5.5. Duality

Duality systems

A category of sets with effective K-duals can see them symmetrically as K-duality systems, made of two sets E,E', with an operation ⋅ : E × E'K that is effective on both sides, so that it can be interpreted both ways as a function evaluator seeing E' as a subset of KE (type of structures over E) and E as a subset of KE'. Their morphisms can be seen symmetrically as

Mor((E,E'),(F,F')) ⊂ Mord((E,E'),(F,F')) = {(f,g)∈FE×E'F'| ∀xE, ∀yF', yf(x) = g(y)⋅x}

where each of f and g is determined by the other, thus can be called its transpose. The category will be qualified as pure duality if this duality somehow "generates" its implicit "structures" in the sense that Mor = Mord.

Dualization of a concrete category

Starting with a concrete category with a chosen space K, each space E defines a K-duality system (E'',E') where E' = Mor(E,K) and E'' is the image of E in KE'. So we have a canonical surjection from E to E'', which is injective when E' is effective on E (so that E'' is identifiable with E). These form a category of duality systems where Mor((E'',E'),(F'',F')) is defined as {(ttf,tf) | f∈Mor(E,F)}.

As K was taken from the class of objects, it keeps the role of base type of this object, but gets a dual K' = End K that is a monoid with a right action on the dual E' of each space E. This K' will be identifiable with K for linear geometry but not for affine geometry.
The same situation on the other side is obtained by picking an object M with a basis of one element (if such an object exists), which is thus a monoid acting on each space E by Mor(M,E) ≡ E. In particular Mor(M,K) ≡ K, giving K the role of K-dual of M.
In the inverse concrete category with Mor'(E',F') = {tf | f∈Mor(F,E)}, we have then E ≡ Mor'(E',K).

Clone of operations

Clones of operations are the concrete version of abstract clones like tranformation monoids are the concrete version of monoids, generalizing these to all arities.
Namely, it is the image of a representation of an abstract clone in a set K, as a subset of OpK.
The clone of operations so defined from the clone of a concrete category with a chosen object K, is the image of this clone in the K-dualized category. And it is the clone of this K-dualized category. Indeed the K-dual of an object Cn with a chosen basis of n elements is Kn ≡ Mor(Cn,K). This defines the interpretation of Cn in K as a family of n-ary operations forming by its image a subspace of OpK(n) in K-duality with Kn.

Coordinate systems

Assume given a concrete category of "spaces", where any family of spaces has a product. Both categories of affine spaces and vector spaces, as we shall later formalize, will be in this case. Fix a choice of an object K (in practice K=ℝ). For any set I, we can give ℝI a status of product space, i.e. such that for any space E, Mor(E,ℝI) ≅ (E')I where E' = Mor(E,ℝ).
Equivalently, first ∀iI, πi ∈ (ℝI)', then this family (πi) is a dual basis of ℝI, which means a basis of its dual (ℝI)' in the opposite concrete category of dual spaces.

This allows to qualify a space as n-dimensional for some n∈ℕ, if it is isomorphic to ℝn (once proven the uniqueness of n ; in some categories this definition is too restrictive to provide existence, and must be modified).
An f∈Mor(E,ℝI) is an isomorphism when the family ∏f = (πif)iIE' I is itself a dual basis, that is called a coordinate system of E.

All ℝI-duality structures are then redundant with the ℝ-duality structure, which will thus suffice for most needs.
Similarly in a concrete category, if a space E has a basis and F has an effective dual Mor(F,K) then Mor(E,F) = Mord(E,F).

For any space E, Mor(E,ℝI) = Mord(E,ℝI). Indeed ∀f∈Mord(E,ℝI), (∀iI, πif = tfi)∈E') ∴ f∈Mor(E,ℝI).
F, (∀E, Mor(E,F) = Mord(E,F)) ⇔ (the morphism ∏F' ∈Mor(F, ℝF') is an embedding).

The duals of ℝn form the algebraic language of ℝ-duals (whose precise form depends on the category) L* = ∐n∈ℕ (ℝn)', so that for any spaces E, F we have ∏E' : E → MorL*(E',ℝ) and ∀f∈Mor(E,F), tf ∈ MorL*(F',E').
any affine space E is an embedded subspace of ℝE'.

We shall prove later the existence and uniqueness of the dimension for subspaces and quotients of finite dimensional spaces. Then, the dimension of any finite product of finite dimensional affine spaces, is the sum of their dimensions.

Duality theories with structures

A duality theory is a theory describing duality systems. Let R be a relational language, that is interpreted in K. We may even see it as R⊂RelK.

For any set E, we have a Galois connection between interpretations of R in E and subsets E'KE, defined as follows:
Now let us consider duality theories, or kinds of K-duality systems (E,E',〈 , 〉), that interpret R in E as defined from E' in this way. (This forms a first-order theory, while the claim of using of E* would form a second order theory).

By the properties of closures of the above Galois connection we have:

Thus, as the system E is a subset of an exponentiation of K (namely KE'), every axiom of the form (∀ variables) (F1 ∧ ... ∧ Fn) ⇒ G (where (F1 ,..., Fn, G) are relation symbols applied to variables) that is true in K, is also true in E.

Theorem. For any two K-duality systems (E,E',〈 , 〉) and (F,F',〈 , 〉') interpreting R in E and F in this way,
Proof.
As (∏yF' y) is an R-embedding from F to KF', we have
fFE, f ∈ MorR(E,F) ⇔ (∏yF' y)०f ∈ MorR(E,KF')

⇔ ∀yF', yf ∈ MorR(E,K)=E*

f ∈ Mor((E,E*),(F,F'))
As E'E*, the condition f∈Mor((E,E'),(F,F')), that is ∀yF', yfE', is more restrictive, thus the inclusion.
Now, let S ⊂ Pol R ⊂ OpK. As seen above, in any K-duality system (E,E') we have E* ∈ SubS(KE).
As the R-structure in E stays unchanged whether it is defined from E' or from E*, it remains also unchanged using any X such that E'XE*. This is the case for X = E'S = the S-algebra generated by E', that is the smallest sub-S-algebra of E* (or equivalently of KE) such that E'X.
conceiving affine and vector spaces as ℝ-duality systems, reduces the issue of formalizing affine and linear geometries, to the following questions:
  1. Describing ℝ with its algebraic structures, and its axioms (including second-order ones);
  2. We shall accept all duality morphisms. Works of functional analysis involving more primitive structures to restrict the class of morphisms, are beyond the scope of this exposition.
  3. Specify the axioms (criteria to accept duality systems as spaces): E and E' will be made algebras by stability axioms E ∈ Sub ℝE' and E' ∈ Sub ℝE, from different algebraic structures on ℝ which we have to describe. That will be all for us.

Clone generated by an algebraic structure

For any set S ⊂ OpE of operations in a set E, the clone of operations generated by S, is the set Cl(S)⊂ OpE of operations equivalently described as

This is a closure, whose closed elements (the elements of Im Cl) are called the clones of operations:
S∈Im Cl ⇔ ∀n∈ℕ, {πi}i<nS(n) ∈ SubS(OpE(n))

The unary part S(1) of a clone of operation, is a transformation monoid. Precisely, the unary part of the clone of operations generated by a given set of transformations, equals the transformation monoid generated by this set. Thus, the concept of clone of operations is an extension of that of transformation monoid.

Theorem. The set Pol E of polymorphisms of an object E of a concrete category, is a clone of operations.

Proof :
n∈ℕ, {πi}i<n ⊂ Mor(En,E) ∈ Sub(Pol E)(OpE(n))
Thus Pol E = ⋃n∈ℕ Mor(En,E) is a clone of operations.

Embeddings and quotients

In a pure duality category, any f∈Mor(E,F) such that tf[F']=E' is an injective embedding in the general sense. The converse is true if for any f∈Mor(E,F) the system (A,A') where A = Im f and A'={u|A | uF'} is an accepted object in the considered category (at least for any algebraic structure, A is a sub-algebra). Otherwise we may take this as a stricter definition of the concept of embedding for duality systems.
The duality structure on a quotient by a morphism f∈Mord(E,F), is defined by Im tf, but may depend on f. The initial one among these for an equivalence relation R on E is defined by {uE' | R⊂∼u}. Its effectivity is implied by the existence of an f∈Mord(E,F) such that R = ∼f and F' is effective.

Semi-morphisms

The notion of semi-morphism, generalizes the above by allowing the type K to be changed but connected by a bijection φ (that will have to be an isomorphism for internal structures given in K by the considered theory). Even if K is the same as a set (and even as a system), this still generalizes the above, as φ may be an automorphism other than IdK.
So, a semi-morphism from a model (E,E',K1) to a model (F, F', K2) will consist in a triple (f,g,φ) where f : EF , g: F'E' and φ:K1K2, such that ∀xE, ∀yF', 〈f(x),y〉' = φ〈x,g(y)〉.

(More developments will be added later)

Homepage : Set Theory and Foundations of Mathematics