5.3. Affine spaces
This page is under construction.
Let us apply the tools just introduced 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
finitedimensional spaces, which fall into one isomorphism class for each
dimension n∈ℕ. We shall not try to classify infinitedimensional 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 finitedimensional
spaces but nonequivalent for infinitedimensional ones. We shall describe two
of them:  The algebraic foundation 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 as concerns the
infinite dimensional ones. It will let all
injective morphisms be sections,
and all surjective morphisms be retractions (which the algebraic foundation only ensures
in finite dimensions).
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 ndimensional affine space E fall into types
E_{p} by their dimension 0 ≤ p ≤ n :

E_{0} ≡ E as zerodimensional subspaces are the singletons ;
 E_{1} is the notion of straight line;
 E_{2} is the notion of plane ;
 E_{n−1} is the notion of hyperplane ;
 E_{n} = {E}, not structuring.
The relation between E_{0} 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} ∈ E_{0}
 if x ≠ y then this l_{x,y} is injective, so
x∨y is a straight line. Thus E_{1} = {a∨ba,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 ℝdual E' = Aff(E,ℝ), whose elements
are called affine forms. Each nonconstant 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 nonconstant 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 ℝdual
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) halfline.
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 4tuples,
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 pdimensional 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.
5. Geometry
5.1. Introduction to the foundations of geometry
5.2. Invariants in concrete categories
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