6.2. Affine spaces

Let us apply the general definitions of second-order invariants in categories, to two geometries (describing distinct but related categories of spaces):

Affine geometry describes affine spaces E, with morphisms called affine maps ∈ Aff(E,F), endomorphisms called affine transformations ∈ Aff(E)=Aff(E,E), and automorphism groups called affine groups GA(E);
Linear geometry describes vector spaces E, with morphisms called linear maps ∈ L(E,F), endomorphisms called linear transformations ∈ L(E), and automorphism groups called linear groups GL(E).

Let us introduce them by first listing some intuitive facts, focusing on finite-dimensional spaces, which fall into one isomorphism class for each dimension n∈ℕ. We shall not try to classify infinite-dimensional spaces, thus not try to define their dimension otherwise than "infinite", especially as it depends on the choice of a precise foundation of these geometries. Indeed multiple foundations are possible (we should say multiple affine geometries, and corresponding multiple linear geometries), which are equivalent for finite-dimensional spaces but not for infinite-dimensional ones. We shall describe two of them:

The algebraic foundation, letting the affine category be a variety, is the weakest and the traditional one;
The duality foundation will turn out to be both more useful to work on finite dimensional spaces, and to fit a "better" category of spaces for the infinite dimensional ones.

Intuitive descriptions of affine maps

An affine map from a plane P to a plane Q can be intuitively understood as the approximated transformation by perspective appearance, over any figure in P, obtained by placing P in the space E, then taking a picture of this, from far away and eventually through a mirror, into the picture plane Q. This is the composite of two affine maps:

An injective one from P to E ;
a surjective one from E to Q (taking a picture of any object Z in E, from a viewpoint as much further away from Z than the size of Z, as needed for the desired accuracy).

Affine subspaces

The notion of (affine) subspace of an affine space E is defined as the set of images of affine maps to E. Intuitively, affine subspaces are straight. In the affine geometries we shall express (while others might differ on infinite dimensional cases), they are affine spaces themselves, thus also images of injective affine maps.
The dimension of subspaces is defined as preserved by injective affine maps. Subspaces of an n-dimensional affine space E fall into types E_p by their dimension 0 ≤ p ≤ n :

E₀ ≡ E as zero-dimensional subspaces are the singletons ;
E₁ is the notion of straight line;
E₂ is the notion of plane ;
E_n−1 is the notion of hyperplane ;
E_n = {E}, not structuring.

The relation between E₀ and any other E_p that is ∈ in the standard set theoretical intepretation (when straight lines are sets of points), is called incidence in generic ones.

Straight lines and algebraic structure

Any two points x ≠ y of an affine space E define one straight line containing them. Precisely,

The pair {0,1} is an affine basis of ℝ, so ∀x,y∈E, {l∈Aff(ℝ,E) | l(0) = x ∧ l(1) = y} = {l_x,y} ;
Denoting x∨y = Im l_x,y, we have x∨x = {x} ∈ E₀
if x ≠ y then this l_x,y is injective, so x∨y is a straight line. Thus E₁ = {a∨b|a,b∈E, a≠b}
if x ≠ y then x∨y is the only straight line containing x and y.

1. gives ℝ a role of set of binary operation symbols, which suffices to define all the algebraic language of affine geometry we shall review later (barycenters).
(1.∧(H)) ⇔ (Bas) ⇒ (4.∧(2.⇔3.)) where

(H) {h∈Aff(ℝ) | h(0) ≠ h(1)} = GA (ℝ)
(Bas) Any pair in ℝ is an affine basis.

Directions

Let us call direction in E, any partition of E defined as the set D = Dir f of preimages of points by some affine map f∈Aff(E,F). Its affine structure, given by that of Im f, is unique for finite dimensional spaces but may otherwise depend on f. In the latter case we have an initial one for the given partition D, i.e. for which the quotient map q∈Aff(E,D) (that is surjective with D = Dir q) satisfies for any affine map f with domain E, ∼_q ⊂ ∼_f ⇒ f/q∈Aff(D,F)).
Elements of directions are subspaces of E. Conversely for the algebraic foundation (thus anyway in finite dimension), each subspace H of a space E belongs to a unique direction D of E, whose dimension is called the codimension of H in E. It satisfies dim E = dim H + codim H. A hyperplane is a subspace with codimension 1.
This defines an equivalence relation among subspaces: two subspaces of an affine space are parallel if, equivalently

They are in the same direction.
They are either equal, or disjoint hyperplanes in a common subspace.

Like any function, any affine map f can be seen as a composite f = g০h of a surjective one h, with an injective one g, namely by regarding the subspace Im f as a space in itself.

In particular, any section g∈Mor(F,E) with an inverse retraction h∈Mor(E,F), i.e. such that h০g = Id_F, define an idempotent affine transformation f = g০h of E, called the projection to Im g = Im f with direction Dir h = Dir f. This is a limit of ( = can be approached by) automorphisms that shrink and progressively "smash" the space onto Im f.

However, we shall see that the (algebraically defined) direction D of a subspace with both infinite dimension and infinite codimension, may be a quite awkward kind of affine space, where some lines (injective l∈ Aff(ℝ,D)) can only be proven to be sections by the axiom of choice, but with only "monstruous" (physically nonsensical) retractions (DC does not suffice). Such spaces D are physically unsatisfactory regardless whether we make all its straight lines to be sections using AC, or not.
The below duality foundation will allow to dismiss such D from its category of spaces. In that way, while parallelism always makes sense (from an algebraic definition), some subspaces will not directly belong to a direction in the sense of an acceptable affine space, but only have one direction which fits them best.

Affine forms

For any affine space E, consider its ℝ-dual E' = Aff(E,ℝ), whose elements are called affine forms. Each non-constant affine form defines a direction of hyperplanes, and can be visualized by the choice of two of these : the preimages of 0 and 1.
Accepting all the above properties of straight lines, all non-constant affine forms are retractions, and the claim that all straight lines are sections is equivalent to the effectiveness of E'.
For this and other reasons, we shall take duality as a foundation (framework) of affine and linear geometries. This means taking the ℝ-dual as the only primitive structure, with the axiom of effectivity, so that all algebraic structures are definable from it. This foundation will be further explained in the next page.

Other affine structures

Structures of affine spaces E with dimension n≥2, are expressible from any notion of subspace E_p for 0 < p < n, such as the notion of straight line, also expressible by the ternary relation of alignment (existence of a line containing 3 given points). The notion of segment, or equivalently the ternary relation of betweenness z∈[xy] = {l_x,y(a)|0≤a≤1}, seems richer but is finally definable from alignment ; it is specific to the use of ℝ (with its order) vs. other systems such as ℂ. Same for the notion of (straight) half-line.
More invariant structures of affine geometry (definable from duals) will be introduced later:

In dimension 2 or more, is the notion of parallelogram, conceived either as 4-tuples, or as whole figures with sides and inside (definable using the order in ℝ), and similarly parallelepipeds of any dimension.
Vectors, i.e. translations, forming a subgroup of the affine group
Ratio of lengths of parallel segments; more generally, ratios of "volumes" of shapes in parallel p-dimensional subspaces (0<p≤n), approximated by ratios of numbers of small parallelepipeds, images of each other by translations, filling/covering these shapes.
In planes: notions of ellipse, parabola, hyperbola; center of an ellipse or hyperbola (but no data on foci, axis and eccentricity). Similar notions exist in higher dimensions.

More detailed study of affine geometry in another page.

6. Geometry
Previous: 6.1. Introduction to the foundations of geometry
Next: 6.3. Duality

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