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categories: Re: graph classifiers

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 
wlawvere@acsu.buffalo.edu               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA