[FOM] 827: Tangible Incompleteness Restarted/1

[Timothy Y. Chow]

On Mon, 30 Sep 2019, Joe Shipman wrote:
> That raises the question of whether any results from graph theory have 
> been applied in category theory in interesting ways. Do graph theorists 
> have anything to tell category theorists about categories?

This question has been raised before on MathOverflow, and to a first 
approximation, the answer seems to be "no":

https://mathoverflow.net/questions/74615/intersection-between-category-theory-and-graph-theory

>> On Sep 30, 2019, at 2:29 PM, Louis H Kauffman <kauffman at uic.edu> wrote:
>> It should be pointed out in this discussion that a category is a 
>> digraph with extra (compositional) structure. Category theorists would 
>> not care to be categorized as studying a subcategory of graph theory.

I'm not sure if you're trying to draw an analogy between quivers and 
categories.  If you are, then one place the analogy breaks down is that 
nobody defines a category as a digraph, pure and simple.  A category has 
extra structure.  On the other hand, a quiver *is* standardly defined as a 
digraph (or perhaps a multidigraph, if you want to emphasize the 
possibility of multiple edges).  Period.  No extra structure.  So why 
introduce a new word?  There's no logical need for it.

Gabriel, in the paper where he introduced quivers, justified his 
introduction of a new word (Köcher) on the grounds that the word "graph" 
already had too many connotations and related concepts ("schon zu viele 
verwandte Begriffe anhaften").  Whether Gabriel was waging a "culture war" 
as Friedman suggested, or whether he just wanted to emphasize that he was 
only interested in a limited circle of questions about the digraph/quiver, 
is unclear to me.  Either way, as I said, I believe that the terminology 
is unfortunate, and generates precisely the type of confusion that Gabriel 
said he was trying to avoid.

Tim