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A rather immediate reaction to seeing the definition of a coarse structure, at least to me, is to be reminded of a uniform structure. The axioms for a coarse structure $\mathcal{C}$ (defined by a collection of controlled sets $U \subset X \times X$) are:

  1. Identity: ($\Delta \in \mathcal{C}$)
  2. Symmetry: ($U \in \mathcal{C}$ implies that $U^{-1} \in \mathcal{C}$)
  3. Transitivity: ($U,V \in \mathcal{C}$ implies that $U \circ V \in \mathcal{C}$)
  4. "Cofilter": ($\mathcal{C}$ is a filter on $(\mathcal{P}(X \times X),\supset)$)

(NOTE: I have combined two of the usual axioms here for the "cofilter" axiom [is cofilter a valid term?]. I usually prefer the weakened versions of axioms 2,3 because of considerations of bases etc.)

The axioms for a uniformity are very similar. In fact they differ as follows: for 1) (reflexivity) we demand that $\Delta \subset U$ for all entourages $U \in \mathcal{U}$. For 3) we demand a sort of "approximate transitivity". This may be interpreted as: every $U \in \mathcal{U}$ has a square-root, that is, some $V \in \mathcal{U}$ with $V^2=U$. Note that 3) above could be replaced by: for all $U \in \mathcal{C}$ we have that $U^2 \in \mathcal{C}$ (this is enough by 4). For 4) we demand the filter condition instead of the cofilter one.

Now, of course as everyone knows, the two definitions have "dual jobs" in a philosophical sense: a coarse structure describes the large scale by saying which covers are "uniformly bounded above" whereas the uniform structure describes the small scale by saying what covers look like when they contain all "sufficiently nearby points".

My question: is there a more mathematically precise way of expressing this "philosophical duality", perhaps in some category-theoretic formalism? Is there an expression of this "philosophical duality" in any results? i.e. are there results from coarse geometry which have counterparts for uniform spaces? I remember seeing a claim that there are some dual results between proper homotopy theory and the theory of fractals/shape theory, but I've never seen anything precise.

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What you call a "cofilter" is sometimes called an "ideal", but I don't mind the term "cofilter" in the least. – James Cranch Mar 27 '14 at 19:56

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