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' × EK 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

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.

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).

In a category of duality systems (where Mor=Mord), 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.

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 meeds.
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.

reducible (dethus making allconceiving affine and vector spaces as ℝ-duality systems. This reduces the issue o, 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.

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)

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