title stringlengths 1 131 | url stringlengths 33 167 | content stringlengths 0 637k |
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!-modality | https://ncatlab.org/nlab/source/%21-modality | +-- {: .rightHandSide}
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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
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#### Modalities, Closure and Reflection
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[[!include modalities - contents]]
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#Contents#
* table of contents
{:toc}
## Idea
{#Idea}
... |
$\infty$-category > history | https://ncatlab.org/nlab/source/%24%5Cinfty%24-category+%3E+history | < [[$\infty$-category]]
[[!redirects $\infty$-category/history]]
[[!redirects $\infty$-category -- history]] |
$n$-category > history | https://ncatlab.org/nlab/source/%24n%24-category+%3E+history | < [[$n$-category]]
[[!redirects $n$-category/history]]
[[!redirects $n$-category -- history]] |
't Hooft anomaly | https://ncatlab.org/nlab/source/%27t+Hooft+anomaly |
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### Context
#### Quantum systems
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[[!include quantum systems -- contents]]
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#### Physics
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}
## Idea
As described in ['t Hooft (198... |
't Hooft coupling | https://ncatlab.org/nlab/source/%27t+Hooft+coupling |
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###Context###
#### Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#Contents#
* table of contents
{:toc}
## Idea
In [[Yang-Mills theory]]/[[QCD]] with [[coupling constant]] $g_{YM}$ and with ... |
't Hooft double line notation | https://ncatlab.org/nlab/source/%27t+Hooft+double+line+notation |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
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#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
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#### String theory
+-- {: .hide}
[[!include string theory - con... |
't Hooft-Polyakov monopole | https://ncatlab.org/nlab/source/%27t+Hooft-Polyakov+monopole |
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+-- {: .toc .clickDown tabindex="0"}
### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
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=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A type of [[monopole]] solution of [[Yang-Mills theory]] with [[Higgs field]] included.
## Properties
#... |
(-1)-category | https://ncatlab.org/nlab/source/%28-1%29-category | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context ###
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
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## Idea
As a degenerate case of the general notion of *[[n-category]]*, _$(-1)$-categories_ may be understood as [[truth value]].... |
(-1)-functor | https://ncatlab.org/nlab/source/%28-1%29-functor |
As a $(-1)$-[[(-1)-category|category]] is simply a [[truth value]], so a __$(-1)$-functor__ is simply [[implication]]. See also $n$-[[n-functor|functor]].
[[!redirects (-1)-functor]]
[[!redirects (-1)-functors]]
[[!redirects (−1)-functor]]
[[!redirects (−1)-functors]]
|
(-1)-groupoid | https://ncatlab.org/nlab/source/%28-1%29-groupoid |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
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=--
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#Contents#
* table of contents
{:toc}
## Definition
A __$(-1)$-groupoid__ or **[[homotopy n-type|(-1)-type]]** is a [[truth... |
(-1)-poset | https://ncatlab.org/nlab/source/%28-1%29-poset | There is just one __$(-1)$-poset__, namely the [[point]]. Compare the concepts of $0$-[[0-poset|poset]] (a [[truth value]]) and $1$-[[1-poset|poset]] (a [[partial order|poset]]). Compare also with $(-2)$-[[(-2)-category|category]] and $(-2)$-[[(-2)-groupoid|groupoid]], which mean the same thing for their own reasons.... |
(-2)-category | https://ncatlab.org/nlab/source/%28-2%29-category | There is just one _$(-2)$-category_, namely the truth value [[True]]. Compare the concepts of [[(−1)-category]] (a [[truth value]] in general) and [[0-category]] (a [[set]]). The point of $(-2)$-categories is that they complete some patterns in the [[periodic table]] of $n$-categories. (They also shed light on the theo... |
(-2)-groupoid | https://ncatlab.org/nlab/source/%28-2%29-groupoid |
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###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
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#Contents#
* table of contents
{:toc}
## Definition
A **$(-2)$-groupoid** or **[[homotopy n-type|(-2)-type]]** is a [[n-tru... |
(0,1)-category | https://ncatlab.org/nlab/source/%280%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(0,1)$-Category theory
+-- {: .hide}
[[!include (0,1)-category theory - contents]]
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=--
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# Contents
* table of contents
{: toc}
## Idea
{#Idea}
Following the general concept of *[[(n,r)-category|$(n,r)$-category... |
(0,1)-category theory | https://ncatlab.org/nlab/source/%280%2C1%29-category+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Idea
In the context of [[higher category theory]] / [[(n,r)-categories]], a [[poset]] is equivalentl... |
(0,1)-category theory - contents | https://ncatlab.org/nlab/source/%280%2C1%29-category+theory+-+contents |
**[[(0,1)-category theory]]**: [[logic]], [[order theory]]
**[[(0,1)-category]]**
* [[relation between preorders and (0,1)-categories]]
* [[proset]], [[partially ordered set]] ([[directed set]], [[total order]], [[linear order]])
* [[top]], [[true]],
* [[bottom]], [[false]]
* [[monotone func... |
(0,1)-presheaf | https://ncatlab.org/nlab/source/%280%2C1%29-presheaf | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### (0,1)-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Topos theory
+--{: .hide}
[[!include topos theory - contents]]
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=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **(0,... |
(0,1)-topos | https://ncatlab.org/nlab/source/%280%2C1%29-topos |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(0,1)$-Category theory
+--{: .hide}
[[!include (0,1)-category theory - contents]]
=--
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes here... |
(1,0)-category | https://ncatlab.org/nlab/source/%281%2C0%29-category | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
By the general rules of $(n,r)$-[[(n,r)-category|categories]], a $(1,0)$-category is an $\infty$-[[infinity-category|category]] such tha... |
(1,1)-category | https://ncatlab.org/nlab/source/%281%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
In the pattern of [[(n,r)-categories]] a **$(1,1)$-category** is just an ordinary [[category]].
[[!redirects (1,1)-categories]]
|
(1,1)-dimensional Euclidean field theories and K-theory | https://ncatlab.org/nlab/source/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory |
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
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[[!include functorial quantum field theory - contents]]
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#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
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+-- {: .standout}
This is a sub-entry of [[geometr... |
(1,2)-topos | https://ncatlab.org/nlab/source/%281%2C2%29-topos | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of *(1,2)-topos* should be the notion of [[higher toposes]] among [[(1,2)-categories]] ... |
(2,1)-algebraic theory of E-infinity algebras | https://ncatlab.org/nlab/source/%282%2C1%29-algebraic+theory+of+E-infinity+algebras | [[!redirects (2,1)-algebraic theeory of E-infinity algebras]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[(∞,1)-algebraic theory]] whose [[∞-algebra ... |
(2,1)-category | https://ncatlab.org/nlab/source/%282%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## I... |
(2,1)-dimensional Euclidean field theories and tmf | https://ncatlab.org/nlab/source/%282%2C1%29-dimensional+Euclidean+field+theories+and+tmf | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functorial quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
+-- {: .standout}
This is a sub-e... |
(2,1)-functor | https://ncatlab.org/nlab/source/%282%2C1%29-functor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
... |
(2,1)-presheaf | https://ncatlab.org/nlab/source/%282%2C1%29-presheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Id... |
(2,1)-sheaf | https://ncatlab.org/nlab/source/%282%2C1%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos -... |
(2,1)-sheafification | https://ncatlab.org/nlab/source/%282%2C1%29-sheafification | [[!redirects (2,1)-sheafififcation]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The higher analogue of [[sheafification]... |
(2,1)-site | https://ncatlab.org/nlab/source/%282%2C1%29-site |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Definition
A _$(2,1)$-site_ is an [[(∞,1)-site]]. whose underlying [[(∞,1)-category]] is a [[(2,1)-cate... |
(2,1)-topos of (2,1)-sheaves | https://ncatlab.org/nlab/source/%282%2C1%29-topos+of+%282%2C1%29-sheaves |
## Idea
A [[(2,1)-topos]] of [[(2,1)-sheaves]]
## Related concepts
* [[sheaf topos]]
* [[(infinity,1)-category of (infinity,1)-sheaves]]
|
(2|1)-dimensional Euclidean field theory | https://ncatlab.org/nlab/source/%282%7C1%29-dimensional+Euclidean+field+theory | [[!redirects (2,1)-dimensional Euclidean field theory]]
[[!redirects (2,1)-dimensional Euclidean field theory]]
[[!redirects (2,1)-dimensional Euclidean field theory]]
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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum f... |
(bo, ff) factorization system | https://ncatlab.org/nlab/source/%28bo%2C+ff%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Defini... |
(classical) axiom of multiple choice | https://ncatlab.org/nlab/source/%28classical%29+axiom+of+multiple+choice |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
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# The axiom of multiple choice
* table of contents
{: toc}
This article is about a [[classical mathematics|classical]] [[set theory]] [[axiom]]. Som... |
(co)isotropic subspaces - table | https://ncatlab.org/nlab/source/%28co%29isotropic+subspaces+-+table |
| type of [[subspace]] $W$ of [[inner product space]] | condition on [[orthogonal]] space $W^\perp$ | |
|--|--|--|
| [[isotropic subspace]] | $W \subset W^\perp $ | |
| [[coisotropic subspace]] | $W^\perp \subset W$ | |
| [[Lagrangian subspace]] | $W = W^\perp$ | (for [[symplectic form]]) |
| [[symplectic v... |
(d,d) Klein space | https://ncatlab.org/nlab/source/%28d%2Cd%29+Klein+space |
+-- {: .rightHandSide}
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###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
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#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
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=--
=--
> not to be confused with [[Klein geometry]]
#C... |
(dense,closed)-factorization | https://ncatlab.org/nlab/source/%28dense%2Cclosed%29-factorization |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[topology|point set topology]], every subspace $X$ of a space $A$ has a unique largest subspac... |
(epi, mono) factorization system | https://ncatlab.org/nlab/source/%28epi%2C+mono%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Defi... |
(eso and full, faithful) factorization system | https://ncatlab.org/nlab/source/%28eso+and+full%2C+faithful%29+factorization+system | [[!redirects (eso+full, faithful) factorization system]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--... |
(eso, fully faithful) factorization system | https://ncatlab.org/nlab/source/%28eso%2C+fully+faithful%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
In the [[2-topo... |
(g-2) anomaly | https://ncatlab.org/nlab/source/%28g-2%29+anomaly |
#Contents#
* table of contents
{:toc}
## Idea
For the moment see at _[[anomalous magnetic moment]]_ _[this section](https://ncatlab.org/nlab/show/anomalous+magnetic+moment#Anomalies)_
## Related concepts
* [[flavour anomaly]]
* [[Cabibbo anomaly]]
* [[V_cb puzzle]]
[[!redirects (g-2)-anomaly]... |
(geometric surjection, embedding) factorization system | https://ncatlab.org/nlab/source/%28geometric+surjection%2C+embedding%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Every [[geometric morphism]] between [[toposes]] factors into a [[geometric surjection]] foll... |
(hyperconnected, localic) factorization system | https://ncatlab.org/nlab/source/%28hyperconnected%2C+localic%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+-- {: .hide}
[[!include factorization systems - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
[[hyperconnected geometric... |
(infinity,0)-category | https://ncatlab.org/nlab/source/%28infinity%2C0%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Following the terminology of *[[(n,r)-category|(n,r)-categories]]*, an __$(\inft... |
(infinity,1)-bimodule | https://ncatlab.org/nlab/source/%28infinity%2C1%29-bimodule |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
## Definition
### $(\infty,1)$-Category of $(\infty,1)$-Bimodules and intertwiners
Write $BMod^\otimes$ for ... |
(infinity,1)-categorical hom-space | https://ncatlab.org/nlab/source/%28infinity%2C1%29-categorical+hom-space |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Model category theory
+--{: .hide}
[[!include model category... |
(infinity,1)-categorification | https://ncatlab.org/nlab/source/%28infinity%2C1%29-categorification |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
[[categorification|Categorification]] is a creative nonmechanical proces... |
(infinity,1)-category | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher catego... |
(infinity,1)-category > history | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+%3E+history | < [[(infinity,1)-category]]
[[!redirects (∞,1)-category - history]]
[[!redirects (infinity,1)-category -- history]] |
(infinity,1)-category of (infinity,1)-categories | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+%28infinity%2C1%29-categories |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The collection of all [[(∞,1)-categories]] forms naturally the [[(∞,2)-cat... |
(infinity,1)-category of (infinity,1)-functors | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The generalization of the notion of [[functor category]] from [[category t... |
(infinity,1)-category of (infinity,1)-presheaves | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of conten... |
(infinity,1)-category of (infinity,1)-sheaves | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
... |
(infinity,1)-category of cartesian sections | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+cartesian+sections |
# Idea #
Let $p : E \to X$ be an [[(∞,1)-functor]] of [[(infinity,1)-category|(∞,1)-categories]]. A _cartesian section_ of $p$ is a [[section]] $\sigma : X \to E$ that sends all 1-morphisms in $X$ to [[Cartesian morphism]]s in $E$.
#Definition#
...
#Remarks#
If $p : E \to X$ is a [[Cartesian fibration]] classifie... |
(infinity,1)-category of chain complexes | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+of+chain+complexes |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
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#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An [[(∞... |
(infinity,1)-category theory | https://ncatlab.org/nlab/source/%28infinity%2C1%29-category+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* auto... |
(infinity,1)-functor | https://ncatlab.org/nlab/source/%28infinity%2C1%29-functor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _$(\infty,1)$-functor_ is a [[homomorphism]] between [[(∞,1)-categori... |
(infinity,1)-geometric morphism | https://ncatlab.org/nlab/source/%28infinity%2C1%29-geometric+morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Idea
By [[categorification]] of the notion of [[geometric morphism]], an _$(\infty,1)... |
(infinity,1)-Grothendieck construction | https://ncatlab.org/nlab/source/%28infinity%2C1%29-Grothendieck+construction |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The **$(\infty,1)$-Grothendieck construction** is the generalization o... |
(infinity,1)-Kan extension | https://ncatlab.org/nlab/source/%28infinity%2C1%29-Kan+extension |
> under construction
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}... |
(infinity,1)-module | https://ncatlab.org/nlab/source/%28infinity%2C1%29-module |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _$(\infty,1)$-modu... |
(infinity,1)-module bundle | https://ncatlab.org/nlab/source/%28infinity%2C1%29-module+bundle | [[!redirects (infinity,1)-vector bundle]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Stable Homotopy theory
+--{: .hide}
[[!include stabl... |
(infinity,1)-monad | https://ncatlab.org/nlab/source/%28infinity%2C1%29-monad |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}
## Idea
The notion of $(\infty,1)$-monad is the [[vertical categorification]] of that of [[monad]... |
(infinity,1)-operad | https://ncatlab.org/nlab/source/%28infinity%2C1%29-operad |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
... |
(infinity,1)-presheaf | https://ncatlab.org/nlab/source/%28infinity%2C1%29-presheaf | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents... |
(infinity,1)-pretopos | https://ncatlab.org/nlab/source/%28infinity%2C1%29-pretopos | [[!redirects (∞,1)-pretopos]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The concept of _$(\infty,1)$-pretopos_ ([Lurie, ... |
(infinity,1)-profunctor | https://ncatlab.org/nlab/source/%28infinity%2C1%29-profunctor |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of content
{:toc}
## Idea
The concept of _$(\infty, 1)$-profunctor_ is the [[categorification]] of th... |
(infinity,1)-pullback | https://ncatlab.org/nlab/source/%28infinity%2C1%29-pullback |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An **$(\infty,1)$-pullback** is a [[limit in an (∞,1)-category]] $\math... |
(infinity,1)-pushout | https://ncatlab.org/nlab/source/%28infinity%2C1%29-pushout |
The dual notion to _[[(infinity,1)-pullback]]_.
[[!redirects (infinity,1)-pushouts]]
|
(infinity,1)-quantity | https://ncatlab.org/nlab/source/%28infinity%2C1%29-quantity | See [[schreiber:∞-quantity]] for now.
[[!redirects (∞,1)-quantity]] |
(infinity,1)-quasitopos | https://ncatlab.org/nlab/source/%28infinity%2C1%29-quasitopos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _$(\infty,1)$-quasitopos is the [[(∞,1)-topos]]-analog of the notion of [[quasitopo... |
(infinity,1)-semitopos | https://ncatlab.org/nlab/source/%28infinity%2C1%29-semitopos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Defini... |
(infinity,1)-sheaf | https://ncatlab.org/nlab/source/%28infinity%2C1%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### Locality and descent
+--{: .hide}
[[!include descent and locality - contents]]
=--
=--
=--
#Contents#
* table of contents
{:t... |
(infinity,1)-sheafification | https://ncatlab.org/nlab/source/%28infinity%2C1%29-sheafification |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
As for ordinary [[sheafification]], _$(\infty,1)$-sheafification_ is the [[exact (∞,1)-functor|le... |
(infinity,1)-site | https://ncatlab.org/nlab/source/%28infinity%2C1%29-site | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The structure of an $(\infty,1)$-site on an [[(∞,1)-category]] $C$ is precisely th... |
(infinity,1)-topos | https://ncatlab.org/nlab/source/%28infinity%2C1%29-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $(\infty,1)$-Category theory
+-- {: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+-- {: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
# Contents
* table of contents
... |
(infinity,1)-topos - contents | https://ncatlab.org/nlab/source/%28infinity%2C1%29-topos+-+contents |
**[[(∞,1)-topos theory]]**
## Background {#sidebar_background}
* [[sheaf and topos theory]]
* [[(∞,1)-category]]
* [[(∞,1)-functor]]
* [[(∞,1)-presheaf]]
* [[(∞,1)-category of (∞,1)-presheaves]]
## Definitions {#sidebar_definitions}
* [[elementary (∞,1)-topos]]
* [[(∞,1)-site]]
* [[refle... |
(infinity,1)-topos theory | https://ncatlab.org/nlab/source/%28infinity%2C1%29-topos+theory | [[!redirects (∞,1)-topos theory]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
... |
(infinity,1)-Yoneda extension | https://ncatlab.org/nlab/source/%28infinity%2C1%29-Yoneda+extension |
#Contents#
* automatic table of contents goes here
{:toc}
## Idea ##
In ordinary [[category theory]] the [[Yoneda extension]] of a [[functor]] $F : C \to D$ is its left [[Kan extension]] through the [[Yoneda embedding]] of its domain to a functor $\hat F : PSh(C) \to D$.
In [[higher category theory]] there should ... |
(infinity,1)Cat | https://ncatlab.org/nlab/source/%28infinity%2C1%29Cat |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### [[categories of categories - contents|categories of categories]]
+-- {: .hide}
[[!include categories of categories - contents]]
=--
=... |
(infinity,1)Prof | https://ncatlab.org/nlab/source/%28infinity%2C1%29Prof | [[!redirects (infinity, 1)Prof]]
#Contents#
* table of contents
{:toc}
## Definition
$\mathbf{(\infty, 1)Prof}$ is the [[(∞,2)-category]] of [[(∞,1)-categories]], [[(\infty, 1)-profunctor|(∞,1)-profunctors]], and [[natural transformations]].
Recall that a (∞,1)-profunctor from $A$ to $B$ is a (∞,1)-func... |
(infinity,1)Topos | https://ncatlab.org/nlab/source/%28infinity%2C1%29Topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
By **$(\infty,1)Topos$** is denoted the collection of all [[(∞,1)-topos]]e... |
(infinity,2)-Categories and the Goodwillie Calculus | https://ncatlab.org/nlab/source/%28infinity%2C2%29-Categories+and+the+Goodwillie+Calculus | [[!redirects (∞,2)-Categories and the Goodwillie Calculus]]
this entry is about the article
* [[Jacob Lurie]], _$(\infty,2)$-categories and the Goodwillie calculus_ ([pdf](http://www.math.harvard.edu/~lurie/papers/GoodwillieI.pdf), [arXiv:0905.0462](http://arxiv.org/abs/0905.0462))
on
1. [[(∞,1)-category|(∞,1)-ca... |
(infinity,2)-category | https://ncatlab.org/nlab/source/%28infinity%2C2%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An __$(\infty,2)$-category__ is the special case of $(\infty,n)$-[[(infinit... |
(infinity,2)-sheaf | https://ncatlab.org/nlab/source/%28infinity%2C2%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An _$(\inft... |
(infinity,2)-site | https://ncatlab.org/nlab/source/%28infinity%2C2%29-site | [[!redirects (∞,2)-site]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,2)$-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--
An [[(n,r)-site]] ... |
(infinity,2)-topos | https://ncatlab.org/nlab/source/%28infinity%2C2%29-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
#... |
(infinity,2)-topos theory - contents | https://ncatlab.org/nlab/source/%28infinity%2C2%29-topos+theory+-+contents |
**[[(∞,2)-topos]]**
* [[(∞,2)-presheaf]], [[(∞,2)-site]],
**[[(∞,2)-sheaf]]**
* [[codomain fibration]], [[tangent (∞,1)-category]], [[quasicoherent ∞-stack]]
## Truncations
**[[2-topos]]**
* [[2-pretopos]]
* [[2-presheaf]], [[2-site]], [[2-sheaf]], [[stack]]
* [[n-localic 2-topos]]
* [[2-Giraud theorem]]
... |
(infinity,n)-category | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea ##
In [[higher category theory]] an _$(\infty,n)$-category_ may be thought ... |
(infinity,n)-category of cobordisms | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category+of+cobordisms |
> under construction
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Manifolds and Cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Cobordism theory... |
(infinity,n)-category of correspondences | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category+of+correspondences | [[!redirects (infinity,n)-category of spans]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The generalization of the [[bi... |
(infinity,n)-category with adjoints | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category+with+adjoints |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An [[(∞,n)-category]] $\mathcal{C}$ is said to _have 1-adjoints_ if in its [[homotopy 2-category... |
(infinity,n)-category with duals | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-category+with+duals | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
#### Monoidal categories
+-- {: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
##... |
(infinity,n)-functor | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-functor |
#Contents#
* table of contents
{:toc}
## Idea
... a morphism between [[(∞,n)-categories]] ...
## Related concepts
* [[function]]
* [[functor]]
* [[2-functor]] / [[pseudofunctor]]
* [[n-functor]]
* [[(∞,1)-functor]]
* **(∞,n)-functor**
* [[monoidal (∞,n)-functor]]
[[!redirects (∞,n)-functor]]
[[!redirec... |
(infinity,n)-module | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-module |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##... |
(infinity,n)-natural transformation | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-natural+transformation | # $(\infty,n)$-natural transformation
## Idea
The analogue of a [[natural transformation]] for [[(∞,n)-categories]]. A special case of a [[transfor]].
## References
* [[Theo Johnson-Freyd]] and [[Claudia Scheimbauer]], *(Op)lax natural transformations, relative field theories, and the "even higher" Morita category... |
(infinity,n)-presheaf | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-presheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
An **$(\infty,n)$-presheaf** on an [[(∞,n)-category]] $C$ is an [[(∞,n)-functor... |
(infinity,n)-sheaf | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
#... |
(infinity,n)-topos | https://ncatlab.org/nlab/source/%28infinity%2Cn%29-topos | [[!redirects (∞,n)-topos]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Higher topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* ta... |
(infinity,n)Cat | https://ncatlab.org/nlab/source/%28infinity%2Cn%29Cat | [[!redirects (∞,n)Cat]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### [[categories of categories - contents|categories of categories]]
+-- {: .hide}
[[!include categories of categorie... |
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