# Comfortable with shehes

A Bag of Contradictions

From: F William Lawvere, To: Posina Venkata Rayudu, Date: 30 Aug 2011

The lack of structure expressed by “completely distinguished from each other but with no distinguishing properties” for the elements of an abstract set is of course not an end in itself, but an attempt to insure that when we model mathematical objects as diagrams of maps in the category of sets, that background contaminates as little as possible and all properties are a result of the explicit assumptions in a particular discussion. The ancient Greeks had a similar concept of arithmos, in some sense the minimal structure derived from a given situation to which one could assign a number; in other words, an abstract set is just one hair less abstract than a cardinal number, but that enables them to potentially carry mappings, which cardinals are too abstract in concept (being equivalence classes) to do.

This sharp contradiction becomes very objective whenever we have a topos defined over another one by a

Unity & Identity of Adjoint Opposites

Discrete …. Points …. Codiscrete

where each is left adjoint to the next. The two opposites map the lower more abstract category to opposite ends of the upper more cohesive or variable category, while the middle abstracts from each object in the upper its denuded version.

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Adjoint cylinders

To: categories@mta.ca, Subject: categories: Adjoint cylinders, From: F W Lawvere wlawvere@acsu.buffalo.edu, Date: 1 Nov 2000

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

Elementary nature of the notion of adjointness

The motivation for introducing 40 years ago the construction, of which “categories of elements” is a special case, was to make clear the elementary nature of the notion of adjointness. Given an opposed pair of functors between two arbitrary given categories, one obviously elementary way of providing them with an adjointness is to give two natural transformations satisfying two equations; but very useful also is the definition in terms of bijections of hom sets which should be equivalent. The frequent mode for expressing the latter in terms of presheaf categories involved the complicated logical notion of “smallness” and the additional axiom that a category of small sets actually exists, but had the disadvantage that it would therefore not apply to arbitrarily given categories. By contrast, a formulation of this bijection in terms of discrete fibrations required no such additional apparatus and was universally applicable.

Unfortunately, since I had given the construction no name, people in reading it began to use the unfortunate term “comma”. It would indeed be desirable to have a more objective name for such a basic construction. (The notation involving the comma was generalized from the very special case when the two functors to B, to which the construction is applied, both have the category 1 as domain, and the result of the construction is the simple hom set in B between the two objects, which is often denoted by placing a comma between the names of the objects and enclosing the whole in parentheses.)

One habit which it would be useful to drop is that of agonizing over the true definition of elements. In any category the elements of an object B are the maps with codomain B, these elements having various forms which are their domains. For example, if the category has a terminal object, we have in particular the elements often called punctiform. On the other hand, it is often appropriate to apply the term point to elements more general than that, for example, in algebraic geometry over a non-algebraically closed field, points are the elements whose forms are the spectra of extensions of the ground field. As Volterra remarked already in the 1880s, the elements of a space are not only points, but also curves, etc.; it is often convenient to use the term “figure” for elements whose forms belong to a given subcategory.

Bill Lawvere

Date: 30 Sep 2003, From: F W Lawvere, Subject: categories: Re: Categories of elements (Pat Donaly)