Subject: categories: Adjoint cylinders

From: F W Lawvere <[email-masked]>
Date: Wed, 1 Nov 2000 21:22:24 -0500 (EST)

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 ?

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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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