[FOM] 827: Tangible Incompleteness Restarted/1

[Patrik Eklund]

No, a category is not just a digraph with extra structure. If you look 
e.g. at categorical algebra, you will understand. For instance, 
investigate tree automata over monoidal closed categories, and 
interesting things happen.

I don't think graph theory is the slums of topology. Graph theory may 
have some relation to universal algebra (tree are terms?!), though 
certainly not being the slums of it. In general, I would be careful 
about saying anything in direction of math A being absorbed by math B. 
It's too sandbox for me, thank you.

Nor would I use "the intersection of math A and math B" (no critics 
against you, Timothy, in any way), and not even "the union of math A and 
math B" or "the complement of of math A with respect to math B", even if 
the latter two to me sound a bit more appealing than just 
"intersection". Union I would see as synergy and complement as 
influence. Maybe intersection then is interest in cooperation and 
communication, interest in learning from each other. Intersection as 
crossroads, so as to say. Then "the intersection of math A and math B" 
very much makes sense,

So math Category Theory is not a subset of math Graph Theory.

Best,

Patrik


On 2019-10-02 21:56, Louis H Kauffman wrote:
> Yes, a category is a digraph with extra structure. And one of the more
> important collections of categories are the categories generated from
> digraphs via the structure of paths on the given digraph.
> My point is that the concept of a graph is involved through and
> through with just about all aspects of mathematics. People often feel
> that they can make a point by contrast, and it gets remembered for too
> long.
> J. H. C. Whitehead famously said “Graph theory is the slums of
> topology.” I’ll not comment on that. But I think it was an intuitive
> avoidance of the sort of complexity in topological invariants that can
> come from
> deep combinatorial definitions. And at the same time Whitehead made
> such definitions with the Whitehead Groups and the concept of CW
> complexes. The dam broke in the 1980’s with advent of the Jones
> polynomial, its relation with the Tutte polynomial and
> contraction/deletion relations, relations with statistical physics,
> topological quantum field theory and categorical formulations that
> span the gamut from graph theoretic structures to braided monodical
> categories, field theory and beyond into present day link homology and
> its relationships with graphs and with the homology of categories.
> Mathematics is a whole and
> it is silly to argue about why some people have prejudices.
> 
> 
>> On Oct 1, 2019, at 4:24 AM, Timothy Y. Chow <tchow at math.princeton.edu> 
>> wrote:
>> 
>> 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_______________________________________________
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