5.5. Duality
Duality systems
A category of sets with effective Kduals can see them symmetrically
as Kduality 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'}.
Their 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',
y⋅f(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 Mor = Mor_{d}, which intuitively means that duality
suffices to define all structures.
Dualization of a concrete category
Starting with a concrete category with a chosen
space K, each space E defines a Kduality 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
{(^{tt}f,^{t}f)  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 Kduality
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
Kdual of M.
In the opposite concrete category with Mor'(E',F') =
{^{t}f  f∈Mor(F,E)},
we have then Mor'(E',K) ≡ E''.
For a category of Kduality systems not given as a fruit of dualization, thus not having K as an object,
any object M with a one element basis allows to reinterpret this doubly concrete category as
a category of M'duality systems.
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 Kdualized category. And it is the clone of this Kdualized category.
Indeed
the Kdual of an object C_{n}
with a chosen basis of n elements is K^{n} ≡ Mor(C_{n},K).
This defines the interpretation of C_{n} in K as a family of nary operations
forming by its image a subspace of Op_{K}^{(n)} in
Kduality with K^{n}.
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 ndimensional for some n∈ℕ,
if it is isomorphic to ℝ^{n} (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}০f)_{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 ℝduality 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}০f =
^{t}f(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 ℝ^{n} form the algebraic language of ℝduals
(whose precise form depends on
the category)
L* = ∐_{n∈ℕ}
(ℝ^{n})', so that for any spaces E, F we have
∏E' : E → Mor_{L*}(E',ℝ) and
∀f∈Mor(E,F), ^{t}f ∈
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.
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⊂Rel_{K}.
For any set E, we have a Galois connection between
interpretations of R in E and subsets E'
⊂ K^{E}, defined as follows:
 Any E' ⊂ K^{E} defines an
interpretation of each nary r∈R in E
as r_{E}={(x_{1},...,
x_{n})∈E^{n}∀y∈E'
, r(y(x_{1}),...,y(x_{n}))}.
This is the only one making (∏_{y}_{∈}_{E'}
y) an embedding
from E to K^{E'}.
 Any interpretation of R in E gives a set E*=
Mor_{R}(E,K) ⊂ K^{E}.
Now let us consider duality theories, or kinds of Kduality
systems (E,E',〈 , 〉), that interpret R in E
as defined from E' in this way. (This forms a
firstorder 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:
 E' ⊂ E*
 As ∏_{y}_{∈}_{E'} y
is an embedding of E into K^{E'}
(injective as E is assumed to be separated by E'),
∏_{y}_{∈}_{E*} y
is also an embedding of E into K^{E}^{*}
(which means that the used Rstructure on E is
closed), i.e. (E,E*) is another Kduality
system giving the same Rstructure on E.
Thus, as the system E is a subset of an exponentiation of K
(namely K^{E'}), every axiom of the form (∀
variables) (F_{1} ∧ ... ∧ F_{n}) ⇒ G
(where (F_{1} ,..., F_{n}, G)
are relation symbols applied to variables) that is true in K,
is also true in E.
Theorem. For any two Kduality systems (E,E',〈
, 〉) and (F,F',〈 , 〉') interpreting R in E
and F in this way,
 Mor((E,E'),(F,F')) ⊂
Mor_{R}(E,F)
 Mor((E,E*),(F,F')) = Mor_{R}(E,F)
Proof.
As (∏_{y}_{∈}_{F'}
y) is an Rembedding from F to K^{F'},
we have
∀f∈F^{E},
f ∈ Mor_{R}(E,F) 
⇔ (∏_{y}_{∈}_{F'}
y)०f ∈ Mor_{R}(E,K^{F'}) 

⇔ ∀y∈F', y०f ∈
Mor_{R}(E,K)=E* 

⇔ f ∈ Mor((E,E*),(F,F')) 
As E'⊂E*, the condition f∈Mor((E,E'),(F,F')),
that is ∀y∈F', y०f ∈ E', is
more restrictive, thus the inclusion.
Now, let S ⊂ Pol R ⊂ Op_{K}. As seen
above, in any Kduality system (E,E') we have
E* ∈ Sub_{S}(K^{E}).
As the Rstructure in E stays unchanged whether it
is defined from E' or from E*, it remains also
unchanged using any X such that E' ⊂ X ⊂ E*.
This is the case for X = E'_{S} = the Salgebra
generated by E', that is the smallest subSalgebra
of E* (or equivalently of K^{E})
such that E' ⊂ X.
conceiving affine and vector
spaces as ℝduality systems, reduces the issue of formalizing
affine and linear geometries, to the following questions:
 Describing ℝ with its algebraic structures, and its axioms (including secondorder ones);
 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.
 Specify the axioms (criteria to accept duality systems as spaces): E
and E' will be made algebras by stability axioms
E ∈ Sub ℝ^{E'} and E' ∈ Sub ℝ^{E},
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 ⊂ 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 nary operations in Cl(S) form the Ssubalgebra
of E^{En} generated by the n
projections π_{i} : E^{n}→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(E^{n},E) ∈
Sub_{(Pol E)}(Op_{E}^{(n)})
Thus Pol E = ⋃_{n∈ℕ} Mor(E^{n},E)
is a clone of operations.
Embeddings and quotients
In a pure duality category, any f∈Mor(E,F) such that ^{t}f[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 subalgebra). 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 ^{t}f,
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.
Semimorphisms
The notion of semimorphism, 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 semimorphism from a model (E,E',K_{1})
to a model (F, F', K_{2}) will
consist in a triple (f,g,φ) where f : E→F
, g: F'→E' and φ:K_{1}↔ K_{2},
such that ∀x∈E, ∀y∈F',
〈f(x),y〉'
= φ〈x,g(y)〉.
(More developments will be added later)
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