Wednesday, November 08, 2006

search and replace job. (projective toposes and the like)

The following text was taken from Wikipedia's page about projective geometry, except that "object " replaces "point", and "morphism" replaces "line" and "topos" replaces geometry.
I hope someone finds it interesting.

non-Euclidean topos that formalizes one of the central principles of perspective art: that parallel morphisms meet at infinity and therefore are to be drawn that way. In essence, a projective topos may be thought of as an extension of Euclidean topos in which the "direction" of each morphism is subsumed within the morphism as an extra "object", and in which a "horizon" of directions corresponding to coplanar morphisms is regarded as a "morphism". Thus, two parallel morphisms will meet on a horizon in virtue of their possessing the same direction.

Idealized directions are referred to as objects at infinity, while idealized horizons are referred to as morphisms at infinity.

However, a projective topos does not single out any object or morphism in this regard -- they are all treated equally. Indeed, with the extension, the axiomatization becomes substantially simpler (based on Whitehead, "The Axioms of Projective topos"):

* G1: Every morphism contains at least 3 objects
* G2: Every two objects, A and B, lie on a unique morphism, AB.
* G3: If morphisms AB and CD intersect, then so do morphisms AC and BD (where it is assumed that A and D are distinct from B and C).

The reason each morphism is assumed to contain at least 3 objects is apparent when thinking of the original motivating example of a Euclidean space supplemented by the morphisms and objects at infinity. The 3rd object is the morphism's direction. Axiom 2 is thus seen to embody a form of Euclid's 5th postulate (which makes the designation of Projective topos as non-Euclidean ironic): given a object and a direction, there is a unique morphism containing the object lying in the given direction.

Because a Euclidean topos is contained within a Projective topos, with Projective topos having a simpler foundation, general results in Euclidean topos may be arrived at in a more transparent fashion. Moreover, as already seen with the preceding interpretation of Axiom 2, separate but similar theorems in Euclidean topos may be handled collectively within the framework of projective topos; for instance, parallel and nonparallel morphisms need not be treated as separate cases.

One can pursue axiomatization in greater depth by postulating a ternary relation, [ABC] to denote when three objects (not all necessarily distinct) are colmorphismar. A relatively simple axiomatization may be written down in terms of this relation as well:

* C0: [ABA]
* C1: If A and B are two objects such that [ABC] and [ABD] then [BDC]
* C2: If A and B are two objects then there is a third object C such that [ABC]
* C3: If A and C are two objects, B and D also, with [BCE], [ADE] but not [ABE] then there is a object F such that [ACF] and [BDF].

For two different objects, A and B, the morphism AB is defined as consisting of all objects C for which [ABC]. The axioms C0 and C1 then provide a formalization of G1; C2 for G2 and C3 for G3.

The concept of morphism generalizes to planes and higher dimensional subspaces. A subspace, AB...XY may thus be recursively defined in terms of the subspace AB...X as that containing all the objects of all morphisms YZ, as Z ranges over AB...X. Colmorphismarity then generalizes to the relation of "independence". A set {A, B,...,Z} of objects is independent, [AB...Z] if {A, B,...,Z} is a minimal generating subset for the subspace AB...Z.

The axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent form as follows. A projective space is of:

* (L1) at least dimension 0 if it has at least 1 object,
* (L2) at least dimension 1 if it has at least 2 distinct objects (and therefore a morphism),
* (L3) at least dimension 2 if it has at least 3 non-colmorphismar objects (or two morphisms, or a morphism and a object not on the morphism),
* (L4) at least dimension 3 if it has at least 4 non-coplanar objects.

The maximum dimension may also be determined in a similar fashion. For the lowest dimensions, they take on the following forms. A projective space is of:

* (M1) at most dimension 0 if it has no more than 1 object,
* (M2) at most dimension 1 if it has no more than 1 morphism,
* (M3) at most dimension 2 if it has no more than 1 plane,

and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar morphisms intersect -- the very principle Projective topos was originally intended to embody. Therefore, property (M3) may be equivalently stated that all morphisms intersect one another.

It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is meant to be on projective planes, a variant of M3 may be postulated. The axioms of (Eves 1997: 111), for instance, include (1), (2), (L3) and (M3). Axiom (3) becomes vacuously true under (M3) and is therefore not needed in this context.

Additional properties of fundamental importance include Desargues' Theorem and the Theorem of Pappus. In projective spaces of dimension 3 or greater there is a construction that allows one to prove Desargues' Theorem. But for dimension 2, it must be separately postulated.

Under Desargues' Theorem, combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically. The resulting operations will satisfy the axioms of a fields -- except that the commutativity of multiplication will require Pappus' Theorem. As a result, the objects of each morphism are in one to one correspondence with a given field, F, supplemented by an additional element, W, such that rW = W, -W = W, r+W = W, r/0 = W, r/W = 0, W-r = r-W = W. However, 0/0, W/W, W+W, W-W, 0W and W0 remain undefined.

The only projective topos of dimension 0 is a single object. A projective topos of dimension 1 consists of a single morphism containing at least 3 objects. The geometric construction of arithmetic operations cannot be carried out in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem. The simplest 2-dimensional projective topos has 3 objects on every morphism, with 7 objects and morphisms in all arranged with the following schedule of colmorphismarities:

* [ABC]
* [ADE]
* [AFG]
* [BDG]
* [BEF]
* [CDF]
* [CEG]

with the coordinates A = {0,0}, B = {0,1}, C = {0,W} = {1,W}, D = {1,0}, E = {W,0} = {W,1}, F = {1,1}, G = {W, W}. For an image review the Fano plane. The coordinates in a Desarguesian plane for the objects designated to be the objects at infinity (in this example: C, E and G) will generally not be unambiguously defined.

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