Subject: categories: Re: Undirected graphs citation

From: Vaughan Pratt <[email-masked]>
Date: Wed, 01 Mar 2006 12:58:14 -0800

Marco Grandis wrote:
 > It is the 2-truncation of "symmetric simplicial sets" as presheaves
 > on finite cardinals, cf (*).

Marco, thanks for that, this is really nice.  It hadn't occurred to me
to extend undirected graphs to higher dimensions but ... of course!

While "symmetric" is technically correct terminology here (and indeed
graph theorists often define undirected graphs as the symmetric case of
directed graphs), "undirected" conveys the appropriate intuition that
the edges and higher-dimensional cells have no specific orientation.
Whereas the automorphism group of a directed n-cell is the trivial
group, that of an undirected n-cell is S_N where N=n+1, i.e. undirected
n-cells are permitted to "flop around" in all N! possible ways.
Moreover the group as a whole behaves like a single cell with regard to
identification: if one of the N! copies of an undirected edge is
identified with a copy of another undirected edge, all copies are
identified bijectively, i.e. the two undirected cells are identified.

So without taking issue with Marco's terminology "symmetric" here, since
it is correct and natural, I would nevertheless like to suggest that in
the context of simplicial complexes, and with ordinary graphs as a
precedent, that "undirected" be considered an acceptable synonym for
"symmetric".

But that connection leads to another that hadn't previously occurred to
me (though again this is unlikely to be news to at least some).  This is
the question of an appropriate language for the signature of simplicial
complexes in general.

Each operation can be named with a lambda-calculus term of the form
\xyz.xyzzy, that is, a string of (distinct of course) variables followed
by another string of the same variables with repetitions or omissions
allowed.  Dually to undirected simplicial complexes being a special case
of (directed) simplicial complexes, the language for the latter is the
special case of that for the former in which the body of the lambda term
preserves the order of the formal parameter list; the smallest term thus
disallowed is \xy.yx.

In particular s and t (source and target) arise as respectively \xy.x
and \xy.y: given an edge, bind x and y to its source and target
respectively and return the designated vertex.  Similarly \x.xx denotes
the distinguished self-loop at a given vertex x (these being reflexive
graphs since we allow contraction).  The lambda terms with N=n+1
parameters have as domain the set of n-cells.

The one operation that undirected graphs have that is absent in the
general directed case is \xy.yx, which names the other member of the
group of automorphic copies of an undirected edge.  These two copies
always travel together (literally as a group), justifying the intuition
that the group of both of them constitutes a single edge (or n-cell).
For general n these copies of a given cell are named by the linear
lambda terms, those with exactly one occurrence of each formal
parameter.  Any given cell of a graph attaches to the rest of the graph
at various points around that cell, but graph homomorphisms cannot
disturb those points of attachment or incidence, though it can certainly
map the cell to any of the N! isomorphic copies of itself.

It should be pointed out that "undirected graph" as a "special case" of
"directed graph" has its syntactic rather than semantic meaning here, in
the sense that UGraph (undirected graphs) does not embed in DGraph
(directed graphs), at least not in the expected way.  Consider a graph
with two vertices x,y, two edges from x to y, and two edges from y to x.
   If a graph homomorphism identifies the two edges from x to y, it need
not identify the other two edges in DGraph, but it does need to identify
them in UGraph.

Unless I've overlooked some subtlety, 2-UGraph does however embed in the
expected way in 2-DGraph, where 2 = {0,1} (= V in enriched parlance) are
the possible cardinalities of "homsets", i.e. at most one edge in each
direction.  This is because the implicit pairing in 2-DGraph perfectly
mimics the explicit pairing in 2-UGraph.   This would explain why graph
theorists, who usually work in 2-DGraph, encounter no ambiguity of the
Set-UGraph < Set-DGraph kind when they define an undirected graph as
simply a symmetric graph, one with no one-way streets.

Vaughan Pratt

Marco Grandis wrote:

 > Vaughan Pratt asked about:
 >
 >>  undirected graphs ...  as presheaves on the full subcategory 1 and
 >> 2 of Set?
 >
 >
 >
 > It is the 2-truncation of "symmetric simplicial sets" as presheaves
 > on finite cardinals, cf (*).
 >
 > Curiously, symmetric simplicial sets have been rarely considered.
 > Even if simplicial complexes (well-known!) are a symmetric notion and
 > have a natural embedding in symmetric simplicial sets.  While
 > simplicial sets are a directed notion, used as an undirected one in
 > classical Algebraic Topology.
 >
 > (*) M. Grandis, Finite sets and symmetric simplicial sets, Theory
 > Appl. Categ. 8 (2001), No. 8, 244-252.
 >
 >
 > Marco Grandis
 >