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*To*: categories@mta.ca*Subject*: categories: Re: graph classifiers*From*: F W Lawvere <wlawvere@acsu.buffalo.edu>*Date*: Wed, 27 Oct 1999 15:59:41 -0400 (EDT)*In-Reply-To*: <38171A91.7CCEF897@cs.keele.ac.uk>*Reply-To*: wlawvere@acsu.buffalo.edu*Sender*: cat-dist@mta.ca

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

**Follow-Ups**:**categories: Re: graph classifiers***From:*John Stell <j.g.stell@cs.keele.ac.uk>

**References**:**categories: graph classifiers***From:*John Stell <j.g.stell@cs.keele.ac.uk>

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