Uniform spaces, coarse spaces, dilation spaces (II)

Background:
(1) Uniform spaces, coarse spaces, dilation spaces

(2) What’s the difference between uniform and coarse spaces (I)

I shall use the idea explained in (1), in the groupoid frame of (2), classical situation of the trivial groupoid over a space X. In this case the uniform and coarse structures are just the classical notions.

This idea says that all we need to have is a field of dilations. With such an object we may construct an uniformity, or a coarse structure, then ask that the field of dilations has some properties with respect to the said uniformity (or coarse structure). If the field of dilations has those properties then it transforms into an emergent algebra  (in the case 0 below; in the other case  there is a new type of emergent algebra which appears in relation to coarse structures).

Remark. Measure can be constructed from a field of dilations, that’s for later.

Fields of dilations. We have a set X, call it “space”. We have the commutative group (0,\infty) with multiplication of reals operation, (other groups work well, let’s concentrate on this one).

A field of dilations is a function which associates to any x \in X and any \varepsilon \in (0,\infty)
an invertible transformation  \delta^{x}_{\varepsilon} : U(x) \subset X \rightarrow U(x,\varepsilon) \subset X which we call “the dilation based at x, of coefficient \varepsilon“.

We ask:

1. x \in U(x) for any point x \in X

2. for any fixed x \in X the function \varepsilon \in (0,\infty) \rightarrow \delta^{x}_{\varepsilon} is a representation of the commutative group (0,\infty).

3. fields of dilations come into 2 flavors (are there more?), depending on the choice between (0,1] and [1,\infty), two important sub-semigroups of (0,\infty).

(case 0) – If you choose (0,1] then we ask that for any \varepsilon \in (0,1] we have U(x,\varepsilon) \subset U(x), for any x,

This case is good for generating uniformities and  for the infinitesimal point of view.

(case \infty) – If your choice is [1,\infty) then we ask that for any \varepsilon \in [1,\infty) we have U(x) \subset U(x,\varepsilon) , for any x,

This case is good for generating coarse structures and for  the asymptotic point of view.

Starting from here, I’m afraid that my latex capabilities on wordpress are below what I need to continue.

Follow this working paper to see more. Thanks for comments!

PS. At some point, at least for the case of uniformities, I shall use “uniform refinements” and what I call “topological derivative” from arXiv:0911.4619, which can be applied for giving alternate proofs for rigidity results, without using Pansu’s Rademacher theorem in Carnot groups.

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