6.3. Duality

Duality systems

A class of sets with effective K-duals can see its objects 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 K^E (type of structures over E) and E as a subset of K^E'. It defines a concrete category where morphisms can be seen symmetrically as

Mor((E,E'),(F,F')) ⊂ Mor_d((E,E'),(F,F')) = {(f,g)∈F^E×E'^F'| ∀x∈E,∀y∈F', f(x) ⋅ y = x ⋅ g(y)}

where each of f and g is determined by the other, thus can be called its transpose. We have a category of pure duality systems if Mor = Mor_d, which intuitively means that duality suffices to define all structures.
While this concept may be elementary enough, let us split it into smaller pieces forming a priori more general cases, and see how they are related.

Co-acting concrete categories and duality operations

A co-acting concrete category is a concrete category with a choice of co-action, thus giving to each object E (that is or plays the role of a set) a dual set E', and to each morphism f∈Mor(E,F) defined as a function f : E → F it gives a transpose ^tf : F' → E' satisfying both axioms of co-action.
A bi-concrete category is a co-acting concrete cateogory whose co-action is faithful (∀f,g∈Mor(E,F), ^tf = ^tg ⇔ f = g) so that the opposite category is also a (bi-)concrete category.

Let C be a co-acting concrete category. Then, a duality operation on C is defined by the choice of a fixed set K and for each object E an operation (abusively all written by the same symbol) ⋅ : E × E' → K, preserved in the sense of the above formula: for any objects E,F,

∀f∈Mor(E,F), ∀x∈E,∀y∈F', f(x) ⋅ y = x ⋅ ^tf(y)

If there is a co-egg (K,k) where K is an object and k∈K', the co-action is E' ≡_k Mor(E,K). It naturally defines a K-duality operation, which is effective on one side:

∀E,∀y≠z∈E',∃x∈E, x ⋅ y ≠ x ⋅ z.

This is an initial object of the following concrete category of all duality operations on the given co-acting concrete category : composing a K-duality operation with any function h:K→K₁ defines a K₁-duality operation, where h is considered the morphism between these duality operations. If the initial duality operation is not effective on a given side then no other one is.

Similarly, any egg (M,e) of a co-acting concrete category, naturally defines an M'-duality operation by x ⋅ y = ^tϵ{f∈Mor(M,E) | f(e)=x}(y), which is also initial among duality operations. But it is not ensured to be effective on either side, except under the assumption of bi-concreteness, by symmetry to the above.
So if a co-acting concrete category has both an egg (M,e) and a co-egg (K,k) then they define the essentially same duality operation (the detailed proof is left as an exercise):

M' ≡_k Mor(M,K) ≡_e K

If moreover the category is bi-concrete then this operation is effective on both sides.

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 K^E'. So we have a canonical surjection from E to E''. This surjection is injective when E' is effective on E, in which case we may identify E'' with E; in particular K''=K. 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, as this dualization process is essentially the general construction of categories of acts on the opposite side. This K' will be identifiable with K for linear geometry but not for affine geometry.
The set K can also appear as the dual an object in the resulting category of K-duality systems in this way: if an object M has a basis of one element, thus is a monoid acting on each space E by Mor(M,E) ≡ E, then Mor(M,K) ≡ K, giving K the role of K-dual of M.
In the opposite concrete category with Mor'(E',F') = {^tf | f∈Mor(F,E)}, we have then Mor'(E',K) ≡ E''.

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 Op_K.
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 C_n with a chosen basis of n elements is Kⁿ ≡ Mor(C_n,K). This defines the interpretation of C_n in K as a family of n-ary operations forming by its image a subspace of Op_K⁽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 ℝ^I a status of product space, i.e. such that for any space E, Mor(E,ℝ^I) ≅ (E')^I where E' = Mor(E,ℝ).
Equivalently, first ∀i∈I, π_i ∈ (ℝ^I)', then this family (π_i) is a dual basis of ℝ^I, which means a basis of its dual (ℝ^I)' 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,ℝ^I) is an isomorphism when the family ∏f = (π_i⚬f)_i∈I ∈E'^I is itself a dual basis, that is called a coordinate system of E.

All ℝ^I-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) = Mor_d(E,F).

For any space E, Mor(E,ℝ^I) = Mor_d(E,ℝ^I). Indeed ∀f∈Mor_d(E,ℝ^I), (∀i∈I, π_i⚬f = ^tf(π_i)∈E') ∴ f∈Mor(E,ℝ^I).
∀F, (∀E, Mor(E,F) = Mor_d(E,F)) ⇔ (the morphism ∏F' ∈Mor(F, ℝ^F') 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 → Mor_L*(E',ℝ) and ∀f∈Mor(E,F), ^tf ∈ Mor_L*(F',E').
any affine space E is an embedded subspace of ℝ^E'.

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.

Clone generated by an algebraic structure

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

defined by terms with variables (arguments, no parameter) ranging over E, and operation symbols in S.
∀n∈ℕ, Cl(S)⁽ⁿ⁾ = 〈{π_i}_i<n 〉_S ⊂ Op_E⁽ⁿ⁾, i.e. the n-ary operations in Cl(S) form the S-subalgebra of E^Eⁿ generated by the n projections π_i : Eⁿ→E (operations defined by each of the n free variables).

This is a closure, whose closed elements (the elements of Im Cl) are called the clones of operations:
S∈Im Cl ⇔ ∀n∈ℕ, {π_i}_i<n ⊂ S⁽ⁿ⁾ ∈ Sub_S(Op_E⁽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, {π_i}_i<n ⊂ Mor(Eⁿ,E) ∈ Sub_{(Pol E)}(Op_E⁽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 | u∈F'} 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∈Mor_d(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 {u∈E' | R⊂∼_u}. Its effectivity is implied by the existence of an f∈Mor_d(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 Id_K.
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 : E→F , g: F'→E' and φ:K₁↔ K₂, such that ∀x∈E, ∀y∈F', 〈f(x),y〉' = φ〈x,g(y)〉.

(More developments will be added later)

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