Mor((*E*,*E'*),(*F*,*F'*)) ⊂
Mor_{d}((*E*,*E'*),(*F*,*F'*)) =
{(*f*,*g*)∈*F ^{E}*×

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.

A

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* ⋅
^{t}*f*(*y*)

∀*E*,∀*y*≠*z*∈*E'*,∃*x*∈*E*, *x* ⋅ *y* ≠ *x* ⋅ *z*.

Similarly, any egg (

So if a co-acting concrete category has both an egg (

*M'* ≡_{k} Mor(*M*,*K*) ≡_{e} *K*

The set

In the opposite concrete category with Mor'(

Namely, it is the image of a representation of an abstract clone in a set

The clone of operations so defined from the clone of a concrete category with a chosen object

Equivalently, first ∀

This allows to qualify a space as *n*-dimensional 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}⚬*f*)_{i∈I}
∈*E'*^{ I} is itself a dual basis, that is called a
*coordinate system* of *E*.

Similarly in a concrete category, if a space

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* =
* ^{t}f*(π

∀

The duals of ℝ^{n} form the algebraic language of ℝ-duals
(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

any affine space

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.

- 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*generated by the^{En}*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.

∀n∈ℕ, {π_{i}}_{i<n}⊂ Mor(E^{n},E) ∈ Sub_{(Pol E)}(Op_{E}^{(n)})

Thus PolE= ⋃_{n∈ℕ}Mor(E^{n},E) is a clone of operations.

The duality structure on a quotient by a morphism

So, a semi-morphism from a model (

(More developments will be added later)

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