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Preamble: My previous education was focused either on classical analysis (which was given in quite old traditions, I guess) or on applied Mathematics. Since I was feeling lack of knowledge in 'modern' maths, I have some time now while doing my PhD to learn things I'm interested in by myself.

My impression is that in the last one-two centuries mathematicians put much effort to categorize their knowledge which led finally to abstract algebra and category theory. I didn't learn deeply non of these subjects so I will understand if your answer/comment will be a link to wikipedia page about category theory. I've already read it and it does not answer question. This is not a lazy interest, it is quite important for my understanding of things.

Question Description: My impression is that there are four clearly distinguishable types of mathematical structures on sets, i.e., ways of thinking of just a collection of elements as something meaningful:

  1. Set-theoretical: relations (order, equivalence, etc.)

  2. Algebraic: groups, algebras, fields, vector spaces etc.

  3. Geometrical: topology, metric, smooth structure etc.

  4. Measure-theoretical: $\sigma$-algebras, independence etc.

Some structure could be combined leading to e.g. $(1,2)$ - cosets, $(1,3)$ - quotient topology, $(2,3)$ - topological groups, $(2,4)$ - Haar measure, $(3,4)$ - Hausdorff measure etc.

I guess that any of these structures can be be restated just a relation on some set, but non-abstracted way of thinking of them is more convenient.

Questions:

  1. if my impression is right?

  2. are there any other structures? e.g. I'm interested if there are any dynamical structures corresponding to directed graphs, Markov Chains and other dynamical systems. These objects endow state space with notions of transitivity, absorbance, recurrence equilibrium etc.

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Concerning the order topology: On a finite set, a topology really is the same as a preorder. So here, (1) and (3) are not combined but turn out to yield the same thing. –  Rasmus Nov 3 '11 at 10:54
    
@Rasmus I see, changed it to quotient topology. I also failed to find a nice example for the combination $(1,4)$ –  Ilya Nov 3 '11 at 11:00
    
@Gortaur: ultrafilters on a set can be seen as finitely additive measures on the power set. They would tie in nicely with 2 and 3 as well via amenable groups and Banach-Tarski. –  t.b. Nov 3 '11 at 11:09
    
@t.b. thanks, I'll take a look –  Ilya Nov 3 '11 at 11:47
    
I threw in three additional tags as I don't like questions only tagged with (soft-question). But unfortunately they may not be 100% appropriate; they are the best I can find. –  Willie Wong Nov 3 '11 at 14:20

3 Answers 3

I have been thinking about similar questions for years and have come to the conclusion that any mathematical structure (set theoretical, algebraic, topological, measure theoretical) can be defined through (combinations of) relations

$ (1) \quad S_X^F\subset F(X)\times X^{I} $

for some underlying set X, some functor $F$: Rel $\rightarrow$ Rel and some set $I$.

Examples:

  • Magmas (monoids, groups,...) are defined through functions $S\subset X^2\times X$.
  • Graphs are defined through relations $S\subset X\times X$.
  • Metric spaces could be defined through relations $S\subset (\mathbb{R}\times X)\times X$, where $((r,x),x')\in S\Leftrightarrow d(x,x')=r$. Or better: through $d(x,x')\le r$, since the category Met have retractions as morphisms.
  • Topological spaces could be defined by $S\subset 2^X\times X$, where $(M,x)\in S\Leftrightarrow x\in M\in\tau$ or by the closure $x\in \overline M$.
  • Measure spaces ($X,\Sigma,\mu$) are combinations of $\sigma$-algebras $\Sigma$ (magmas on $2^{X}$) and measure function structures defined through relations $S\subset (\mathbb{R}\times 2^X)\times X$, where $((r,E),x)\in S\Leftrightarrow x\in E\in \Sigma \wedge \mu(E)=r$. The $x\in E$-part might be nonsense and a no operation, but formally the formula $(1)$ works.
  • Uniform spaces could be defined through relations $S\subset 2^{X\times X}\times X^2$, where $(U,(x,y))\in S \Leftrightarrow (x,y)$ is U-close. (Wikipedia)

This works for any structure I know and there even seems to be a general rule to generate the morphisms between the structures, showed by the (in general not commuting, if the relations not are functions) diagram of sets and relations: $\require{AMScd}$ \begin{CD} F(X) @>F(f)>> F(Y)\\ @V S_X^F V V(2) @VV S_Y^F V\\ X^{I} @>>f^{I}> Y^{I} \end{CD} $(2)\quad (\phi_X,\phi_Y)\in F(f)\Rightarrow [(\phi_X,(x_i)_I)\in S_X^F \Rightarrow (\phi_Y,((f(x_i))_I)\in S_Y^F$].

Example: If $I=1$, $F$ is the (contravariant) functor defined as $2^X\overset{2^f}\longrightarrow 2^Y$, where $ (M,M')\in 2^f\Leftrightarrow M=f^{-1}(M')$ and $S_X^F$ is defined as $(M,x)\in S_X^{F}\Leftrightarrow x\in \overline{M}$.

Then due to $(2)$:

$M=f^{-1}(M')\Rightarrow (x\in \overline{M}\Rightarrow f(x)\in \overline{M\,'})$, so $x\in \overline{f^{-1}(M\,')}\Rightarrow f(x)\in \overline{M\,'}$. (Continuity).

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You might want to read "Die Architektur der Mathematik" by Pierre Basieux where a similar point of view is developed. In fact, he argues that there are only three types of structures.

I don't know if an English translation is available.

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Speaking for myself, I'm not very fond of the "structural" view of pure mathematics which your classification 1 through 4 suggests. It is not just that the different structures can be combined, but that the same idea can be encoded in apparently different structures.

E.g. Klein's Erlangen program puts forward the point of view that geometries should be described in terms of the symmetry groups that act on them. From a modern point of view, this leads (among other things) to the notion of symmetric spaces, which are fundamental objects in many areas of modern mathematics, from topology to mathematical physics to Lie theory.

E.g. In number theory, the property of congruence of numbers (in the sense of modular arithmetic) can be encoded in terms of considering the different possible metric completions of the field $\mathbb Q$ (this is Ostrowski's theorem).

Regarding your question about other structures, you probably know that dynamical systems is a thriving area of pure mathematics. Many different structures can appear there: discrete or Lie group actions on manifolds or measure spaces, cocycles, objects from descriptive set theory, etc. Like most active areas of mathematics, thinking in terms of structures probably isn't the best way to understand the goals of and developments in the subject.

I don't mean to downplay the importance of structure; I just don't think that it provides the preeminent lens through which to view modern pure mathematics.

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+1. Nice answer! I wonder if structures don't "provides the preeminent lens through which to view modern pure mathematics", what else do(es)? Thanks! –  Tim Mar 28 '12 at 23:26

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