRAPH by the word QUIVER. According to [1], a quiver 
> is a collection of edges which may stretch between (ordered) pairs of 
> points, called vertices. Hence a quiver is a kind of graph, often called 
> a directed graph. So, a mathematician may say: I do not know what a 
> directed graph [2] is, I work with quivers [3]. Why Gabriel did not use 
> the word GRAPH?

It is true that everywhere in the literature, a quiver is defined as a 
directed graph.  I find this unfortunate because it does raise the 
question of why you would introduce a new word for something that already 
has a name.

Associated to a quiver is something called its path algebra.  When people 
use the word "quiver," they're signalling the fact that they're primarily 
interested in the representation theory of the path algebra, and not in 
the directed graph for its own sake.  Graph-theoretic facts about the 
underlying directed graph are interesting only insofar as they lend 
insight into the representation theory of the path algebra.  That's a very 
narrow set of graph-theoretic facts, compared to the sort of things that 
graph theorists might be interested in.

In my opinion, it would have been better to use the word "quiver" to refer 
to the path algebra rather than to the directed graph itself, but it's too 
late to change established terminology.

Tim