5.3. Second-order invariants in concrete categories

A language L ⊂ Inv G of first-order invariants may not suffice to approach the concept of invariance defined by a given group : For this and more reasons which will appear later, the formalization of geometries needs second-order invariants. Here are ways to systematically produce them from concrete categories.

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.

For example, Euclidean space geometry admits the type "plane", each plane is a set of points, and automorphisms can move any plane to any other plane. On Earth, "the sea level" names a plane, distinguishing its points from those above or below it, but Euclidean geometry does not see it as invariant, i.e. does not accept the name "sea level" in its language. A language accepting it, would form a different geometry.
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).

Endomorphisms

The transformation monoid M = End E is an invariant second-order structure on E, containing the group G of automorphisms defined as its subset of invertible elements. Both are second-order constructible if the language is finite (or once its infinity of symbols is usable as type). In spaces of classical geometries with given finite dimensions, they are even first-order constructible, by a few parameters, such as the image of one tuple with enough elements depending on the dimension. For example, rotations (automorphisms in Euclidean geometry) can be specified by their center or axis, angle...
As invariants, they are preserved by isomorphisms, but not by other morphisms. The next ones will be preserved by other morphisms as well.

Parametered subspaces

For a fixed set K, giving to any set E in a class, a type of structures KEEK, defines a concrete category of the fFE preserving it in the sense that cf[KE] ⊂ KF, denoting ∀xEK, cf(x) = fx.

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, ℝn, or any subset of ℝn which would form a space. Actually ℝ and ℝ² will suffice as grounds to formalize geometry.

Following the concept of concretization, the type KE = Mor(K,E) of K-parametered subspaces of E is an invariant second-order structure as ∀f∈Mor(E,F), cf coincides on KE with Hom(K,f) : KEKF.
If K has a basis B then this structure is synonymous with a structure of K-algebra seeing K as the set of all B-ary algebraic symbols relevant to this category.

Dual types

A K-dual type is a type of structures E'KE, on each space E. It defines a concrete category whose duality morphisms f∈Mord(E,F)⊂FE are the functions preserving this duality structure by acting there on the opposite side as their transpose tf : F'E'

fFE, ∀uF', tf(u) = uf
Mord(E,F) = {fFE | tf[F']⊂E'}
f∈Mord(E,F), ∀g∈Mord(F,G), t(gf) = tftg : G'E'
f∈Mord(E,F), Im f = F ⇒ Inj tf.

The effectiveness of a dual E' (∀xyE, ∃uE', uxuy), which means the injectivity of

h = ∏E' : EKE'

implies that of Mord(F,E) on the dual type (∀fg∈Mord(F,E), tftg).
In any concrete category, any choice of an object K defines a K-dual type as E' = Mor(E,K).
This is an invariant of this category (Mor(E,F) ⊂ Mord(E,F)) because ∀f∈Mor(E,F), tf|F' = HomF(f,K) : F'E'.
Moreover Mord(E,K) = E' because IdKK' ∴ ∀f∈Mord(E,K), f = IdKf = tf(IdK) ∈ E'.
When effective, the duality structure suffices to define all algebraic structures on any object E as determined by those on K: E'⊂ MorL(E,K) ⇒ E∈SubLKE' as a product of algebras:

E'⊂ MorL(E,K) ⇔ (∀uE', φKLu = u০φE) ⇔ h০φE = ∏uE' φKLu = φKE'Lh.

Naturally as they are defined from duality, algebraic structures are also preserved by duality morphisms (Mord(E,F) ⊂ MorL(E,F)): as F' is effective,

f∈Mord(E,F), ∀uF', ufE'uf০φE = φKLuLf = u০φFLf.

Subspaces, embeddings and sections

From Mor(K,E), further invariant second-order structures on E can be defined, depending on the isomorphism class of K but no more using K as a base type or a list of symbols. First is the set {Im f | f ∈ Mor(K,E)} of images of K-parametered subspaces. Invariant subsets of this come by putting conditions on f: being injective, or an embedding, or a section. As already discussed, these are successive qualities for Im f to be a space, which means (instead of structures) to give its morphisms with other spaces.

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 form HE = {Im f| f ∈ Mor(K,E)} are preserved in the sense that f∈Mor(E,F), ∀AHE, f[A]∈HF.
But 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.

Quotients

Reversing the sides on concepts of subspaces which may be images of morphisms, injective morphisms, embeddings or sections, we get diverse ranges of equivalence relations. Let us further comment the one reversing the concept of embedding which gave statuses of sub-objects to some subsets.

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) P = E/R of an object E by an equivalence relation R, receives a status of quotient object (P,π) when P exists as object in this category (or can be added by isomorphism with an existing object) with π∈ Mor(E,P) that is initial among all f∈Mor(E,F) (for any object F) such that R ⊂ ∼f.
Then any f∈Mor(E,F), can be said to give to its image Im f an image structure by transport from the quotient object E/∼f when it exists; this structure is generally lower than (or equal to) that of the sub-object Im f (when it exists) which is then the greatest of them. So, the inverse transport on the latter defines another structure on the quotient E/∼f, greater than that of the quotient object which is thus minimal among such. In some categories such as any category of algebras, all such structures on any given sub-object (and thus also on any quotient) with variable f coincide.

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