5.2. Invariants in concrete categories

Multiple automorphisms and models

Most geometries, except RCF, have a big group of automorphisms in each model. To produce such theories from a framework without automorphisms (ZF with only sets and no pure elements, or second-order arithmetic), requires not only developments but also to forget some structures, and notice how some permutations then become automorphisms.

Like in a set theoretical framework, for geometries with nontrivial automorphisms, multiple models (spaces) may be interestingly considered in each isomorphic class (instead of taking one to represent them all), for the following reason. Any isomorphism f between 2 systems E and F induces a bijection gfg from Aut(E) to the set of isomorphisms between E and F. Thus, a plurality of automorphisms of E means a plurality of isomorphisms between E and F. This plurality makes the difference between E and F meaningful, as an object in E may correspond to several possible ones in F, depending on the choice of isomorphism, so choosing an object in E does not mean choosing an object in F in any invariant way. By contrast, the uniqueness of isomorphism between standard models of ℕ or ℝ makes them unambiguously play the role of each other (copies of each other), rendering their plurality superfluous.

The Galois connection (Aut, sInv)

For any set E, the relation of strong preservation between RelE and the symmetric groupE induces a Galois connection (sInv, Aut) between their powersets (similar to (Inv, End) previously seen): for any set G⊂ ⤹E of permutations, the set of strong invariants of G is

sInv (G) = Inv (G ∪ -G) = Inv (G) ∩ Inv(-G)
L⊂RelE, ∀G⊂ ⤹E, L⊂sInv (G) ⇔ G⊂AutL(E).

If G is a group then G = -G, thus sInv (G) = Inv (G).

The double meaning of "invariance"

We initially conceived invariant structures as those defined by expressions without parameters. This implies invariance by automorphisms.

Like proofs explore truths (theorems), that is the closure of the set of axioms, whose addition to axioms preserves the class of models, definitions from a language LR explore its closure sInv(AutL(E)) = Inv(AutL(E)) ⊃ L, whose addition to L leaves unchanged the set of automorphisms of each model (among other isomorphisms between models). Constructions leave unchanged the groups of automorphisms as abstract groups, just providing more types on which these groups act.
Our last scheme of constructions (set of all structures defined by an expression with all values of parameters), was that of an invariant second-order structure that is a set of non-invariant first-order structures. Well-defined by that formula, its invariance by automorphisms means that automorphisms preserve that set while acting as permutations on it. 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.

But unlike provability for which the completeness theorem gave a converse, a subtle difference separates both concepts of invariance as follows.
A tuple t can be distinguished by a structure r ∈ Inv(AutL(E)) as tr from tuples outside r, when no element of AutL(E) can move it out. This intuitively suggests that t is not similar to anyone outside r with respect to L, unless the trouble is to choose an automorphism witnessing the similarity. But this dissimilarity, and thus r itself, may be inexpressible by formulas with language L if this would require an infinitely complex description. For example, there is no nontrivial automorphism in ℝ, or in a model of second-order arithmetic, yet objects of an uncountable type (real numbers or sets of integers) cannot be all defined without parameter (but, as objects, they are definable with a parameter: themselves).
Both concepts of invariance can be still reconciled in different ways:

Using morphisms to form geometries

By its general definition, the automorphism group for any theory with a finite language is a second-order construction. In classical geometries (but not topology), it is even first-order constructible from the space. For example, rotations are automorphisms in Euclidean geometries, and can be specified by just a few parameters (depending on the dimension: center or axis, angle... or the image of one tuple). Anyway geometries generally admit their own automorphism group either as a type, or as a subset of a type (especially the type of endomorphisms, if different).

A core idea of Felix Klein's Erlangen Program for the foundations of geometry, was to rebuild the geometry of a space E from a permutation group G of E intended as its group of automorphisms: can we find a good list L of a few structures from its set of invariants (Inv G), which would suffice to define other useful invariant structures (yet not the uncountable set of them all), at least so that G=AutL ?
As we saw with relations defined by tuples, the set Inv(G) is made of orbits of tuples, and any unions of these.

More generally, a convenient approach to some important geometries is to start with the concrete category of their intended spaces, whose class of morphisms first comes from intuition, then draw from this the needed list of structures. Let us review more methods to produce from there useful invariants for the formalization of geometries.

Basis and algebraic structures

Algebraic structures come as relations defined by tuples, which happen to be functional (like with the interpretations of drafts in algebras): let us call basis of a space K any subset BK such that (K,IdB) is an initial object of the category of all (M,u) where M is a space and uMB:

uMB, ∃!f∈Mor(K,M), f|B=u.

This gives K the role of a set of B-ary operation symbols, interpreted in each M by re-currying this family of morphisms f indexed by u.

Parametered subspaces

Equivalently we may regard this K as a type. The introduction of K to the list of types can be admitted as a construction ex nihilo since the version of the theory that admits K as an additional type (instead of as a model of the initial theory) does not let any endomorphism act nontrivially on it. This may be understood as well by taking the basis B as a set of (values of) constants, or by taking K itself as a type of constants (which remains possible if there is no basis). Anyway all possible structures on K are considered "invariants" in this theory. This role of K may be played by ℝ or more generally any ℝn, or any subset of it which would form a space in the considered category. Usually K=ℝ or ℝ² already suffices to define all structures that could be brought by further ℝn. The set Mor(K,M) then constitutes a second-order structure, that is the set of K-parametered subspaces of M.
The category has a natural action on this type on the same side as it acts on the space: for any spaces M and N, any f ∈ Mor(M,N) defines a function from the type M0=Mor(K,M) to N0 = Mor(K,N) by composition: ∀xM0, fxN0.
Such constructions are interestingly similar to those of the representation theorem for categories.

Subspaces, embeddings and sections

More second-order structures on M that are invariant by the automorphisms of M, can be defined from the sets Mor(K,M) but removing K from what these structures are in the sense that they can be taken as primitive in the formalism (not using K as either base set or set of symbols): just types of subsets of M.
First is the set {Im f| f ∈ Mor(K,M)} which is a certain set of subspaces of M : those which can be parametered by K. The dependence on K in this definition (while any definability issue is ignored when structures are taken as primitive) is more precisely a dependence on the isomorphism class of K.

Two invariant subsets of this (but which may not bring anything new depending on the geometry) are the embeddings and the sections. As already discussed, these are successive qualities for subspaces A = Im f to become spaces on their own right, which means (instead of structures for A) to give its morphisms with other spaces. For example, if a geometry admits a circle and a plane as spaces, the circle may have an embedding into the plane but that is not a section. (In the case of differential geometry, this concept of embedding may be less strict than the standard one, I did not check)

Sets of subpaces of the form HM = {Im f| f ∈ Mor(K,M)} are preserved in the sense that

f∈Mor(M,N), ∀AHM, f[A]∈HN

Otherwise, the sets of embeddings, or sections, or embedded subspaces, or section subspaces, are preserved by isomorphisms, but not necessarily by other morphisms depending on the geometry.

Dual types

Except for algebraic structures, the above works can be repeated with sides reversed, using "dual types" of the form Mor(M,K) instead of Mor(K,M). Morphisms act on these dual types by the oppose side to their natural action on spaces: any morphism from M to N defines a function from Mor(N,K) to Mor(M,K).
The similar concept to embeddings with reversed sides, is the concept of quotient space by a given relation.

(*)This bridge to Inv(AutL(E)) by means of extended kinds of "definitions" (with infinite amounts of data) has been investigated by Borner, Martin Goldstern, and Saharon Shelah : short presentation in .ps - full article.

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