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algebraic approaches to differential calculus | https://ncatlab.org/nlab/source/algebraic+approaches+to+differential+calculus | Derivatives and differentials are usually expressed in terms of limits in the sense of analysis. However it became clear in about the last half century that much of the knowledge on usual [[differential calculus]] can be inferred from using just algebraic properties of differentials and derivatives, most notably the Le... |
algebraic category | https://ncatlab.org/nlab/source/algebraic+category |
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# Algebraic categories
* table of contents
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## Idea
An algebraic category is a [[concrete category]] which behaves very much like... |
algebraic cobordism | https://ncatlab.org/nlab/source/algebraic+cobordism |
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## Idea
_Algebraic cobordism_ is the bigraded [[generalized cohomology theory]] represented by... |
algebraic correspondences | https://ncatlab.org/nlab/source/algebraic+correspondences | See [[pure motives]] for now. |
algebraic curve | https://ncatlab.org/nlab/source/algebraic+curve |
#Contents#
* table of contents
{:toc}
## Idea
An algebraic curve is an [[algebraic variety]] of dimension $1$. Typically
one restricts considerations to either affine or projective algebraic curves. Most often one treats the plane algebraic curves, i.e. curves
with an embedding into $\mathbf{A}^2$ or $\mathbf{P}^2$;... |
algebraic cycle | https://ncatlab.org/nlab/source/algebraic+cycle |
#Contents#
* table of contents
{:toc}
## Definition
Let $X$ be a [[noetherian scheme]]. One defines a **$k$-dimensional algebraic [[cycle]]** as an element of the [[free abelian group]] $Z_k(X)$ generated by the [[closed subspace|closed]] [[integral scheme|integral]] subschemes of [[dimension]] $k$, and dual... |
algebraic definition of higher categories | https://ncatlab.org/nlab/source/algebraic+definition+of+higher+categories | +-- {: .rightHandSide}
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* [[geometric definition of higher categories]]
* **algebraic definition of higher categories**
***
# Contents
* automatic ta... |
algebraic dynamics | https://ncatlab.org/nlab/source/algebraic+dynamics | While standard dynamics studies processes in real time, sometimes it also considers discrete dynamical systems. The step of discrete dynamics is in the appropriate category.
In __algebraic dynamics__ one typically studies discrete dynamical systems on algebraic varieties. Such a system is given by a regular endomorphi... |
algebraic effect | https://ncatlab.org/nlab/source/algebraic+effect | [[!redirects algebraic side effect]]
[[!redirects algebraic side effects]]
[[!redirects algebraic effects]]
#Contents#
* table of contents
{:toc}
## Idea
**Algebraic effects** are computational [[side effect|effects]] that can be represented by an equational theory, or [[algebraic theory]], whose operations produc... |
algebraic element | https://ncatlab.org/nlab/source/algebraic+element |
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## Definition
Given a [[field]] $K$ and a field extension $K \subseteq F$, an element $\alpha \in F$ is **algebraic** if the subfield... |
algebraic equation | https://ncatlab.org/nlab/source/algebraic+equation |
#Contents#
* table of contents
{:toc}
## Idea
An [[algebraic equation]] is an [[equation]] of the form
$$
P(x_1, \cdots, x_n) = 0
$$
where $P(x_1, \cdots, x_n)$ is a [[polynomial]] with [[coefficients]] in some [[field]].
## Related concepts
* [[linear equation]]
* [[Diophantine equation]]
## References
* W... |
algebraic extension | https://ncatlab.org/nlab/source/algebraic+extension |
#Contents#
* table of contents
{:toc}
## Idea
A kind of [[field extension]]
## Related concepts
* [[perfect field]]
* [[transcendental extension]]
## References
* Wikipedia, _[Algebraic extension](http://en.wikipedia.org/wiki/Algebraic_extension)_
[[!redirects algebraic extensions]]
|
algebraic function | https://ncatlab.org/nlab/source/algebraic+function |
## References
* Wikipedia, _[Algebraic function](http://en.wikipedia.org/wiki/Algebraic_function)_
[[!redirects algebraic functions]] |
algebraic fundamental group | https://ncatlab.org/nlab/source/algebraic+fundamental+group |
#Contents#
* table of contents
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## Idea
The _algebraic fundamental group_ is the [[fundamental group]] of a [[scheme]], as defined by [[Grothendieck]] in [[SGA1]]. It is essentially the fundamental group as seen by [[étale homotopy]], the _[[étale fundamental group]]_.
For [[fields]] this is essentia... |
algebraic geometry | https://ncatlab.org/nlab/source/algebraic+geometry |
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## Overview
__Algebraic geometry_... |
algebraic group | https://ncatlab.org/nlab/source/algebraic+group |
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## Definition
Given a (typically [[algebraically closed field|algebraically closed]]) field $k$, an __algebr... |
Algebraic Homotopy | https://ncatlab.org/nlab/source/Algebraic+Homotopy |
The book
* [[Hans Baues]],
_Algebraic Homotopy_,
Cambridge studies in advanced mathematics 15,
Cambridge University Press, (1989)
[doi:10.1017/CBO9780511662522](https://doi.org/10.1017/CBO9780511662522)
gives an account of the author's interpretation of [[Henry Whitehead]]'s [[Algebra... |
algebraic homotopy | https://ncatlab.org/nlab/source/algebraic+homotopy |
#Contents#
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## Whitehead's algebraic homotopy programme
In his talk at the 1950 ICM in Harvard, [[Henry Whitehead]] introduced the idea of *algebraic homotopy theory* and said
_"The ultimate aim of algebraic homotopy is to construct a purely
algebraic theory, which ... |
algebraic integer | https://ncatlab.org/nlab/source/algebraic+integer | +-- {: .rightHandSide}
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## Idea
In [[number theory]], the concept of _algebraic integer_ is a generalization of that of [[integer]] to m... |
algebraic K-theory | https://ncatlab.org/nlab/source/algebraic+K-theory |
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## Idea
Algebraic K-the... |
algebraic K-theory of smooth manifolds | https://ncatlab.org/nlab/source/algebraic+K-theory+of+smooth+manifolds |
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## Definition
The constru... |
algebraic K-theory spectrum | https://ncatlab.org/nlab/source/algebraic+K-theory+spectrum | The term _algebraic K-theory spectrum_ refers to one of the following two notions:
* an object $K(C) \in \Spt$ in the [[stable (infinity,1)-category of spectra]], associated to a [[stable (infinity,1)-category]] or [[symmetric monoidal (infinity,1)-category]] $C$, whose [[homotopy groups]] give the [[algebraic K-theor... |
algebraic K-theory, a historical perspective | https://ncatlab.org/nlab/source/algebraic+K-theory%2C+a+historical+perspective | [[!redirects Algebraic K-theory, a historical perspective]]
## A little history: the beginnings
[[algebraic K-theory|Algebraic K-theory]] grew out of two apparently unrelated areas of algebraic geometry and algebraic
topology. The second of these, historically, was the development by Grothendieck of (geometric
a... |
algebraic Kan complex | https://ncatlab.org/nlab/source/algebraic+Kan+complex | +-- {: .rightHandSide}
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## Idea
The notion of _algebraic Kan complex_ is an [[algebraic definition of higher cate... |
algebraic lattice | https://ncatlab.org/nlab/source/algebraic+lattice |
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## Definition {#Def}
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###### Definition
An **algebraic lattice** ... |
algebraic Lefschetz formula | https://ncatlab.org/nlab/source/algebraic+Lefschetz+formula | Let $(C,d)$ be a nonnegative [[cochain complex]] of [[vector space]]s over a [[field]] of (total) finite dimension $dim C = \sum_{p=0}^\infty dim C^p \lt \infty$ and $f = (f^p)_{p\geq 0} :(C,d)\to (C,d)$ an endomorphism of cochain complexes.
The **algebraic Lefschetz formula** is the statement
$$
\sum_{p\geq 0} (-1)... |
algebraic limit field | https://ncatlab.org/nlab/source/algebraic+limit+field | [[!redirects algebraic limit fields]]
[[!redirects algebraic limit theorem]]
[[!redirects algebraic limit theorems]]
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... |
algebraic limit vector space | https://ncatlab.org/nlab/source/algebraic+limit+vector+space | [[!redirects algebraic limit vector spaces]]
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## Definition ##
Given an [[algebraic limit field]] $F$, an __algebrai... |
algebraic line bundle | https://ncatlab.org/nlab/source/algebraic+line+bundle |
#Contents#
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## Idea
The generalization of the concept of _[[holomorphic line bundle]]_ from [[complex analytic geometry]] to more general [[algebraic geometry]].
Algebraic line bundles on some [[variety]]/[[scheme]] form its [[Picard group]]/[[Picard scheme]].
## Related concepts
* [[div... |
algebraic line n-bundle | https://ncatlab.org/nlab/source/algebraic+line+n-bundle |
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## Idea
The concept of [[line n-bundle]] in [[algebrai... |
algebraic microlocalization | https://ncatlab.org/nlab/source/algebraic+microlocalization | #Contents#
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# Idea
As an analogue of the [[microlocalization]] in operator theory, T. Springer has introduced an algebraic microlocalization in the theory of filtered noncommutative rings.
[[microlocal analysis|Microlocal analysis]] using [[hyperfunctions]] inst... |
algebraic model category | https://ncatlab.org/nlab/source/algebraic+model+category |
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## Idea
The structure of an _algebraic model category_ is a refinement of that of a [[mod... |
algebraic model for modal logics | https://ncatlab.org/nlab/source/algebraic+model+for+modal+logics |
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# Algebras for modal logics
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... |
algebraic model structures - table | https://ncatlab.org/nlab/source/algebraic+model+structures+-+table | **Algebraic model structures:** [[Quillen model structures]], mainly on [[locally presentable categories]], and their constituent [[categories with weak equivalences]] and [[weak factorization systems]], that can be equipped with further algebraic structure and "freely generated" by small data.
| structure | small-s... |
algebraic number | https://ncatlab.org/nlab/source/algebraic+number |
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## Definition
An **algebraic number** is a [[root]] of a non-zero [[polynomial]] with [[integer]] [[co... |
algebraic number theory | https://ncatlab.org/nlab/source/algebraic+number+theory |
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## Idea
Algebraic number ... |
algebraic quasi-category | https://ncatlab.org/nlab/source/algebraic+quasi-category |
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An **algebraic quasi-category** is a [[quasi-category]]... |
algebraic set theory | https://ncatlab.org/nlab/source/algebraic+set+theory | [[!redirects Algebraic set theory]]
#Contents#
* table of contents
{:toc}
## Idea
The new insight taken as a starting point in _algebraic set theory_ (AST) is that [[models]] of [[set theory]] are in fact [[algebra over an algebraic theory|algebras]] for a suitably presented [[algebraic theory]], and that ma... |
algebraic small object argument | https://ncatlab.org/nlab/source/algebraic+small+object+argument | # The algebraic small object argument
* table of contents
{: toc}
## Idea
The ordinary [[small object argument]] is a way of constructing (often [[combinatorial wfs|combinatorial]]) [[weak factorization systems]] from a set of "generators". While powerful and useful, it has several defects, such as:
* Its... |
algebraic space | https://ncatlab.org/nlab/source/algebraic+space |
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## Idea
An _algebraic space_ is an [[object]] in the [[sheaf topos]] over the [[fppf-site]], that h... |
algebraic spaces > history | https://ncatlab.org/nlab/source/algebraic+spaces+%3E+history | < [[algebraic spaces]]
[[!redirects algebraic spaces -- history]] |
algebraic stack | https://ncatlab.org/nlab/source/algebraic+stack |
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## Idea
A... |
algebraic structure | https://ncatlab.org/nlab/source/algebraic+structure | Algebraic structure may mean either
* the same as [[algebra]] in the sense of [[universal algebra]];
or
* the [[extra structure]] which such an algebra has with respect to the underlying object.
There are several notions of an __algebraic structure__ on an object of some category or higher category, wh... |
algebraic surface | https://ncatlab.org/nlab/source/algebraic+surface |
## Idea
An [[algebraic variety]] of [[dimension]] 2.
## Examples
* [[K3 surface]]
## Related concepts
* dimension 1-case: [[algebraic curve]]
* [[complex surface]]
[[!redirects algebraic surfaces]]
|
algebraic theories in functional analysis | https://ncatlab.org/nlab/source/algebraic+theories+in+functional+analysis |
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At the moment, this is a "place holder" page. I ([[Andrew Stacey]]) want to learn about the appearance of [[algebraic theories]] in [[fun... |
algebraic theory | https://ncatlab.org/nlab/source/algebraic+theory |
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## Idea
An _algebraic [... |
algebraic topology | https://ncatlab.org/nlab/source/algebraic+topology |
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algebraic topology - contents | https://ncatlab.org/nlab/source/algebraic+topology+-+contents |
**[[algebraic topology]]** -- application of [[higher algebra]] and [[higher category theory]] to the study of ([[stable homotopy theory|stable]]) [[homotopy theory]]
* [[topological space]], [[homotopy type]]
* [[homotopy group|homotopy]]
[[cohomology]]
[[homology]]
* [[spectral sequence]]
|
algebraic triangulated category | https://ncatlab.org/nlab/source/algebraic+triangulated+category | A __triangulated category__ is called __algebraic__ (in the sense of [[B. Keller]]) if it is equivalent to the stable category of a [[Quillen exact category]] of a Frobenius category (a Quillen exact category is Frobenius if it has enough injectives and enough projectives and the two classes coincide).
* [[B. Kelle... |
algebraic variety | https://ncatlab.org/nlab/source/algebraic+variety |
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## Idea
In [[algebraic geometry]], __algebraic variety__ (not to be confused with [[variety of algebras]]) ... |
algebraic vector bundle | https://ncatlab.org/nlab/source/algebraic+vector+bundle |
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#### Linear algebra
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{:toc}
## Idea
The notion of [[vector bundle]] in [[algebraic geomet... |
algebraic weak factorization system | https://ncatlab.org/nlab/source/algebraic+weak+factorization+system | [[!redirects natural weak factorization system]]
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algebraically closed field | https://ncatlab.org/nlab/source/algebraically+closed+field |
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## Idea
A [[field]] $k$ is __algebraically closed__ if every non-constant [[polynomial]] (with one variable ... |
algebraically compact category | https://ncatlab.org/nlab/source/algebraically+compact+category |
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* table of contents
{: toc}
## Idea
A [[category]] is ... |
algebraically injective object | https://ncatlab.org/nlab/source/algebraically+injective+object | * table of contents
{: toc}
## Definition
Let $J$ be a set of [[morphisms]] in a [[category]] $C$. An **algebraically $J$-injective object** is an object $X\in C$ equipped with the [[stuff, structure, property|structure]] of, for every morphism $i:A\to B$ in $J$ and every morphism $f:A\to X$, a specified morphi... |
Algebras > history | https://ncatlab.org/nlab/source/Algebras+%3E+history | < [[Algebras]]
[[!redirects Algebras -- history]] |
algebroid | https://ncatlab.org/nlab/source/algebroid |
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# Algebroids (linear categories)
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## Idea
A *linear category*, or *algebroid*, is a [[category]... |
Algimantas Adolfas Jucys | https://ncatlab.org/nlab/source/Algimantas+Adolfas+Jucys |
* [Wikipedia entry](https://en.wikipedia.org/wiki/Jucys%E2%80%93Murphy_element)
## Selected writings
Introducing the [[Jucys-Murphy elements]]:
* [[Algimantas Adolfas Jucys]], *Symmetric polynomials and the center of the symmetric group ring*, Rep. Mathematical Phys.5 (1974), pp. 107–112 (<a href="https://do... |
algorithm | https://ncatlab.org/nlab/source/algorithm |
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* table of contents
{:toc}
## Idea
(...)
## Appl... |
Algèbres Enveloppantes | https://ncatlab.org/nlab/source/Alg%C3%A8bres+Enveloppantes |
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This pages provides links concerning the book:
* [[Jacques Dixmier]]:
\linebreak
**Algèbres Enveloppantes**
\linebreak
Cahiers sc... |
Ali Caglayan | https://ncatlab.org/nlab/source/Ali+Caglayan | a.k.a Alizter
[mathoverflow](https://mathoverflow.net/users/54401/ali-caglayan) |
Ali Chamseddine | https://ncatlab.org/nlab/source/Ali+Chamseddine |
* [website](http://sites.google.com/site/achamseddine2/home)
category: people |
Ali Mozaffari | https://ncatlab.org/nlab/source/Ali+Mozaffari |
* [website](https://www.ucl.ac.uk/natural-sciences/dr-ali-mozaffari)
## Selected writings
On [[cosmic structure formation]] in view of [[MOND]]:
* [[Daniel Thomas]], [[Ali Mozaffari]], [[Tom Zlosnik]], *Consistent cosmological structure formation on all scales in relativistic extensions of MOND* ([arXiv:23... |
Ali Ovgun > history | https://ncatlab.org/nlab/source/Ali+Ovgun+%3E+history |
* see _[[Ali Övgün]]_
|
Alice Rogers | https://ncatlab.org/nlab/source/Alice+Rogers |
Alice Rogers is emeritus professor of pure mathematics at King's college London.
## related $n$Lab entries
* [[supermanifold]]
category: people |
Alisa Govzmann | https://ncatlab.org/nlab/source/Alisa+Govzmann |
* [institute page](https://wwwde.uni.lu/research/fstm/dmath/people/alisa_govzmann)
## Selected writings
On [[homotopy limits]], [[homotopy pullbacks]] and their [[pasting law for pullbacks|pasting law]] in/via [[model categories]]:
* [[Alisa Govzmann]], [[Damjan Pištalo]], [[Norbert Poncin]], *Indeterminacie... |
Alissa Crans | https://ncatlab.org/nlab/source/Alissa+Crans |
* [website](http://myweb.lmu.edu/acrans/)
## related entries
* [[Lie 2-algebra]]
* [[2-vector space]]
category: people
[[!redirects Alissa S. Crans]]
|
Alistair Savage | https://ncatlab.org/nlab/source/Alistair+Savage |
* [personal page](https://alistairsavage.ca/)
## Selected writings
Introduction to [[categorification]]:
* [[Alistair Savage]], *Introduction to categorification* [[arXiv:1401.6037](https://arxiv.org/abs/1401.6037), slides:[pdf](https://alistairsavage.ca/talks/2014-savage-intro-to-categorification.pdf... |
all arrows monic > history | https://ncatlab.org/nlab/source/all+arrows+monic+%3E+history | < [[all arrows monic]] |
all changes | https://ncatlab.org/nlab/source/all+changes |
* [current](http://www.math.ntnu.no/~stacey/Mathforge/nForum/?CategoryID=5)
* [[2009 September changes|2009 September]]
* [[2009 August changes|2009 August]]
* [[2009 July changes|2009 July]]
* [[2009 June changes|2009 June]]
* [[2009 May changes|2009 May]]
* [[2009 April changes|2009 April]]
* [[2009 March cha... |
all horizontal weight systems are partitioned Lie algebra weight systems | https://ncatlab.org/nlab/source/all+horizontal+weight+systems+are+partitioned+Lie+algebra+weight+systems |
+-- {: .rightHandSide}
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### Context
#### Knot theory
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[[!include knot theory - contents]]
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#### Lie theory
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[[!include infinity-Lie theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
All [[weight syst... |
All Pages > history | https://ncatlab.org/nlab/source/All+Pages+%3E+history | |
Allan J. Silberger | https://ncatlab.org/nlab/source/Allan+J.+Silberger |
* [personal page](https://academic.csuohio.edu/silberger/)
## Related concepts
On [[L-parameters]] in the [[local Langlands correspondence]]
* {#SilbergerZink14} [[Allan J. Silberger]], [[Ernst-Wilhelm Zink]], Section 3 of: *Langlands Classification for L-Parameters*, Journal of Algebra Volume 511, 1 Octob... |
Allan L. Edelson | https://ncatlab.org/nlab/source/Allan+L.+Edelson |
* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=1087)
## Selected writings
On [[Real vector bundles]] and their [[KR-theory]]:
* [[Allan L. Edelson]], *Real Vector Bundles and Spaces with Free Involutions*, Transactions of the American Mathematical Society **157** (1971) 179-188 ... |
Allan Merino | https://ncatlab.org/nlab/source/Allan+Merino |
* [personal page](http://allanmerino.com/)
## Selected writings
On [[reductive dual pair|reductive dual pairs]]/[[Howe duality]] for [[super Lie algebras]]:
* [[Allan Merino]], Hadi Salmasian, *Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup* [[arXiv:... |
allegorical set theory | https://ncatlab.org/nlab/source/allegorical+set+theory |
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### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
There are two general approaches to [[structural set theory]]; those that attempt to axiomatise the [... |
allegory | https://ncatlab.org/nlab/source/allegory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea ##
In [[category th... |
Allen Hatcher | https://ncatlab.org/nlab/source/Allen+Hatcher |
* [website](https://math.cornell.edu/allen-hatcher)
## Selected writings
On [[algebraic topology]]:
* {#Hatcher02} [[Allen Hatcher]], *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic... |
Allen Knutson | https://ncatlab.org/nlab/source/Allen+Knutson |
* [webpage](http://www.math.cornell.edu/m/People/Faculty/knutson)
category: people
|
almost connected topological group | https://ncatlab.org/nlab/source/almost+connected+topological+group |
+-- {: .rightHandSide}
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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
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#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
+-- {: .num_defn #Almo... |
almost equality | https://ncatlab.org/nlab/source/almost+equality |
In [[measure theory]], two [[functions]] $f, g\colon X \to Y$ are __almost-everywhere equal__, or __almost equal__, if their [[equaliser]]
$$ \{ a\colon X \;|\; f(a) = g(a) \} $$
is a [[full set]]. We are usually only interested when the functions $f$ and $g$ are [[measurable function|measurable]], although this is n... |
almost function | https://ncatlab.org/nlab/source/almost+function |
In [[measure theory]], an __almost-everywhere-defined function__, or __almost function__, is a [[partial function]] whose domain is a [[full set]]. We are usually only interested in *[[measurable function|measurable]]* almost functions. Typically, we consider almost functions up to the [[equivalence relation]] of al... |
almost Hermitian structure | https://ncatlab.org/nlab/source/almost+Hermitian+structure |
+-- {: .rightHandSide}
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###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
A... |
almost Kähler geometric quantization | https://ncatlab.org/nlab/source/almost+K%C3%A4hler+geometric+quantization |
#Contents#
* table of contents
{:toc}
## Idea
A [[Kähler structure]] on a [[symplectic manifold]] induces a [[polarization]], and [[geometric quantization]] with respect to such [[Kähler polarizations]] works particularly well and has numerous examples (e.g. the [[orbit method]] and [[quantization of Chern-Simons th... |
almost module | https://ncatlab.org/nlab/source/almost+module |
#Contents#
* table of contents
{:toc}
_Under construction: Extracted from a series of tweets by Syzygay_
## Idea
Let $R$ be a (commutative unital) [[ring]]. Suppose $I$ is a flat idempotent $R$-ideal. To be a flat ideal means $-\otimes I$ is an [[exact functor]]. The tensor product is always right exact, s... |
almost open subspace | https://ncatlab.org/nlab/source/almost+open+subspace |
A [[subspace]] $A$ of a [[space]] $X$ is __almost open__ if it is [[open subspace|open]] modulo the $\sigma$-[[sigma-ideal|ideal]] of [[meagre subspace|meagre subspaces]]. We also say that $A$ has the __Baire property__.
Explicitly, $A$ is almost open if there exist an open subspace $G$ and an [[infinite sequence]] ... |
almost scheme | https://ncatlab.org/nlab/source/almost+scheme |
#Contents#
* table of contents
{:toc}
## Motivation
Gabber and Ramero have developed a chapter in abstract algebra and [[algebraic geometry]] which should, in the style of Grothendieck's theory of [[schemes]], capture some of the ideas of
* [[Gerd Faltings]], _Almost étale extensions_, Cohomologie... |
almost surely | https://ncatlab.org/nlab/source/almost+surely |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
In [[probability theory]] an [[event (probability theory)|event]] of [[p... |
Alonso Botero | https://ncatlab.org/nlab/source/Alonso+Botero |
* [Institute page](https://fisica.uniandes.edu.co/en/professors/alonso-botero-mejia)
* [GoogleScholar page](https://scholar.google.com/citations?user=e06A7mUAAAAJ&hl=en)
## Selected writings
Relating [[quantum information theory]] to the [[representation theory of the symmetric group]]:
* [[Alonso Botero]... |
Alonso Perez-Lona | https://ncatlab.org/nlab/source/Alonso+Perez-Lona |
doctoral student at Virginia Tech in the group of [[Eric Sharpe]].
* [MO page](https://mathoverflow.net/users/172910/alonso-perez-lona)
## Selected writings
On [[orbifolds]] by [[2-groups]] in view of [[sigma-models]] inspired from [[string theory]]:
* [[Alonso Perez-Lona]], [[Eric Sharpe]], *Three-dimens... |
Alonzo Church | https://ncatlab.org/nlab/source/Alonzo+Church | * [Wikipedia entry](http://en.wikipedia.org/wiki/Alonzo_Church)
## Selected writings
On [[simple type theory]]:
* {#Church40} [[Alonzo Church]], §5 of: *A Formulation of the Simple Theory of Types*, The Journal of Symbolic Logic **5** 2 (1940) 56-68 [[doi:10.2307/2266170](https://doi.org/10.2307/226617... |
alpha particle | https://ncatlab.org/nlab/source/alpha+particle | [[!redirects alpha particles]]
## Idea
The [[atomic nucleus]] of [[Helium]] is also called the _alpha-particle.
## References
* Wilipedia, _[Alpha particle](https://en.m.wikipedia.org/wiki/Alpha_particle)
|
alpha-equivalence | https://ncatlab.org/nlab/source/alpha-equivalence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--
\tableofcontents
## Idea
In [[logic]] and [[type t... |
alternating group | https://ncatlab.org/nlab/source/alternating+group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An [[alternating group]] $A_n$ is a [[subgroup]] of a [[symmetric group]] $S_n$ consisting of th... |
alternating multifunction | https://ncatlab.org/nlab/source/alternating+multifunction |
# Alternating functions
* table of contents
{: toc}
## Idea
In [[linear algebra]], alternating [[multilinear functions]] are well known, and are in many cases (over the [[real numbers]], for example) are equivalent to [[antisymmetric multifunction|antisymmetric functions]]. In cases where they differ (such as in [[... |
alternating representation | https://ncatlab.org/nlab/source/alternating+representation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $S_n$ a [[symmetric group]], its _alternating representation_ is the... |
alternative algebra | https://ncatlab.org/nlab/source/alternative+algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definitions
+-- {: .num_defn #AlternativeAlgebra}
###### Definition
Consider the following equational l... |
alternative experimental definition of commutative diagram | https://ncatlab.org/nlab/source/alternative+experimental+definition+of+commutative+diagram | ##Discussion
[[Eric]]: Here is a degenerate loop:
$$\array{
X & \stackrel{e}{\to} & X \\
{} & \mathllap{\scriptsize{e}}{\nwarrow} & \darr\scriptsize{e} \\
{} & {} & X
}$$
To say this loop commutes, is to say $e^3 = 1_X$. Or is it?
>[[David Roberts]]: That is correct.
[[Eric]]: Since this loop is degenerate, we may ... |
alternative magmoid | https://ncatlab.org/nlab/source/alternative+magmoid | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Categorification
+-- {: .hide}
[[!include categorification - contents]]
=--
=--... |
Alvaro Restuccia | https://ncatlab.org/nlab/source/Alvaro+Restuccia |
* [InSpire page](https://inspirehep.net/authors/991923)
* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=194082)
## Selected writings
On the [[super 2-brane in 4d]]-[[D=4 supergravity]]:
* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], _$\mathcal{N}=1$ 4D Superm... |
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