- A well-describable
*L*independent of*E*may not fill Inv*G*(which is usually uncountable, but an uncountable*L*may be accepted either directly or as an additional type), in which case invariants outside*L*may be undefinable from*L*. - We need
*G*= Aut_{L}*E*, which will fail even for*L*= Inv*G*in the case of topology.

A type of structures *P*,
in its role of second-order structure, is invariant by automorphisms when it is stable by
the action of automorphisms on the full type of structures including *P*, thus when
automorphisms preserve *P* while acting as permutations on it.
Still each element of *P*, which is a first-order structure, can be non-invariant
as it does not stay fixed by this action.

Our last scheme of constructions (type of structures), was that of an invariant type of structures that is well-defined from first-order structures (by the defining formula).

As invariants, they are preserved by isomorphisms, but not by other morphisms. The next ones will be preserved by other morphisms as well.

As a development of any
geometry (we may call *constructions ex nihilo*), consider adding any space *K* as
a type with enough additional structures to not let endomorphisms act nontrivially on it.
This may be done by taking as set of constants, either all *K* (anyway), or any basis
of it (if one exists, which may have the advantage of being finite, or countable).
Then all elements and structures purely over *K* are invariants as well.
This role of *K* may be played by ℝ, ℝ^{n},
or any subset of ℝ^{n} which would form a space.
Actually ℝ and ℝ² will suffice as grounds to formalize geometry.

If

Such constructions are interestingly similar to those of the representation theorem for categories.

∀*f*∈*F ^{E}*, ∀

Mor

∀

Moreover Mor

If

The effectiveness of a dual

When effective, the duality structure contains all algebraic structures: it defines any

*E'*⊂ Mor_{L}(*E*,*K*) ⇔
(∀*u*∈*E'*, φ_{K}০* ^{L}u*
=

Depending on the geometry, these types may differ or not : images of non-injective morphisms may be images of sections with another domain. In topology or differential geometry, a circle and a plane are spaces, the circle has embeddings into the plane (the official concept of embedding there may be more strict than the general one) but they are not sections.

Sets of the formBut depending on the geometry, other sets of embeddings, or sections, or embedded subspaces, or section subspaces, may not be anyhow preserved by morphisms which are not embeddings or sections.

In categories of relational systems, any set may form diverse systems depending on the choice of interpretation of a given relational language, and this set of possible choices is ordered by inclusion. More generally in concrete categories, a given set may receive multiple statuses as object, defined (transferred) by bijections serving as isomorphisms with existing objects, and these form an ordered set: reflexivity and transitivity are what axioms of concrete categories give in this case of uniqueness of the requested morphism (that is Id on the set), while antisymmetry comes by definition of the equality of statuses as object.

Then a partition (quotient)Then any

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