# Tagged Questions

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39k views

### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
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### Can a mathematical definition be wrong?

This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...
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### Category Theoretic Interpretation of Matroids?

Hello everyone! First time poster, long time lurker here. I have a really basic question that has been bugging me for sometime. Specifically, I'm not exactly sure what the 'correct' category ...
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### What does the adjective “natural” actually mean?

Terms like "in the natural way" or "the natural X" are used frequently in mathematical writing. While it is certainly clear most of the time what is meant, on occasion, I have been confounded. The ...
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### What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
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### When is a classification problem “wild”?

I hope someone can point me to a quick definition of the following terminology. I keep coming across wild and tame in the context of classification problems, often adorned with quotes, leading me to ...
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### What is a Kelley ring?

I've heard that in some book by someone named Kelley, perhaps an early edition of John L. Kelley's General Topology, the author gave a definition of a ring which turned out to be weaker than the usual ...
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### Surreal exponentiation — are the varying definitions equivalent? If not, is there agreement on which ones are better?

The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...
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### Definition of area

I am looking for an attractive, but rigorous definition of area; say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...
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### Is this a vertex algebroid?… What is vertex algebroid?

A couple of day ago, I was lamenting to a friend about the fact that I have no idea what vertex algebroids are. During our discussion, I came up with a guess of what a vertex algebroid might be. I'm ...
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### What are examples of theorems which were once “valid”, then became “invalid” as standard definitions shifted?

That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...
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### Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?

Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain ...
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### Definition of ind-schemes

What is the correct definition of an ind-scheme? I ask this because there are (at least) two definitions in the literature, and they really differ. Definition 1. An ind-scheme is a directed colimit ...
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### What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space?

Could anyone tell me what the mathematical structure is called if we replace commutative group by commutative monoid in the definition of linear space? Also, are there any names for "commutative ...
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### Does the notion of graphs with vertex multiplicity exist?

I need to use graphs where each vertex gets a natural number, $b(v)$, its multiplicity. These numbers indicate how many 'replications' of the vertex we have. It is actually a way to write in a ...
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### What makes a distance?

In the answers to my previous question, I learned that there are different concepts of distance, that is of distance-like functions with the usual metric being only the most popular and important one. ...
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### What are the relationship between various definitions for quasi coherent sheaves?

It seems that there are many definitions of quasi coherent sheaves(modules). There is a nice page on nLab quasi coherent sheaves My questions are: Are there any other definitions of quasi coherent ...
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### Grothendieck Topologies versus Pretopologies

The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that ...
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### What are Penrose Tilings, and how do they relate to Quasicrystals?

The question is in the title, but let me elaborate a little. Background Penrose Tilings are really pretty and satisfy some remarkable properties. For instance, I believe the following is true: even ...
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### Definition of forgetful functor

I was wondering if it is possible to make a formal definition of what it means for a functor to be forgetful. That is, using only the terminology of categories. I have seen so many examples of ...
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### Who first introduced the functional definition of symmetry?

Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves ...
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### BRST cohomology definition

Is there written anywhere a full definition of BRST cohomology? All I have found so far is BRST cohomology in _______. As far as I can see, BRST cohomology is the ...
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### Why is Pic^0(C) of a curve C a variety?

Let $C$ be an abstract non-singular curve. I'm having a hard time finding a reference for why $\text{Pic}^0(C)$ is a variety. Any pointers towards a reference would be appreciated.
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### Lattices: why require bilinear form to be integral?

This is a quite localized question, but I hope it won't be closed as unfit to MO. Well, a lattice $\Lambda$ in $\mathbb{R}^n$ is a discrete subgroup generated by a basis. Such a lattice gets a ...
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### Is there a mathematical object called “ivy”?

As the title says, is there a mathematical object referred to as "ivy" or "ivy type" or similar? I have a type of graph where this name fits perfectly, but I don't want it to clash with something ...
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### procedure-based (as opposed to definition-based) concepts

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated ...
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### how to define the injectivity radius of manifolds with boundary?

For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...
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### End points of continua

Whyburn (1942) defined an end point $x$ of a continuum $X$ to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines an end point ...
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### On the definition of ‘smooth vectors’ in Rieffel's “Deformation Quantization for Actions of $\mathbb{R}^{d}$”.

On the first page of Chapter 1 of Rieffel's Deformation Quantization for Actions of $\mathbb{R}^{d}$, Rieffel defines a family of seminorms on the space $A^{\infty}$ of smooth vectors of a Fréchet ...
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### A scheme simple over Spec(A)?

What does it mean to say that a scheme $X$ is simple over $Spec(A)$ ? I stumbled on this terminology in a paper of S. Lubkin entitled "On a conjecture of Andre Weil".
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### Q-construction and Gabriel-Zisman Localization

It might be a stupid question. When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the same as $P$ but ...
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### Strings and “co-subsequences”

Let $S$ be a string over some alphabet $\Sigma$. It is well known that a substring of $S$ is commonly defined as a sequence of contiguous elements from $S$, while a subsequence of $S$ is a sequence ...
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### Set theory definition of addition, negative numbers, and subtraction? [closed]

Using the definition of natural numbers $0 = \emptyset$ and $S(n) = n \cup \lbrace n \rbrace$ where S is the successor function, what is the definition of addition on natural numbers? Concerning the ...
As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...