Subject: categories: graph classifiers

From: John Stell <[email-masked]>
Date: Wed, 27 Oct 1999 16:30:25 +0100

The classifier, \Omega, for subgraphs has two nodes and five edges.
The bi-Heyting algebra of subgraphs of a graph has two complement
like operators, $\neg$ and $\suppl$. These satisfy
$G \meet \neg G = \bot$ and $G \join \suppl G = \top$.
One of these ($\neg$) arises from 
an endomorphism of $\Omega$ but the other does not.

The operations $\neg$ and $\suppl$ are nicely dual in some ways
but only one can be described in terms of $\Omega$.
Is there some kind of explanation for this?

It is possible to find a graph $\Delta$ and to describe a given 
subgraph as a morphism to $\Delta$ and to obtain $\suppl$ as an
endomorphism of $\Delta$. But although there is a morphism
from $\Delta$ to $\Omega$, we don't get $\neg$ via any endomorphism
of $\Delta$. $\Delta$ has four nodes corresponding to a classification
of nodes as (1) out (2) in and with all incident edges in (3) in and
with all incident edges out (4) in and with some incident edges in
and some out. Of course this is analogous to the edges in 
$\Omega$. There are four kinds of edge identified by $\Omega$
(working with a version of $\Omega$ for undirected graphs)
(1) in (2) out and with all incident nodes out (3) out and
with all incident nodes in (4) out and with some incident nodes out
and some in. Is there a formal sense in which $\Delta$ and $\Omega$
are dual to each other? Is there a better way to view $\suppl$ as
coming from some kind of classifier of graphs?

Does anyone have any comments or suggestions for relevant literature?

John Stell

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Dr John Stell
Department of Computer Science         email     [email-masked]
Keele University
Keele, Staffordshire,                  telephone +44 1782 584083
ST5 5BG   U. K.                        fax       +44 1782 713082
http://www.keele.ac.uk/depts/cs/Staff/Homes/John/homepage.html
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From: F W Lawvere <[email-masked]>
Date: Wed, 27 Oct 1999 15:59:41 -0400 (EDT)

The coHeyting complement (i.e. the "least supplement") is not a natural
endomorphism of the subobject functor, hence is not implemented by an 
endomap of the (unique) representing object for that functor.This
productive contradiction was apparently known already to medieval
logicians in the sense that certain logical operators are not preserved by
substitution (here one substitutes along any map in the topos). That
there is still information to be found about this was hinted at by my 1990
result presented at Como (see Springer Lecture Notes in Math 1488) where a
nontrivial class of presheaf toposes was shown to satisfy the Leibniz
product rule for the coHeyting boundary ("A and not A" where not means the
least supplement); this rule is equivalent to substitutivity along
projection maps but not all maps !

Unfortunately I don't fully grasp which is the graph Delta that Dr. Stell
is working with but it doesn't seem to be the following. Some
relevant concepts richer than a single subobject may also be
representable,for example, the concept of a subobject together with
another subobject whose union with it is the whole. The union map from
omega cross omega to omega classifies a subobject which does that representing
job and which has an obvious endomap which switches.This general
construction gives in the case of graphs a 9-edge graph with 3 nodes, I
believe. There seems to be no way to make these pseudo-supplements any
smaller for the general graphs since the top element is isolated in the
lattice of truth values) 

Bill Lawvere

 ***************************************************************
F. William Lawvere			Mathematics Dept. SUNY 
[email-masked]               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

*****************************************************************

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From: John Stell <[email-masked]>
Date: Fri, 29 Oct 1999 12:53:34 +0100

To clarify what I meant by the graph Delta in my earlier message,
we can proceed via hypergraphs. In what follows all graphs are
undirected, so Omega has two nodes and four edges. Define a hypergraph
to be a function h : E -> PN, where E and N are sets of edges and nodes
and PN is the powerset of N. Each hypergraph has dual h* : N -> PE
where h* n = {e \in E | n \in h e}. 

A morphism of hypergraphs is a pair of functions phi_N : N_1 -> N_2
and phi_E : E_1 -> E_2 such that P phi_N h_1 subseteq h_2 phi_E.
If H and K are graphs a hypergraph morphism from H to K may not
be a graph morphism since a loop can be mapped to an edge which is not
a loop. However, given any hypergraph h we can define a graph having
the same nodes as h but with an edge joining node x to y for every
edge in h incident with both x and y. If this graph is denoted
by G(h), hypergraph morphisms from a graph K to a hypergraph h
correspond to graph morphisms from K to G(h). The graph I called 
Delta before is G(Omega*) it has four nodes and nine edges.

To explain the interpretation of Omega*, let's denote the nodes of
Omega by 0 and 1 and the four edges by {1}1, {1}0, {0,1}0, {0}0.
Given a subgraph gamma : H -> Omega, the nodes and edges have the
following interpretations.
  0      nodes not in the subgraph
  1      nodes in the subgraph
  {1}1   edges in the subgraph
  {1}0   edges not in the subgraph but with both end nodes in 
  {0,1}0 edges not in the subgraph but with one end in and one out
  {0}0   edges not in the subgraph with both end nodes out.

Now give Omega* the following interpretation
  0      edges in the subgraph
  1      edges not in the subgraph
  {1}1   nodes not in the subgraph
  {1}0   nodes in the subgraph which are ends of a non-empty set of
         edges all of which are out of the subgraph.
  {0,1}0 nodes in the subgraph which are ends of some edges in the 
         subgraph and ends of some edges which are out of the subgraph.
  {0}0   nodes in the subgraph having all their incident edges in the 
         subgraph, or having no incident edges.

Given any subgraph gamma : H -> Omega, we get neg gamma from
the endomorphism of Omega switching 0 and 1 and taking {0}0
to {1}1. Using the above interpretation of Omega* we can construct a
hypergraph morphism gamma! : H -> Omega*. I don't have a neat
construction for gamma! except via the above interpretation of Omega*.
Now the endomorphism neg : Omega -> Omega dualizes to
neg* : Omega* -> Omega* and we compose this with gamma! to
get a hypergraph morphism from H to Omega* which represents suppl gamma.

John Stell