# Mathematical structures

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

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
Indeed, there is no apparent reason to think that there is a simplifying viewpoint to understand the vast and varied activities in modern mathematics. No meta- "royal road". It is true that various meta-notions do help organize various things, but a good suitcase does not express meaning of its contents, etc., however useful it is. –  paul garrett Nov 30 '14 at 1:23

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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I have been thinking about similar questions for years and have come to the conclusion that many mathematical constructs (categories where objects are sets and morphisms are functions) can be defined through (combinations of) relations

$(1) \quad S_X^F\subseteq F(X)\times X$

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

Examples:

• Magmas (monoids, groups,...) are defined through functions $S\subseteq X^2\times X$.
• Directed graphs are defined through relations $S\subseteq X\times X$.
• Metric spaces could be defined through relations $S\subseteq (\mathbb{R}\times X)\times X$, 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 $S\subseteq 2^X\times X$, where $(M,x)\in S\Leftrightarrow x\in M\in\tau$ or by the closure $x\in \overline M$.

This works for the most common constructs 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: $\require{cancel}$ $\require{AMScd}$ \begin{CD} F(X) @>F(f)>>F(Y)\\ @V S_X^F V V(2) @VV S_Y^F V\\ X @>>f> Y \end{CD} $(2)\quad (\phi_X,\phi_Y)\in F(f)\Rightarrow [(\phi_X,x)\in S_X^F \Rightarrow (\phi_Y,f(x))\in S_Y^F$].

Example: $F$ 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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This captures only a tiny bit of all mathematical structures. –  Martin Brandenburg Oct 7 '14 at 9:31
@Martin: yes but by combining such structures the common structures fits in and hopefully some newly invented. –  Lehs Oct 7 '14 at 9:35
@Martin: Clarifying that I meant mathematical structures on a set. –  Lehs Oct 8 '14 at 15:06
@Martin Brandenburg: I'll edit. –  Lehs Nov 30 '14 at 7:54