What’s the difference between uniform and coarse structures? (I)

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

Given a set $X$, endowed with an uniform structure or with a coarse structure, Instead of working with subsets of $X^{2}$, I prefer to think about the trivial groupoid $X \times X$.

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 $G$ endowed with a partially defined operation $m: G^{(2)} \subset G \times G \rightarrow G$, $m(g,h) = gh$, and an inverse operation $inv: G \rightarrow G$, $inv(g) = g^{-1}$, satisfying some well known properties (associativity, inverse, identity).

The set $Ob(G)$ is the set of objects of the groupoid, $\alpha , \omega: G \rightarrow Ob(G)$ are the source and target functions, defined as:

– the source fonction is $\alpha(g) = g^{-1} g$,

– the target function is $\omega(g) = g g^{-1}$,

– the set of objects is $Ob(G)$ consists of all elements of $G$ with the form $g g^{-1}$.

Trivial groupoid. An example of a groupoid is $G = X \times X$, the trivial groupoid of the set $X$. The operations , source, target and objects are

$(x,y) (y,z) = (x,z)$, $(x,y)^{-1} = (y,x)$

$\alpha(x,y) = (y,y)$, $\omega(x,y) = (x,x)$,

– objects $(x,x)$ for all $x \in X$.

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

Definition 1. (groupoid with uniform structure) Let $G$ be a groupoid. An uniformity $\Phi$ on $G$ is a set $\Phi \subset 2^{G}$ such that:

1. for any $U \in \Phi$ we have $Ob(G) \subset U$,

2. if $U \in \Phi$ and $U \subset V \subset G$ then $V \in \Phi$,

3. if $U,V \in \Phi$ then $U \cap V \in \Phi$,

4. for any $U \in \Phi$ there is $V \in \Phi$ such that $VV \subset U$,  where $VV$ is the set of all $gh$ with $g, h \in V$ and moreover $(g,h) \in G^{(2)}$,

5. if $U \in \Phi$ then $U^{-1} \in \Phi$.

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 $G$ be a groupoid. A coarse structure  $\chi$ on $G$ is a set $\chi \subset 2^{G}$ such that:

1′. $Ob(G) \in \chi$,

2′. if $U \in \chi$ and $V \subset U$ then $V \in \chi$,

3′. if $U,V \in \chi$ then $U \cup V \in \chi$,

4′. for any $U \in \chi$ there is $V \in \chi$ such that $UU \subset V$,  where $UU$ is the set of all $gh$ with $g, h \in U$ and moreover $(g,h) \in G^{(2)}$,

5′. if $U \in \chi$ then $U^{-1} \in \chi$.

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.