See this, if you need: uniform structure and coarse structure.

Given a set , endowed with an uniform structure or with a coarse structure, Instead of working with subsets of , I prefer to think about the trivial groupoid .

* Groupoids.* Recall that a groupoid is a small category with all arrows invertible. We may define a groupoid in term of it’s collection of arrows, as follows: a set endowed with a partially defined operation , , and an inverse operation , , satisfying some well known properties (associativity, inverse, identity).

The set is the set of objects of the groupoid, are the source and target functions, defined as:

– the source fonction is ,

– the target function is ,

– the set of objects is consists of all elements of with the form .

* Trivial groupoid.* An example of a groupoid is , the trivial groupoid of the set . The operations , source, target and objects are

– ,

– , ,

– objects for all .

In terms of groupoids, here is the definition of an uniform structure.

* Definition 1.* (groupoid with uniform structure) Let be a groupoid. An uniformity on is a set such that:

1. for any we have ,

2. if and then ,

3. if then ,

4. for any there is such that , where is the set of all with and moreover ,

5. if then .

Here is a definition for a coarse structure on a groupoid (adapted from the usual one, seen on the trivial groupoid).

* Definition 2.*(groupoid with coarse structure) Let be a groupoid. A coarse structure on is a set such that:

1′. ,

2′. if and then ,

3′. if then ,

4′. for any there is such that , where is the set of all with and moreover ,

5′. if then .

Look pretty much the same, right? But they are not, we shall see the difference when we take into consideration how we generate such structures.

* Question.* Have you seen before such structures defined like this, on groupoids? It seems trivial to do so, but I cannot find this anywhere (maybe because I am ignorant).

* UPDATE (01.08.2012):* Just found this interesting paper Coarse structures on groups, by Andrew Nicas and David Rosenthal, where they already did on groups something related to what I want to explain on groupoids. I shall be back on this.

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