Questions tagged [definitions]
For questions of the kind "What is the correct definition of property or object X?" or questions about how to define terms within specific theories.
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Lax natural transformations and adjunctions between strict 2-functors
Note: This is a repost of a question I asked on Math Stack Exchange. A friend of mine suggested that maybe the question is more suitable for MathOverflow.
I'm relatively new to higher category theory, ...
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Definition of length structure in Burago, Burago, Ivanov's "A Course in Metric Geometry" does not imply Exercise 2.1.4?
In Burago, Burago, Ivanov's "A Course in Metric Geometry" (Definition 2.1.1, page 26 and 27) a length structure on a topological space $X$ is defined as a pair $(A,L)$ where $A$ is a set of ...
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A misunderstanding in Gorensteiness
I am reading the definition of a Gorenstein ring in Bruns-Herzog.
A Noetherian local ring $R$ is called Gorenstein if
$$
\operatorname{inj,円dim}_R R < \infty,
$$
that is, if the injective ...
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user569831
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literature request: morphisms of plane trees as natural transformations in $[\omega^{op},\Delta_+]$?
This is a more open-ended followup to a related question: On Joyal's definition of a category of plane trees.
That question recalled how rooted plane trees can be represented as contravariant ...
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On Joyal's definition of a category of plane trees
I'm trying to understand better the motivation for the definition of the category $Trees$ of finite plane trees in Joyal's unpublished manuscript "Disks, duality and Θ-categories" (1997, ...
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Why is Galois cohomology usually defined only for discrete modules?
$\DeclareMathOperator{\Aut}{Aut}$If $K$ is a field, the Galois cohomology $H^n(K, M)$ can be defined using cochains (or by a derived construction, etc.). The absolute Galois group $G_K$ naturally ...
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Epimorphisms with kernel pairs
I am a bit lost understanding some subtleties in various form of epimorphy.
The nLab reports that an effective epimorphism is one that coequalizes its kernel pair. A regular epimorphism is simply one ...
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Reference request: generalized Jacobian variety for higher dimensional variety
Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate ...
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Is there a syntactic proof that first-order positive inductive definitions are conservative?
Every first-order positive inductive definition has a fixed point. It follows that, if the biconditional is thought of as an axiom in the language obtained from the background language by adding a new ...
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How to define Dedekind reals and Eudoxus reals such that they are equivalent to unmodulated Cauchy reals
In constructive mathematics without choice, we have three different versions of the real numbers (each embedding into the next).
Regular Cauchy reals (functions $f : \mathbb N \to \mathbb Q$ such ...
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Is there a name for finite unions of intervals?
Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is ...
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Looking for definition of function spaces appearing in article of DiPerna & Lions
I am looking for the definition of various function spaces appearing in the following article, preferably with references to other sources where such spaces are discussed in greater detail:
Article: ...
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Definition of "interval of continuity" for function defined on sets
At the beginning of Chapter 8 of Kubilius's Probabilistic Methods in the Theory of Numbers, the author defines $Q=Q(E)$ to be a completely additive nonnegative function defined for all Borel subsets $...
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Sequential definitions of continuity and related classes
It is well-known that the usual 'epsilon-delta' definition of continuity is equivalent to the sequential definition (assuming countable choice). Less well-known is the sequential definition of ...
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What is the "weight" of an automorphic form for $\mathrm{PGL}_2$?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PGL{PGL}$I'm trying to understand what the notion of "weight" is for automorphic forms over $\GL_2(F)$ where $F$ is some number field, in ...