title stringlengths 1 131 | url stringlengths 33 167 | content stringlengths 0 637k |
|---|---|---|
(infinity,r)-category > history | https://ncatlab.org/nlab/source/%28infinity%2Cr%29-category+%3E+history | < [[(infinity,r)-category]]
[[!redirects (infinity,r)-category -- history]] |
(n Γ k)-category | https://ncatlab.org/nlab/source/%28n+%C3%97+k%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Internal categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
=--
=--
#Contents#
* table of contents
{... |
(n+1,1)-category of n-truncated objects | https://ncatlab.org/nlab/source/%28n%2B1%2C1%29-category+of+n-truncated+objects | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### 2-ategory theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
... |
(n,0)-category | https://ncatlab.org/nlab/source/%28n%2C0%29-category |
An _$(n,0)$-category_ is an [[(n,r)-category]] that is an [[n-groupoid]].
By the general rules of $(n,r)$-[[(n,r)-category|categories]], an __$(n,0)$-category__ is an $\infty$-[[infinity-category|category]] such that
* any $j$-morphism is an [[equivalence]], for $j \gt 0$;
* any two parallel $j$-morphisms are equi... |
(n,1)-category | https://ncatlab.org/nlab/source/%28n%2C1%29-category |
+-- {: .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#
* table of contents
{:toc}
## Idea
T... |
(n,1)-sheaf | https://ncatlab.org/nlab/source/%28n%2C1%29-sheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
For $n \in \mathbb{Z}$, $-2 \leq n \lt \infty$,
an _$(n,1)$-sheaf_ is a... |
(n,1)-topos | https://ncatlab.org/nlab/source/%28n%2C1%29-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A (Grothendieck) $(n,1)$-topos is the [[(n,r)-category|(n,1)-category]] version ... |
(n,n)-category | https://ncatlab.org/nlab/source/%28n%2Cn%29-category | Following the pattern of the notion of [[(n,r)-category]], an _$(n,n)$-category_ is a [[higher category theory|higher category]] with non-trivial cells of at most dimension $n$ and none of them guaranteed to be reversible.
So this is what is usually simply called an [[n-category]].
Note that it is possible to go on ... |
(n,r)-category | https://ncatlab.org/nlab/source/%28n%2Cr%29-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
An $(n,r)$-category is a [[higher category theory|higher category]] such that, ... |
(n,r)-congruence | https://ncatlab.org/nlab/source/%28n%2Cr%29-congruence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Internal $(\infty,1)$-Categories
+--{: .hide}
[[!include internal i... |
(n,r)-site | https://ncatlab.org/nlab/source/%28n%2Cr%29-site |
An [[(n,r)-site]] is an [[(n,r)-category]] equipped with a notion of [[coverage]].
## Examples
* [[(0,1)-site]]
* [[(1,1)-site]]
* [[(2,1)-site]], [[(2,2)-site]]
* [[(β,1)-site]]
[[!redirects (n,r)-site]]
|
(n-connected, n-truncated) factorization system | https://ncatlab.org/nlab/source/%28n-connected%2C+n-truncated%29+factorization+system |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!in... |
(p,q)-string | https://ncatlab.org/nlab/source/%28p%2Cq%29-string |
> under construction
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Quantu... |
(p,q)5-brane | https://ncatlab.org/nlab/source/%28p%2Cq%295-brane |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Quantum field theory
+--{: .h... |
(sub)object classifier in an (infinity,1)-topos | https://ncatlab.org/nlab/source/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos |
> This page is about object classifier objects in [[(β,1)-toposes]]. For the unrelated notion of the [[classifying topos]] of the [[theory of objects]] see at _[[classifying topos for the theory of objects]]_.
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Universes
+-- {: .hide... |
(β%2C1)-category > history | https://ncatlab.org/nlab/source/%28%E2%88%9E%252C1%29-category+%3E+history | < [[(?%2C1)-category]]
[[!redirects (β,1)-category/history]]
[[!redirects (?%2C1)-category -- history]] |
(β,0)-category > history | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C0%29-category+%3E+history | < [[(β,0)-category]]
[[!redirects (β,0)-category/history]]
[[!redirects (β,0)-category -- history]] |
(β,1)-algebra over an (β,1)-operad > history | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-algebra+over+an+%28%E2%88%9E%2C1%29-operad+%3E+history |
> [[infinity-algebra over an (infinity,1)-operad]] |
(β,1)-algebraic theory | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-algebraic+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--
#Contents#
* automatic table of contents ... |
(β,1)-category of (β,1)-categories > history | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-categories+%3E+history | < [[(β,1)-category of (β,1)-categories]] |
(β,1)-category of (β,1)-modules | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-modules |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
For $R \in CRing_\infty$ an [[E-β ring]], the [[(β,1)-modules]] over $R$ with [[homomorph... |
(β,1)-comparison lemma | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-comparison+lemma |
+-- {: .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]]
=--
=--
=--
# The $(\infty,1)$-comparison lemma... |
(β,1)-end | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-end |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{: toc}
## Idea
This is the generalization of the notion of *[[ends]]* from [[category theory]] to [[(β,1)-ca... |
(β,1)-limit | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-limit |
+-- {: .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}
... |
(β,1)-local geometric morphism | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-local+geometric+morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The analog in the context of [[(β,1)-topos]] theory of a [[local geometric morphism]] in [[to... |
(β,1)-logic | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29-logic |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Type theory
+--{: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+--{: .hide}
[[!include foundations - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
As a [[category]] has [[internal logic]], so a [... |
(β,1)Operad | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C1%29Operad | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### [[categories of categories - contents|categories of categories]]
+-- {: .hide}
[[!include categories of categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[(β,2)-category]] of [[(β,1)-operads]]. The [[oper... |
(β,2)-category of cobordisms | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C2%29-category+of+cobordisms |
## Idea
An [[(β,n)-category of cobordisms]] for $n = 2$.
## Related concepts
* [[2d TQFT]], [[TCFT]]
* [[Calabi-Yau object]]
[[!redirects (infinity,2)-category of cobordisms]] |
(β,2)-category theory | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C2%29-category+theory |
## Idea
The theory of [[(β,2)-categories]]...
## Related concepts
[[!include table of category theories]] |
(β,2)-functor | https://ncatlab.org/nlab/source/%28%E2%88%9E%2C2%29-functor |
## Idea
[[(infinity,n)-functor]] for $n = 2$
## Related entries
* [[2d TQFT]], [[TCFT]]
[[!redirects (β,2)-functors]]
[[!redirects (infinity,2)-functor]]
[[!redirects (infinity,2)-functors]]
|
(β,n)-category theory | https://ncatlab.org/nlab/source/%28%E2%88%9E%2Cn%29-category+theory |
## Idea
The theory of [[(β,n)-categories]]
## Related concepts
[[!include table of category theories]]
|
0-bundle | https://ncatlab.org/nlab/source/0-bundle |
## Idea
In the context of [[principal infinity-bundles]] a 0-bundle is equivalently a [[function]].
## Examples
* [[prequantum 0-bundle]].
[[!redirects 0-bundles]]
|
0-category | https://ncatlab.org/nlab/source/0-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context ###
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Definition
A __$0$-category__ (or __$(0,0)$-category__) is, up to [[equivalence ... |
0-dimensional TQFT | https://ncatlab.org/nlab/source/0-dimensional+TQFT | #Contents#
* automatic table of contents goes here
{:toc}
## Idea
A _0-dimensional TQFT_ is a [[TQFT]] regarded in the sense of [[FQFT]] as a [[representation]] of the [[category]] of 0-dimensional [[cobordism]]s.
This degenerate case turns out to exhibit a nontrivial amount of interesting information, in particula... |
0-functor | https://ncatlab.org/nlab/source/0-functor |
As a $0$-[[0-category|category]] is simply a [[set]], so a __$0$-functor__ is simply a [[function]]. See also $n$-[[n-functor|functor]].
[[!redirects 0-functor]]
[[!redirects 0-functors]]
|
0-groupoid | https://ncatlab.org/nlab/source/0-groupoid |
#Contents#
* table of contents
{:toc}
## Idea
A _0-groupoid_ or **[[homotopy n-type|0-type]]** is a [[set]]. This terminology may seem strange at first, but it is very helpful to see sets as the beginning of a sequence of concepts: sets, [[groupoid|groupoids]], [[2-groupoid|2-groupoids]], [[3-groupoid|3-groupoids]],... |
0-morphism | https://ncatlab.org/nlab/source/0-morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
Following the patterns of [[k-morphism]]s in a [[category]] or [[higher category]] it make... |
0-object > history | https://ncatlab.org/nlab/source/0-object+%3E+history | < [[0-object]]
[[!redirects 0-object -- history]] |
0-poset | https://ncatlab.org/nlab/source/0-poset | A __0-poset__ is a [[truth value]]. Compare the concept of $1$[[1-poset|poset]] (a [[partial order|poset]]) and $(-1)$-[[(-1)-poset|poset]] (which is trivial); compare also with $(-1)$-[[(-1)-category|category]] and $0$-[[0-groupoid|groupoid]], which mean the same thing for different reasons.
The point of 0-posets is ... |
0-sphere | https://ncatlab.org/nlab/source/0-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _0-sphere_ $S^0$ is the [[n-sphere]] for $n = 0$: The [[disjoint union]] of two [[points]],
$$
S^0... |
1-category | https://ncatlab.org/nlab/source/1-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea... |
1-category equipped with relations | https://ncatlab.org/nlab/source/1-category+equipped+with+relations | #Contents#
* automatic table of contents goes here
{:toc}
## Idea ##
A [[2-category equipped with proarrows]] is a 2-category together with a 2-category of "proarrows" which are intended to generalize the arrows of $K$ in the same way that [[profunctors]] generalize the [[functors]] in [[Cat]]. Since profunc... |
1-dimensional Chern-Simons theory | https://ncatlab.org/nlab/source/1-dimensional+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 physic... |
1-functor | https://ncatlab.org/nlab/source/1-functor |
As a $1$-[[1-category|category]] is simply a [[category]], so a __$1$-functor__ is simply a [[functor]]. See also $n$-[[n-functor|functor]].
[[!redirects 1-functor]]
[[!redirects 1-functors]]
|
1-groupoid | https://ncatlab.org/nlab/source/1-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
By _1-groupoid_ one means -- for empha... |
1-morphism | https://ncatlab.org/nlab/source/1-morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
* [[object]]
* [[morphism]] / **1-morphism**
[[horizontal morphism]], [[vertical morphism]]
* [[2-morphism]]
* [[3-mo... |
1-poset | https://ncatlab.org/nlab/source/1-poset | Fix a meaning of $\infty$-[[infinity-category|category]], however weak or strict you wish. Then a __$1$-poset__ is an $\infty$-category such that every $2$-morphism is an [[equivalence]] and all [[parallel pair]]s of $j$-morphisms are equivalent for $j \geq 1$. Thus, up to equivalence, there is no point in mentioning a... |
1-topos | https://ncatlab.org/nlab/source/1-topos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A __$1$-topos__, or __$(1,1)$-topos__, is simply a [[topos]] in the usual sense of the word. The pr... |
10d supergravity | https://ncatlab.org/nlab/source/10d+supergravity |
[[supergravity]] in [[dimension]] 10.
* [[type II supergravity]], [[double field theory]]
* [[heterotic supergravity]] |
10j symbol | https://ncatlab.org/nlab/source/10j+symbol |
In the context of fixed [[triangulations]] of spacetime arising in [[spin foam models]], the [[10j symbol]] is a spin network that arises in the partition function for the [[Barrett-Crane model]] of [[Riemannian quantum gravity]]. [[Greg Egan]], [[J. Daniel Christensen]], and [[John Baez]] formulated an algorithm in ... |
11-dimensional Chern-Simons theory | https://ncatlab.org/nlab/source/11-dimensional+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]]
=--
#### String theory
+-- {: .hide... |
11d supergravity Lie 3-algebra | https://ncatlab.org/nlab/source/11d+supergravity+Lie+3-algebra | [[!redirects supergravity Lie 3-algebra]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--
#Contents#
* t... |
120-cell | https://ncatlab.org/nlab/source/120-cell |
#Contents#
* table of contents
{:toc}
## Idea
...one of the [[regular polytopes]] in [[dimension]] 4...
...hence a higher dimensional analogs of the [[Platonic solids]]...
\begin{centre}
\begin{imagefromfile}
"file_name": "Chilton120Cell.jpg",
"width": 360
\end{imagefromfile}
\end{cen... |
13-sphere | https://ncatlab.org/nlab/source/13-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[n-sphere]] of [[dimension]] $n = 13$.
## Properties
### AKM-Theorem
In higher-dimensional an... |
15-sphere | https://ncatlab.org/nlab/source/15-sphere |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The [[sphere]] of [[dimension]] 15.
## Properties
### Octonionic Hopf fibration
{#Octo... |
1d Dijkgraaf-Witten theory | https://ncatlab.org/nlab/source/1d+Dijkgraaf-Witten+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 physi... |
1d WZW model | https://ncatlab.org/nlab/source/1d+WZW+model |
+-- {: .rightHandSide}
+-- {: .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... |
1lab | https://ncatlab.org/nlab/source/1lab | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--
\tableofcontents
## Idea
The **1lab** is an online reference maintain... |
1T relation | https://ncatlab.org/nlab/source/1T+relation |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Knot theory
+--{: .hide}
[[!include knot theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
In [[knot theory]], by the _1-term relations_, or _1T relations_ for short, one means the [[relation]... |
2-adjunction | https://ncatlab.org/nlab/source/2-adjunction | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **2-adjunction** is a common name for various kinds of adjunctions in [[2-category theory... |
2-algebraic geometry | https://ncatlab.org/nlab/source/2-algebraic+geometry |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
W... |
2-categorical limit > history | https://ncatlab.org/nlab/source/2-categorical+limit+%3E+history | < [[2-categorical limit]]
[[!redirects 2-categorical limit -- history]] |
2-category | https://ncatlab.org/nlab/source/2-category |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
The notion of a _2-category_ generalizes that of [[category]]: a 2-category is a [[... |
2-category equipped with proarrows | https://ncatlab.org/nlab/source/2-category+equipped+with+proarrows |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Contents
* automatic table of contents goes here
{: toc}
## Idea
A **2-category equipped with proarrows**, also called a *proarrow equ... |
2-category of 2-dimensional cobordisms | https://ncatlab.org/nlab/source/2-category+of+2-dimensional+cobordisms | This is a special case of the $(\infty,n)$-[[(infinity,n)-category of cobordisms|category of cobordisms]].
##References#
A complete [[generators and relations]] presentation of the 2-dimensional extended unoriented oriented bordism [[bicategory]] as a [[symmetric monoidal bicategory]] is given in
* Chris Schommer-P... |
2-category of adjunctions | https://ncatlab.org/nlab/source/2-category+of+adjunctions |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
##Idea
For any [[2-category]] $K$, there are two 2-categories (each with several variants) th... |
2-category theory | https://ncatlab.org/nlab/source/2-category+theory |
+-- {: .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}
... |
2-category theory - contents | https://ncatlab.org/nlab/source/2-category+theory+-+contents | **[[2-category theory]]**
**Definitions**
* [[2-category]]
* [[strict 2-category]]
* [[bicategory]]
* [[enriched bicategory]]
**Transfors between 2-categories**
* [[2-functor]]
* [[pseudofunctor]]
* [[lax functor]]
* [[equivalence of 2-categories]]
* [[2-natural transformation]]... |
2-category with contravariance | https://ncatlab.org/nlab/source/2-category+with+contravariance | # 2-categories with contravariance
* table of contents
{:toc}
## Idea
A *2-category with contravariance* is a [[2-category]]-like structure that contains a basic notion of [[contravariant functor|contravariant morphism]]. There are multiple ways to make this precise, depending on what sort of natural transformation... |
2-congruence | https://ncatlab.org/nlab/source/2-congruence |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Internal $(\infty,1)$-Categories
+--{: .hide}
[[!include internal infinity-categories conte... |
2-crossed complex | https://ncatlab.org/nlab/source/2-crossed+complex |
# Contents
* automatic table of contents goes here
{:toc}
##Idea#
[[crossed complex|Crossed complexes]] are a useful extension of [[crossed module]]s allowing not only the encoding of an algebraic model for the [[homotopy 2-type]], but also information on the '[[complex of chains on the universal cover]]'. The... |
2-crossed module | https://ncatlab.org/nlab/source/2-crossed+module |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}... |
2-dinatural transformation | https://ncatlab.org/nlab/source/2-dinatural+transformation | # 2-dinatural transformations
* table of contents
{: toc}
## Idea
A *2-dinatural transformation* is a [[categorification]] of a [[dinatural transformation]]. It is more complicated since there are three kinds of [[opposite 2-category]], and also because it could be [[lax natural transformation|lax]], colax, or pseu... |
2-functor | https://ncatlab.org/nlab/source/2-functor | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A __$2$-functor__ is the [[categorification]] of the notion of a [[functor]] to the setti... |
2-gerbe | https://ncatlab.org/nlab/source/2-gerbe |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
### Nonabelian 2-gerbes
... |
2-Giraud theorem | https://ncatlab.org/nlab/source/2-Giraud+theorem |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The _2-... |
2-Grothendieck construction | https://ncatlab.org/nlab/source/2-Grothendieck+construction |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
Generalization of the [[Grothendieck construction]] from [[categories]] to [[2-categori... |
2-group | https://ncatlab.org/nlab/source/2-group |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## I... |
2-groupoid | https://ncatlab.org/nlab/source/2-groupoid |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{:toc}
## Idea
A **$2$-groupoid** is
* a [[(n,r)-category|(2,0)-category]]
* a [[2-ca... |
2-groupoid of Lie 2-algebra valued forms | https://ncatlab.org/nlab/source/2-groupoid+of+Lie+2-algebra+valued+forms |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--
#Contents#
* automatic table of contents goes her... |
2-Hilbert space | https://ncatlab.org/nlab/source/2-Hilbert+space |
#Contents#
* table of contents
{:toc}
## Idea
The concept of $2$-Hilbert space is supposed to be a [[categorification]] of that of [[Hilbert space]], or at least of [[finite-dimensional vector space|finite dimensional]] such: [[inner product spaces]].
One way to define this is as a [[Kapranov-Voevodsky 2-... |
2-isomorphism | https://ncatlab.org/nlab/source/2-isomorphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
\tableofcontents
\section{Idea}
In a [[higher category theory|higher category]], [[inverse|invertibility]] of [[n-morphisms]] in... |
2-Lawvere theory | https://ncatlab.org/nlab/source/2-Lawvere+theory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The notion of _2-Lawvere theory_ or _Lawvere 2-theory_ is the [[categorification]] of [[Law... |
2-limit | https://ncatlab.org/nlab/source/2-limit |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### $\infty$-Limits
+-- {: .hide}
[[!include infinity-limits - contents]]
=--
----
#### 2-Category theory
+-- {: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
# Contents
* table of contents
{: toc}
## Idea
... |
2-limits and 2-colimits in 2-categories of 2-algebras | https://ncatlab.org/nlab/source/2-limits+and+2-colimits+in+2-categories+of+2-algebras | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* tic
{: toc}
## Introduction
In moving from o... |
2-line | https://ncatlab.org/nlab/source/2-line |
#Contents#
* table of contents
{:toc}
## Idea
Given a [[2-ring]] $R$, a _2-line_ over $R$ is an $R$-[[2-module]] which is [[equivalence|equivalent]] to $R$ as a 2-module.
## Related concepts
* [[line]]
* [[2-line bundle]]
[[!redirects 2-lines]] |
2-logic | https://ncatlab.org/nlab/source/2-logic |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Type theory
+--{: .hide}
[[!include type theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
As a [[category]] has an [[internal logic]] so a [[2-category]] has internal _2-logic_
## Related concepts
* [[type ... |
2-monad | https://ncatlab.org/nlab/source/2-monad |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Definition
... |
2-morphism | https://ncatlab.org/nlab/source/2-morphism |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--
* [[object]]
* [[morphism]]
* **2-morphism**
* [[3-morphism]]
* [[k-morphism]]
***
#Contents#
* automatic table of content... |
2-morphism - SVG | https://ncatlab.org/nlab/source/2-morphism+-+SVG | <svg width="100" height="75" xmlns="http://www.w3.org/2000/svg" xmlns:se="http://svg-edit.googlecode.com" se:nonce="47584">
<g>
<title>Layer 1</title>
<path marker-end="url(#se_marker_end_svg_47584_1)" id="svg_47584_1" d="m20.4,29c12.400003,-21.8 45.6,-22.799997 57.6,-0.6" fill-opacity="0" stroke-width="2" stroke=... |
2-morphisms > history | https://ncatlab.org/nlab/source/2-morphisms+%3E+history | < [[2-morphisms]]
[[!redirects 2-morphisms -- history]] |
2-periodic Morava K-theory | https://ncatlab.org/nlab/source/2-periodic+Morava+K-theory |
## Idea
The 2-[[periodic ring spectrum]]-version of [[Morava K-theory]].
Let $C$ be any [[elliptic curve]] over the [[prime field]] $\mathbb{F}_p$. This will have a [[formal group]] of either [[height of a formal group law|height]] $h=1$ ("ordinary curve") or height $h=2$ ("[[supersingular elliptic curve]]"). For ... |
2-periodic sphere spectrum | https://ncatlab.org/nlab/source/2-periodic+sphere+spectrum |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
The 2-[[periodic ring spectrum]]-version of the [[sphere spectrum]]. It is formed by taking the [[Tho... |
2-plectic geometry | https://ncatlab.org/nlab/source/2-plectic+geometry |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
_2-Plectic geometry_ is the higher generalization of [[symplectic geometry]], the special case... |
2-plethory | https://ncatlab.org/nlab/source/2-plethory |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Combinatorics
+-- {: .hide}
[[!include combinatorics - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
## Idea
A **2-plethory** ([... |
2-polycategory | https://ncatlab.org/nlab/source/2-polycategory | A **2-polycategory** is a [[polycategory]] enriched over [[Cat]]. It can be identified with a [[double polycategory]] with only identity vertical arrows. |
2-poset | https://ncatlab.org/nlab/source/2-poset |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
# $2$-posets
* table of contents
{: toc}
A **2-poset** is any of several concepts that generalize ([[categorify]]) the not... |
2-poset of partial maps | https://ncatlab.org/nlab/source/2-poset+of+partial+maps | +-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--
## Contents ##
* table of contents
{:toc}
## Definition ##
Given a [[dagger 2-poset]] $A$, the **2-poset of partial maps** $Map... |
2-presheaf | https://ncatlab.org/nlab/source/2-presheaf |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--
## Idea
The generalization of the noti... |
2-pretopos | https://ncatlab.org/nlab/source/2-pretopos |
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--
#Contents#
* table of contents
{:toc}
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.