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.

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

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'.

From any concrete category having K as an object, the K-dual type E' = Mor(E,K) is invariant (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'.
If f∈Mord(E,F) is surjective (Im f = F) then tf is injective from F' to E'.
The effectiveness of a dual E' (∀xyE, ∃uE', uxuy) implies that of Mord(F,E) on the dual type (∀fg∈Mord(F,E), tftg).
When effective, the duality structure contains all algebraic structures: it defines any L-algebra structure on E from that on K by E'⊂ MorL(E,K) ⇒ E∈SubLKE' as a products of algebras: using the injection h = ∏E' : EKE',

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

The effectiveness of F' gives Mord(E,F) ⊂ MorL(E,F): ∀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 concepts of quotients by equivalence relations. Here again depending on the category, a partition (quotient) of a space defined by a morphism f may either always or sometimes have a space structure (identified with that of Im f, when it can be considered a space) ; this structure may be either unique or dependent on f giving the same partition; among these, there may be an initial one, which will then be preferred.
For algebraic structures, the one on a quotient exists and is unique (independent of the morphism defining the given partition, provided that one exists).

Homepage : Set Theory and Foundations of Mathematics