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*To*: categories@mta.ca*Subject*: categories: Adjoint cylinders*From*: F W Lawvere <wlawvere@acsu.buffalo.edu>*Date*: Wed, 1 Nov 2000 21:22:24 -0500 (EST)*Reply-To*: wlawvere@acsu.buffalo.edu*Sender*: cat-dist@mta.ca

I would be happy to learn the results which Till Mossakowski has found concerning those situations involving an Adjoint Unity and Identity of Opposites as I discussed in my "Unity and Identity of Opposites in Calculus and Physics",in Applied Categorical Structures vol.4, 167-174, 1996. Two parallel functors are adjointly opposite if they are full and faithful and if there is a third functor left adjoint to one and right adjoint to the other; the two subcategories are opposite as such but identical if one neglects the inclusions. A simple example which I recently noted is even vs odd. That is, taking both the top category and the smaller category to be the poset of natural numbers, let L(n)=2n but R(n)=2n+1. Then the required middle functor exists; a surprising formula for it can be found by solving a third-order linear difference equation. I hope that Till Mossakowski's results may help to compute some other number-theoretic functions that arise by confronting Hegel's Aufhebung idea (or one mathematical version of it) with multi-dimensional combinatorial topology. Consider the set of all such AUIO situations within a fixed top category. This set of "levels" is obviously ordered by any of the three equivalent conditions : L1 belongs to L2, R1 belongs to R2, F2 depends on F1. (Here "belongs" and "depends" just mean the existence of factorizations, but in dual senses). However there is also the stronger relation that L1 might belong to R2; for a given level, there may be a smallest higher level which is strongly higher in that sense, and if so it may be called the Aufhebung of the given level. In case the given containing category is such that the set of all levels is isomorphic to the natural numbers with infinity (the top) and minus infinity (the initial object=L and terminal object=R), then the Aufhebung exists, but the specific function depends on the category. Mike Roy in his 1997 U. of Buffalo thesis studied the topos of ball complexes, finding in particular that both Aufhebung and coAufhebung exist and that they are both equal to the successor function on dimensions. Still not calculated is that function for the topos of presheaves on the category of nonempty finite sets. This category is important logically because the presheaf represented by 2 is generic among all Boolean algebra objects in all toposes defined over the same base topos of sets, and topologically because of its close relation with classical simplicial complexes. Here the levels or dimensions just correspond to those subcategories of finite sets that are closed under retract. It is easy to see that the Aufhebung of dimension 0 (the one point set) is dimension 1 (the two-point set and its retracts), but what is the general formula ? ************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************

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