title stringlengths 1 131 | url stringlengths 33 167 | content stringlengths 0 637k |
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2-pro-object | https://ncatlab.org/nlab/source/2-pro-object | +-- {: .rightHandSide}
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###Context###
#### 2-Category theory
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[[!include 2-category theory - contents]]
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# Contents
* table of contents
{:toc}
## Idea
In the context of [[2-category theory]], the concept of a **2-pro-object** is the [[cat... |
2-pullback | https://ncatlab.org/nlab/source/2-pullback | +-- {: .rightHandSide}
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###Context###
#### 2-category theory
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[[!include 2-category theory - contents]]
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# $2$-pullbacks
* table of contents
{: toc}
An ordinary [[pullback]] is a [[limit]] over a [[diagram]] of the form $A \to C \leftarrow B$. ... |
2-representation | https://ncatlab.org/nlab/source/2-representation | [[!redirects 2-representations]]
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###Context###
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
A _2-representation_ of a [[2-group]] $G$ is a... |
2-rig | https://ncatlab.org/nlab/source/2-rig |
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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
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# $2$-rigs
* table of contents
{: toc}
## Idea
The notion of _$2$-rig_ is supposed to be a [[categorification]] of that of a [[rig]]. Sever... |
2-sheaf | https://ncatlab.org/nlab/source/2-sheaf |
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###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
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#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
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#### 2-Category theory
+--{: .hide}
[[!includ... |
2-site | https://ncatlab.org/nlab/source/2-site |
This entry is about the notion of [[site]] in [[2-category]] theory. For the notion "bisite" of a 1-categorical site equipped with two coverages see instead [[separated presheaf]].
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###Context###
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include ... |
2-spectral triple | https://ncatlab.org/nlab/source/2-spectral+triple |
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###Context###
#### Noncommutative geometry
+--{: .hide}
[[!include noncommutative geometry - contents]]
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#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
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#Contents#
* tab... |
2-topos | https://ncatlab.org/nlab/source/2-topos |
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###Context###
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
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#Contents#
* table of contents
{:toc}... |
2-topos theory | https://ncatlab.org/nlab/source/2-topos+theory |
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###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Idea
The theory of [[2-topos]]es, generalizing [[topos theory]] from [[category theory]] to [[2-category theor... |
2-trivial model structure | https://ncatlab.org/nlab/source/2-trivial+model+structure |
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###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
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#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
##... |
2-type > history | https://ncatlab.org/nlab/source/2-type+%3E+history | < [[2-type]]
[[!redirects 2-type -- history]] |
2-type theory | https://ncatlab.org/nlab/source/2-type+theory |
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###Context###
#### Type theory
+--{: .hide}
[[!include type theory - contents]]
=--
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Idea
As [[type theor... |
2-vector bundle | https://ncatlab.org/nlab/source/2-vector+bundle |
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###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _2-module bundle_ / _2-vector bun... |
2-vector space | https://ncatlab.org/nlab/source/2-vector+space |
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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
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#Content#
* table of contents
{:toc}
## Idea
... |
2008 changes | https://ncatlab.org/nlab/source/2008+changes |
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### Archive
+-- {: .hide}
[[!include all changes]]
=--
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=--
Archive of changes made during 2008.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/nlab/re... |
2009 April changes | https://ncatlab.org/nlab/source/2009+April+changes | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during April 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://nca... |
2009 August changes | https://ncatlab.org/nlab/source/2009+August+changes |
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### Archive
+-- {: .hide}
[[!include all changes]]
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Archive of changes made during August 2009.
The substantive content of this page should **not** be altered.
***
## 2009-08-31
* [[Urs Schreiber]]
* made [[infinitesimal space]] and [[infin... |
2009 February changes | https://ncatlab.org/nlab/source/2009+February+changes |
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### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during February 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.or... |
2009 January changes | https://ncatlab.org/nlab/source/2009+January+changes |
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+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during January 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org... |
2009 July changes | https://ncatlab.org/nlab/source/2009+July+changes |
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+-- {: .toc .clickDown tabindex="0"}
### Archive
+-- {: .hide}
[[!include all changes]]
=--
=--
=--
Archive of changes made during July 2009.
The substantive content of this page should **not** be altered.
***
## 2009-07-31
* [[Mike Shulman]] fixed a mistake at [[axiom of foundation]].
* ... |
2009 June changes | https://ncatlab.org/nlab/source/2009+June+changes |
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### Archive
+-- {: .hide}
[[!include all changes]]
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Archive of changes made during June 2009.
The substantive content of this page should **not** be altered.
***
# 2009-06-30 #
* [[Urs Schreiber|Urs]]
* created [[Eilenberg-MacLane object]] ... |
2009 March changes | https://ncatlab.org/nlab/source/2009+March+changes |
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### Archive
+-- {: .hide}
[[!include all changes]]
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=--
Archive of changes made during March 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/n... |
2009 May changes | https://ncatlab.org/nlab/source/2009+May+changes |
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### Archive
+-- {: .hide}
[[!include all changes]]
=--
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=--
Archive of changes made during May 2009.
The substantive content of this page should **not** be altered.
For past versions of this page beyond its own history, start [here](http://ncatlab.org/nla... |
2009 September changes | https://ncatlab.org/nlab/source/2009+September+changes |
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### Archive
+-- {: .hide}
[[!include all changes]]
=--
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=--
Archive of changes made during September 2009.
The substantive content of this page should **not** be altered.
The announcement of the change to [the Forum](http://www.math.ntnu.no/~s... |
24 | https://ncatlab.org/nlab/source/24 |
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###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
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#### Exceptional structures
+-- {: .hide}
[[!include exceptional structures -- contents]]
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=--
#Contents#
* table of contents
{:toc}
## ... |
24 branes transverse to K3 | https://ncatlab.org/nlab/source/24+branes+transverse+to+K3 |
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Several classes of [[string theory vacua]] require the presence of exactly [[24]] [[branes]]... |
24-cell | https://ncatlab.org/nlab/source/24-cell |
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### Context
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
...one of the [[regular polytopes]] in [[dimension]] 4...
...hence a higher dimensional analo... |
2Cat | https://ncatlab.org/nlab/source/2Cat |
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###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### [[categories of categories - contents|categories of categories]]
+... |
2d Chern-Simons theory | https://ncatlab.org/nlab/source/2d+Chern-Simons+theory |
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###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
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#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physi... |
2d quantum gravity | https://ncatlab.org/nlab/source/2d+quantum+gravity |
#Contents#
* table of contents
{:toc}
## Idea
In the very low number of just 2 dimensions, [[quantum gravity]] is comparatively trivial and can be completely understood. While trivial as a theory of [[quantum gravity]] on a 2d spacetime, when that spacetime is regarded instead as the [[worldsheet]] of a string, then... |
2d SCFT | https://ncatlab.org/nlab/source/2d+SCFT |
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###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
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#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry... |
2d TQFT | https://ncatlab.org/nlab/source/2d+TQFT |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _2-dim... |
2d TQFT -- table | https://ncatlab.org/nlab/source/2d+TQFT+--+table |
| [[2d TQFT]] ("[[TCFT]]") | [[coefficients]] | [[algebra]] structure on [[space of quantum states]] | |
|-------------|-------------|--------------------|--------|
| [[open string|open]] [[topological string]] | [[Vect]]${}_k$ | [[Frobenius algebra]] $A$ | [[folklore]]+([Abrams 96](2d+TQFT#Abrams96)) |
| [[open ... |
2d Wess-Zumino-Witten theory | https://ncatlab.org/nlab/source/2d+Wess-Zumino-Witten+theory |
* [[WZW model]]
## Related concepts
* [[2d Chern-Simons theory]]
|
2Mod | https://ncatlab.org/nlab/source/2Mod |
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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
The [[2-category]] of [[2-modules]]/[[2-vector spaces]].
Fix some [[commutative ring]] or more generally an [[... |
2T relation | https://ncatlab.org/nlab/source/2T+relation |
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### Context
#### Knot theory
+-- {: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[knot theory]] the _2-term relations_ on [[horizontal chord diagrams]] are the following [[rel... |
3-brane in 6d | https://ncatlab.org/nlab/source/3-brane+in+6d |
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
There is supposed to be a $(p=3)$-[[brane]] in 6-dimensional [[supersymmetry|super]]-[[spaceti... |
3-category | https://ncatlab.org/nlab/source/3-category |
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###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
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=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A $3$-category is any of several concepts that generalize $2$-[[2-category|categories]] one ste... |
3-category of fermionic conformal nets | https://ncatlab.org/nlab/source/3-category+of+fermionic+conformal+nets |
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###Context###
#### Super-Algebra and Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - con... |
3-category with contravariance | https://ncatlab.org/nlab/source/3-category+with+contravariance | # 3-categories with contravariance
* table of contents
{: toc}
## Idea
A *3-category with contravariance* is a [[categorification]] of a [[2-category with contravariance]]. It is an abstract structure that generalizes the structure present on the collection of [[2-categories]] and all four sorts of [[contravariant ... |
3-group | https://ncatlab.org/nlab/source/3-group |
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###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
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#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definit... |
3-groupoid | https://ncatlab.org/nlab/source/3-groupoid |
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###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of **3-groupoid** is the next higher generalization in [[higher... |
3-groupoid of Lie 3-algebra valued forms | https://ncatlab.org/nlab/source/3-groupoid+of+Lie+3-algebra+valued+forms |
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###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--
#Contents#
* table of contents... |
3-manifold | https://ncatlab.org/nlab/source/3-manifold |
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###Context###
#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A _3-manifold_ is a [[manifold]] of [[dimension]] 3. (Our default meaning... |
3-morphism | https://ncatlab.org/nlab/source/3-morphism |
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###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
* [[object]]
* [[morphism]]
* [[2-morphism]]
* **3-morphism**
* [[k-morphism]]
***
#Contents#
* table of contents
{:toc}
## Definition
A... |
3-poset | https://ncatlab.org/nlab/source/3-poset | A $3$-poset is any of several concepts that generalize $2$-[[2-poset|posets]] one step in [[higher category theory]]. One does not usually here about $3$-posets by themselves but instead as special cases of $3$-[[3-category|categories]].
$3$-posets can also be called $(2,3)$-[[(n,r)-category|categories]]. The concept... |
3-sphere | https://ncatlab.org/nlab/source/3-sphere |
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###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The $n$-[[sphere]] for $n = 3$.
## Properties
### Isomorphisms
The underlying manifold of the [[s... |
3-strand-braid-1-SVG | https://ncatlab.org/nlab/source/3-strand-braid-1-SVG |
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="403.2646pt" height="143.25488pt" viewBox="-18.65686 -71.62744 403.2646 143.25488 ">
<defs
id="defs4">
<inkscape:perspective
id="perspective10"
inkscape:persp3d-origin="372.04724 : 350.78739 :... |
3-theory | https://ncatlab.org/nlab/source/3-theory |
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###Context###
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
* table of contents
{: toc}
## Idea
In the context of an accoun... |
3d Calabi-Yau object | https://ncatlab.org/nlab/source/3d+Calabi-Yau+object |
#Contents#
* table of contents
{:toc}
## Idea
A [[Calabi-Yau object]] ([[Calabi-Yau manifold]], [[Calabi-Yau category]]) of (complex) [[dimension]] 3.
## Properties
### As exceptional geometry
[[!include Spin(8)-subgroups and reductions -- table]]
### Moduli spaces of line bundles
[[!includ... |
3d Chern-Simons theory | https://ncatlab.org/nlab/source/3d+Chern-Simons+theory |
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###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physi... |
3d gravity and Chern-Simons theory -- references | https://ncatlab.org/nlab/source/3d+gravity+and+Chern-Simons+theory+--+references | ### 3d Gravity and Chern-Simons theory
{#3dGravityAndChernSimonsTheoryReferences}
On 3-dimensional ([[quantum gravity|quantum]]) [[gravity]] ([[general relativity]]) with [[cosmological constant]], and its (non-)relation to [[Chern-Simons theory]] with non-[[compact Lie group|compact]] [[gauge groups]]:
The ori... |
3d mirror symmetry | https://ncatlab.org/nlab/source/3d+mirror+symmetry |
## Related concepts
* [[symplectic duality]]
* [[duality in physics]]
## References
The original article is
* [[Ken Intriligator]], [[Nathan Seiberg]], _Mirror Symmetry in Three Dimensional Gauge Theories_ ([arXiv:hep-th/9607207](http://arxiv.org/abs/hep-th/9607207))
based on the [[ADE classificatio... |
3d quantum gravity | https://ncatlab.org/nlab/source/3d+quantum+gravity |
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###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Solving the problem of [[quantization]] of s... |
3d superconformal gauge field theory | https://ncatlab.org/nlab/source/3d+superconformal+gauge+field+theory |
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###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory... |
3d toric code | https://ncatlab.org/nlab/source/3d+toric+code |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
What has come to be called the "3d toric code" is a [[4d ... |
3d TQFT | https://ncatlab.org/nlab/source/3d+TQFT |
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###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
... |
3d-3d correspondence | https://ncatlab.org/nlab/source/3d-3d+correspondence |
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###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Id... |
3x3 lemma | https://ncatlab.org/nlab/source/3x3+lemma |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Diagram chasing lemmas
+-- {: .hide}
[[!include diagram chasing lemmas - contents]]
=--
#### Homological algebra
+-- {: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##... |
4-3-2 8-7-6 | https://ncatlab.org/nlab/source/4-3-2+8-7-6 |
The talk
* [[Daniel Freed]],
**4-3-2 8-7-6**
talk at _[ASPECTS of Topology](https://people.maths.ox.ac.uk/tillmann/ASPECTS.html)_ Dec 2012
[pdf](https://people.maths.ox.ac.uk/tillmann/ASPECTSfreed.pdf), [[Freed432876.pdf:file]]
discusses aspects and open problems of understanding
* [[3d Chern... |
4-manifold | https://ncatlab.org/nlab/source/4-manifold |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
a [[manifold]] of [[dimension]] 4.
## Examples
* [[4-sphere]]
* [[spacetime]]
## Prope... |
4-sphere | https://ncatlab.org/nlab/source/4-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[sphere]] of [[dimension]] 4.
## Properties
### Basic differential geometry
{#Differenti... |
4d Chern-Simons theory | https://ncatlab.org/nlab/source/4d+Chern-Simons+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
... |
4d superconformal gauge field theory | https://ncatlab.org/nlab/source/4d+superconformal+gauge+field+theory |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theo... |
4d supergravity Lie 2-algebra | https://ncatlab.org/nlab/source/4d+supergravity+Lie+2-algebra |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
#Contents#
* ta... |
4d TQFT | https://ncatlab.org/nlab/source/4d+TQFT |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
... |
4d WZW model | https://ncatlab.org/nlab/source/4d+WZW+model |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Wess-Zumino-Witten theory
+--{: .hide}
[[!include infinity-Wess-Zumino-Witten theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!i... |
4T relation | https://ncatlab.org/nlab/source/4T+relation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
### For round chord diagrams
{#ForCircularChordDiagrams}
In [[knot theory]] by the _4-te... |
5-manifold | https://ncatlab.org/nlab/source/5-manifold |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[manifold]] of [[dimension]] 5
## Related concepts
* [[Sasakian m... |
5-sphere | https://ncatlab.org/nlab/source/5-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[n-sphere]] for $n = 5$.
## Properties
### Free group actions
\begin{example}
There are, up... |
6-sphere | https://ncatlab.org/nlab/source/6-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[n-sphere]] of [[dimension]] $n = 6$.
## Properties
### Coset structure
The 6-sphere, as a [... |
600-cell | https://ncatlab.org/nlab/source/600-cell |
#Contents#
* table of contents
{:toc}
## Idea
...one of the [[regular polytopes]] in [[dimension]] 4...
...hence a higher dimensional analog of the [[Platonic solids]]...
The 120 vertices of the 600-cell form the [[binary icosahedral group]].
## Related concepts
* [[24-cell]]
* [[120-cell]]
... |
6d Chern-Simons theory | https://ncatlab.org/nlab/source/6d+Chern-Simons+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include phy... |
6j symbol | https://ncatlab.org/nlab/source/6j+symbol | [[!redirects Wigner 6-j symbol]]
[[!redirects Wigner 6-j symbol]]
#Contents#
* table of contents
{:toc}
## Idea
$6j$ symbols are pieces of combinatorial data which can be associated to [[fusion categories]], encoding their [[associator]] with a finite amount of algebraic data. The $6j$ symbols along with th... |
7-sphere | https://ncatlab.org/nlab/source/7-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[sphere]] of [[dimension]] 7.
This is one of the [[parallelizable manifold|parallelizable]] sp... |
7d Chern-Simons theory | https://ncatlab.org/nlab/source/7d+Chern-Simons+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[... |
7d spherical space forms -- table | https://ncatlab.org/nlab/source/7d+spherical+space+forms+--+table |
| $N$ [[Killing spinors]] on <br/> [[spherical space form]] $S^7/\widehat{G}$ | $\phantom{AA}\widehat{G} =$ | [[spin group|spin]]-lift of [[subgroup]] of <br/>[[isometry group]] of [[7-sphere]] | [[3d superconformal gauge field theory]] <br/> on [[black brane|back]] [[M2-branes]] <br/> with [[near horizon geometry]]... |
8-manifold | https://ncatlab.org/nlab/source/8-manifold |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A [[manifold]] of [[dimension]] 8.
## Properties
### Signature
{#Signature}
Let $X$ be ... |
8-sphere | https://ncatlab.org/nlab/source/8-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[n-sphere]] for $n = 8$.
## Properties
+-- {: .num_prop}
###### Proposition
There is a [[hom... |
9-dimensional supergravity | https://ncatlab.org/nlab/source/9-dimensional+supergravity |
#Contents#
* table of contents
{:toc}
## Idea
[[supergravity]] in [[dimension]] 9
## Properties
### U-duality
[[!include U-duality -- table]]
## References
* [[Maria P. Garcia del Moral]], J. M. Pena, [[Alvaro Restuccia]], _Supermembrane origin of type II gauged supergravities in 9D_, JHEP 12... |
A Categorical Manifesto | https://ncatlab.org/nlab/source/A+Categorical+Manifesto |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--
[_A Categorical Manifesto_]( ht... |
A Concise Course in Algebraic Topology | https://ncatlab.org/nlab/source/A+Concise+Course+in+Algebraic+Topology |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic topology
+--{: .hide}
[[!include algebraic topology - contents]]
=--
=--
=--
This page collects material related to:
* [[Peter May]]:
\linebreak
**A Concise Course in Algebraic Topology**
\line... |
A crash course in topos theory -- The big picture | https://ncatlab.org/nlab/source/A+crash+course+in+topos+theory+--+The+big+picture |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
* [[André Joyal]],
_A crash course in topos theory:... |
A Cubical Approach to Synthetic Homotopy Theory | https://ncatlab.org/nlab/source/A+Cubical+Approach+to+Synthetic+Homotopy+Theory |
* [[Dan Licata]], [[Guillaume Brunerie]]:
**A Cubical Approach to Synthetic Homotopy Theory**
LICS 2015,
30th Annual ACM/IEEE Symposium on Logic in Computer Science
[doi:10.1109/LICS.2015.19](https://doi.org/10.1109/LICS.2015.19)
[PDF from Dan's homepage](http://dlicata.web.wesleyan.edu/pub... |
A first idea of quantum field theory -- Feynman diagrams | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Feynman+diagrams | ## Feynman diagrams
{#FeynmanDiagrams}
So far we considered only the [[axioms]] on a consistent perturbative S-matrix /[[time-ordered products]]
and its formal consequences. Now we discuss the actual construction of [[time-ordered products]],
hence of perturbative S-matrices, by the process called _[[renormaliza... |
A first idea of quantum field theory -- Field variations | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Field+variations | ## Field variations
{#FieldVariations}
In this chapter we discuss these topics:
* _[Jet bundles](##JetBundles)_
* _[Differential operators](#DifferentialOperators)_
* _[Variational calculus and the Variational bicomplex](#VariationaCalculusAndVariationalBicomplex)_
$\,$
Given a [[field bundle]] as i... |
A first idea of quantum field theory -- Fields | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Fields | ## Fields
{#Fields}
In this chapter we discuss these topics:
* _[Field bundles](#FieldBundles)_
* _[Spaces of field histories](#NonFiniteDimensionalGeometry)_
* _[Infinitesimal geometry](#InfinitesimalGeometry)_
* _[Fermion fields and Supergeometry](#Supergeometry)_
$\,$
A [[field history]] on ... |
A first idea of quantum field theory -- Free quantum fields | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Free+quantum+fields | ## Free quantum fields
{#FreeQuantumFields}
In this chapter we discuss the following topics:
* _[Wick algebra](#WickAlgebraAbstract)_
* _[Time-ordered product](#AbstractTimeOrderedProduct)_
* _[Operator product notation](#OperatorProductAndNormalOrderedProduct)_
* _[Hadamard vacuum state](#HadamardVacu... |
A first idea of quantum field theory -- Gauge fixing | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Gauge+fixing |
## Gauge fixing
{#GaugeFixing}
In this chapter we discuss the following topics:
1. _[Quasi-isomorphisms between local BV-BRST complexes](#QuasiIsomorphismsBetweenBVBRSTComplexes)_
1. [gauge fixing chain maps](#GaugeFixingChainMaps);
1. [adjoining contractible complexes of auxiliary fields](#Adjo... |
A first idea of quantum field theory -- Gauge symmetries | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Gauge+symmetries |
## Gauge symmetries
{#GaugeSymmetries}
In this chapter we discuss these topics:
* [compactly supported infinitesimal symmetries obstruct covariant phase space](#CompactlySupportedInfinitesimalSymmetriesObstructTheCovariantPhaseSpace)
* [Infinitesimal gauge symmetries](#InfinitesimalGauge)
* [Lie a... |
A first idea of quantum field theory -- Geometry | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Geometry |
## Geometry
{#Geometry}
The [[geometry of physics]] is _[[differential geometry]]_. This is the flavor of [[geometry]] which is modeled on
[[Cartesian spaces]] $\mathbb{R}^n$ with [[smooth functions]] between them. Here we briefly review the basics of
[[differential geometry]] on [[Cartesian spaces]].
In ... |
A first idea of quantum field theory -- Interacting quantum fields | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Interacting+quantum+fields |
## Interacting quantum fields
{#QuantumObservables}
In this chapter we discuss the following topics:
* _[Free field vacua](#FreeFieldVacua)_
* _[Perturbative S-matrices](#PerturbativeSMatrixAndTimeOrderedProducts)_
* _[Conceptual remarks](#RemarksOnCausalPerturbationTheoryAxioms)_
* _[Interacting fie... |
A first idea of quantum field theory -- Lagrangians | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Lagrangians | ## Lagrangians
{#Lagrangians}
In this chapter we discuss the following topics:
* _[Lagrangian densities](#LagrangianDensities)_
* _[Euler-Lagrange forms and Presymplectic currents](#ELFormsAndPresymplecticCurrents)_
* _[Euler-Lagrange equations of motion](#ELEquationsOfMotion)_
$\,$
Given any [[t... |
A first idea of quantum field theory -- Observables | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Observables |
## Observables
{#Observables}
In this chapter we discuss these topics:
* _[General observables](#GeneralObservables)_
* _[Polynomial off-shell observables and Distributions](#LinearOffShellObservablesAreDistributions)_
* _[Polynomial on-shell observables and Distributional solutions to PDEs](#Poly... |
A first idea of quantum field theory -- Phase space | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Phase+space | ## Phase space
{#PhaseSpace}
In this chapter we discuss these topics:
* _[Covariant phase space](#PreymplecticPhaseSpace)_
* _[BV-Resolution of the covariant phase space](#BVResolutionOfTheCovariantPhaseSpace)_
* _[Hamiltonian local observables](#HamiltonianLocalObservablesOnACauchySurface)_
$\,$
... |
A first idea of quantum field theory -- Propagators | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Propagators | ## Propagators
{#Propagators}
In this chapter we discuss the following topics:
* _Background_
* _[Fourier analysis and Plane wave modes](#FourierAnalysis)_
* _[Microlocal analysis and UV-Divergences](#MicrolocalAnalysisAndUltravioletDivergence)_
* _[Cauchy principal values](#CauchyPrincipalValues... |
A first idea of quantum field theory -- Quantization | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Quantization | ## Quantization
{#Quantization}
In this chapter we discuss the following topics:
* _[Motivation from Lie theory](#MotivationFromLieTheory)_
* _[Geometric quantization](#GeometricQuantization)_
* _[Moyal star products](#StarProducts)_
* _[Moyal star product as deformation quantization](#MoyalStarProdu... |
A first idea of quantum field theory -- Quantum electrodynamics | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Quantum+electrodynamics |
## Quantum Electrodynamics
{#QED}
* [[quantum electrodynamics]]
|
A first idea of quantum field theory -- Reduced phase space | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Reduced+phase+space | ## Reduced phase space
{#ReducedPhaseSpace}
In this chapter we discuss these topics:
* Global gauge reduction for strictly [[invariant]] functions ([[action functionals]]):
* _[Derived critical loci inside Lie algebroids](#DerivedCriticalLocusInsideLieAlgebroids)_
* _[Schouten bracket on Lie algeb... |
A first idea of quantum field theory -- Renormalization | https://ncatlab.org/nlab/source/A+first+idea+of+quantum+field+theory+--+Renormalization |
## Renormalization
{#Renormalization}
In this chapter we discuss the following topics:
* _[Epstein-Glaser normalization](#EpsteinGlaserRenormalization)_
* _[Stückelberg-Petermann re-normalization](#SPRenormalizationGroup)_
* _[UV-Regularization via Counterterms](#UVRegularizationViaZ)_
* _[Wilson... |
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