Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some familiar traits with the Tukey's definition of uniformity, which is also given in terms of covers. But The Tukey's definitions is far less general, as it was designed only with 'traditional' point-set topology in mind.
The only article I found, which explores this similarity, is "Uniform sheaves and differential equations" by Yves André. In this paper André states, that Tukey uniformity on a topological space $X$ is a Grothendieck topology on a small category of open sets $\mathcal{O}(X)$ with inclusions as morphisms only if the space $X$ is precompact (totally bounded). Although, André's interest here lies primary in the domain of non-Archimedean algebraic geometry and specifically of differential equations over punctured p-adic domains. So, The main focus are not sites in general, but the sites which are open sets $\mathcal{O}(X)$. However, there is Appendix I of this article, which describes a very general construction (claimed to be of use in representation theory). But this construction can be viewed as an universal version of uniformity defined by entourages (connectors). My question:
I want a treatment of uniformity for small categories. Specifically, is there a meaningful way of translating the Entourage definition below in the language of sieves, understood as generalized covers? For me sieves over object $A$ are subfunctors of $\mathrm{Hom}(\bullet,A)$. Note, that Yves André also uses some other notion of sieve in his definition, which I don't understand. So, I also would be glad if you explain this new notion to me. Is it just a synonym of a presheaf?
I assume that classical notions like Grothendieck topology as defined in MacLane and Moerdijk. Here I list other definitions in order of logical dependency and "importance".
(Yves André) The entourage (connector) on an object $T$ with finite powers of category $\mathcal{C}$ is an object $E$ of $\mathcal{C}$ equipped with three morphisms $\delta : T \to E$ and $p_1,p_2 : E \to T$ such that $p_1 \circ \delta = \mathrm{id}_T$. The author notes that totality of such objects is a category with a final object $T^2$. Which is evidently equipped projection $\pi_1,\pi_2$ and what happens to be the version of the diagonal map $\triangle$ in category $\mathcal{C}$.
(Yves André) Uniformity $\mathcal{U}$ on $T$ is a sieve of entourages of $T$ such that following holds: (cotransitivity) for any $E \in \mathcal{U}$ there is $E' \in \mathcal{U}$, such that $(E' \times T) \times_{T^2} (T \times E') \in \mathcal{U}$ and maps to $E$; (symmetry) For any $E \in \mathcal{U}$ the entourage obtained by switching $p_1$ and $p_2$ is in $\mathcal{U}$.
For comparison:
- (John Tukey) Uniformity $\mathcal{U}$ on a set $X$ is a filter of covers of $X$, such that for any $U \in \mathcal{U}$ there is some $V \in \mathcal{U}$ with $V^\star \subset U$. Notion of star refinement can be clearly stated in a point-set context: For a cover $U$ of $X$ and $u \in U$ the star operation is defined as $\star(U,u) = \bigcup \{ v \in U : v \cap u \neq \emptyset\}$ and $U^\star=\{ \star(U,u)| u \in U \}$. But I'm not so sure how to generalize this notion of star-refinement to sieves and small categories.
Some afterthoughts:
In this context a Grothendieck topology $\mathbf{J}$ can be thought as a filter of sieves (you may have $\emptyset$ in $\mathbf{J}(A)$ but it is a matter of definition), which is closed by (functorial) pullbacks and an operation of replacing elements by covers of elements. I don't see any problems with a "Grothendieck" uniformity $\mathbf{U}$ being closed under pullbacks (as a uniform cover restricted to a subset with a subset uniformity is still a uniform cover). But the second property as shown by Yves André holds only if the uniform space is precompact, otherwise there are some limitation on how much the cover may be refined. So in general a compatible "Grothendieck Uniformity" $\mathbf{U} \subset \mathbf{J}$ must be viewed as a subpresheaf of Grothendieck topology $\mathbf{J}$. And uniform sites can be viewed as triple $(\mathcal{C},\mathbf{J},\mathbf{U})$ with $\mathbf{J}$ being minimal Grothendieck topology containing $\mathbf{U}$. So, $\mathbf{J}$ is determined by $\mathbf{U}$ and can be dropped.
Regarding the star-refinement. I thought that set-theoretic operations can be replaced with taking pullbacks and pushforwards in the category $\mathcal{C}$, then $\star(S,f)$ for a sieve $S$ and $f$ would be a morphism defined by universal property. Of course, this means that Grothendieck uniform structures may not exist in any small category. But they would exists for ones with complete lattice-like structure. At least this definition should cover uniform locales. In fact the construction of André is more abstract then my idea here, as $T$ can be taken to be a topos. But it is too abstract for me. After writing this question, I think that members of André entourage category are building blocks in the diagram for $\star(S,f)$.
Sorry, if this question is too wordy and vague.
