Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and ...

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49
votes
16answers
7k views

Why is it a good idea to study a ring by studying its modules?

This is related to another question of mine. Suppose you met someone who was well-acquainted with the basic properties of rings, but who had never heard of a module. You tell him that modules ...
5
votes
3answers
374 views

Abstract Relation between Presehaves and Simplicial Sets

Every presheaf (let's say on a topological space) comes with restriction maps. The open sets of a topological space are ordered by inclusion and these inclusions yield the restrictions. Now a sheaf ...
112
votes
130answers
26k views

Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please) I'd love to learn about ...
9
votes
3answers
2k views

What is the relationship between algebraic geometry and quantum mechanics?

The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of ...
43
votes
11answers
5k views

What precisely Is “Categorification”?

(And what's it good for.) Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
8
votes
3answers
332 views

How does one identify properties of objects with good “inheritance”?

When you are dealing with a very general object like a topological space or a ring, usually you impose an additional condition (such as compact Hausdorff or Noetherian) with the property that the ...
7
votes
2answers
458 views

What is the conceptual significance of supercommutativity?

A $\mathbb{Z}/2\mathbb{Z}$-graded algebra is said to supercommute if $xy = (-1)^{|x| |y|} yx$; in other words, odd elements anticommute. Why is this the "right" definition of supercommutativity? ...
30
votes
6answers
2k views

Universal definition of tangent spaces (for schemes and manifolds)

Both schemes and manifolds are local ringed spaces which are locally isomorphic to spaces in some full subcategory of local ringed spaces (local models). Now, there is the inherent notion of the ...
3
votes
1answer
191 views

limits of algebraic varieties

I'm looking for a reference which deals with limits of families of algebraic varieties as the degree increases (or at least keywords from this subject). For the kind of example I have in mind, ...
13
votes
2answers
646 views

Is there any meaning to a “nice bijective proof?”

From Zeilberger's PCM article on enumerative combinatorics: The reaction of the combinatorial enumeration community to the involution principle was mixed. On the one hand it had the universal ...
7
votes
1answer
880 views

What is Drinfeld's manuscript “Best Dream” (in Russian!) about?

I would like to know what Drinfeld's scanned manuscript "Best Dream" is about: the title makes me curious. It's in Russian.
5
votes
5answers
705 views

Visual representation of mathematical research interrelationships

I remember seeing a visualization in the form of a 2d (nodal) graph of all areas of academia, with math, physics and engineering over in one section, connecting in an arc to the central area of ...
4
votes
7answers
1k views

What can't be described by categories?

I've been reading some "introduction to categories" type materials and have been impressed with the all-encompassing nature, but the skeptic in me wonders: is there any mathematical object that ...
42
votes
50answers
10k views

Describe a topic in one sentence. [closed]

When you study a topic for the first time, it can be difficult to pick up the motivations and to understand where everything is going. Once you have some experience, however, you get that good ...
14
votes
3answers
2k views

K(F_1) = sphere spectrum?

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?
22
votes
11answers
2k views

What are examples of good toy models in mathematics?

This post is community wiki. A comment on another question reminded me of this old post of Terence Tao's about toy models. I really like the idea of using toy models of a difficult object to ...
5
votes
7answers
511 views

Given a sequence defined on the positive integers, how should it be extended to be defined at zero?

This question is inspired by a lecture Bjorn Poonen gave at MIT last year. I have ideas of my own, but I'm interested in what other people have to say, so I'll make this community wiki and post my ...
34
votes
8answers
5k views

Sheaf cohomology and injective resolutions

In defining sheaf cohomology (say in Hartshorne), a common approach seems to be defining the cohomology functors as derived functors. Is there any conceptual reason for injective resolution to come ...
38
votes
21answers
5k views

What's a groupoid? What's a good example of a groupoid?

Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?
42
votes
12answers
4k views

Is there a high-concept explanation for why characteristic 2 is special?

The structure of the multiplicative groups of Z/pZ or of Zp is the same for odd primes, but not for 2. Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for 2. So ...
8
votes
6answers
1k views

Why the search for ever larger primes?

I understand why primes are useful numbers and also why the product of large primes are useful such as for application in public key cryptography, but I am wondering why it is useful to continue the ...