n-connected continuous function

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### Context
#### Homotopy theory
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[[!include homotopy - contents]]
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# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

In [[algebraic topology]], a [[continuous function]] between [[topological spaces]] is called **$n$-connected** if it induces [[isomorphisms]] on all [[homotopy groups]] in degree $\lt n$ and [[epimorphisms]] in degree $n$.

In older literature this is often called an **$n$-equivalence**, since an $\infty$-equivalence in this sense is a  _[[weak homotopy equivalence]]_.

In terms of the [[homotopy theory]] presented by the [[classical model structure on topological spaces]], an $n$-connected function represents an [[n-connected morphism]] in the [[(∞,1)-topos]] [[∞Gpd]].

## Definition

+-- {: .num_defn }
###### Definition

A map of [[topological space|topological spaces]] $f \colon X \to Y$ is called __$n$-connected__ (e.g. [tomDieck 08, p. 144](#tomDieck08)) or an __$n$-equivalence__ (older literature) if the following equivalent definitions hold:

1. The induced morphism on [[homotopy groups]] $\pi_\bullet(X,x)\to \pi_\bullet(Y,f(x))$ is, for all $x\in X$

   1. an [[isomorphism]] in degree $\lt n$;

   1. an [[epimorphism]] in degree $n$.

1. for all $i \le n$ and all commutative squares

  $$
    \begin{matrix}
      S^{i-1} & \overset{u}{\longrightarrow} & X
      \\
      \big\downarrow & & \, \big\downarrow^\mathrlap{f}
      \\
      D^i & \underset{v}{\longrightarrow} & Y
    \end{matrix}
  $$

  there exists a map $w \colon D^i \to X$ such that $w | S^{i-1} = u$ and $f w$ is homotopic to $v$ relative to $S^{i-1}$.

=--

Hence an $\infty$-connected map is a [[weak homotopy equivalence]].

## Properties

+-- {: .num_prop }
###### Proposition

For a map $f \colon X \to Y$ and an integer $n \ge -1$ the following conditions are equivalent.

1. $f$ is $n$-connected.

1. All [[homotopy fibers]] of $f$ are $(n-1)$-[[n-connected space|connected]].

=--

+-- {: .num_prop }
###### Proposition

Let $f \colon X \to Y$ and $g \colon Y \to Z$ be maps of spaces.

1. If $f$ and $g$ are $n$-connected, then so is $g f$.

1. If $f$ is $(n-1)$-connected and $g f$ is $n$-connected, then $g$ is $n$-connected.

1. If $g$ is $(n+1)$-connected and $g f$ is $n$-connected, then $f$ is $n$-connected.

=--

+-- {: .num_prop }
###### Proposition

Let
$$
\begin{matrix}
  B & \longleftarrow & A & \longrightarrow & C \\
  \mathllap{{}^g}\big\downarrow & & \big\downarrow^\mathrlap{f} & & \, \big\downarrow^\mathrlap{h} \\
  B' & \longleftarrow & A' & \longrightarrow & C'
\end{matrix}
$$
be a commutative diagram of maps of spaces. If $f$ is $(n-1)$-connected and $g$ and $h$ are $n$-connected, then the induced map between [[homotopy pushout|homotopy pushouts]] $B \sqcup_A^h C \to B' \sqcup_{A'}^h C'$ is $n$-connected.

=--

This is ([tom Dieck, Theorem 6.7.9](#tomDieck)).

+-- {: .num_prop }
###### Proposition

Let
$$
\begin{matrix}
  Y & \longrightarrow & X & \longleftarrow & Z \\
   \mathllap{{}^g} \big\downarrow & & \big\downarrow^\mathrlap{f} & & \, \big\downarrow^\mathrlap{h} \\
  Y' & \longrightarrow & X' & \longleftarrow & Z'
\end{matrix}
$$
be a commutative diagram of maps of spaces. If $f$ is $(n+1)$-connected and $g$ and $h$ are $n$-connected, then the induced map between [[homotopy pullback|homotopy pullbacks]] $Y \times_X^h Z \to Y' \times_{X'}^h Z'$ is $n$-connected.

=--

## Related concepts

* [[Postnikov tower]]

* [[n-image]]

## References

Textbook accounts:

* {#tomDieck08}[[Tammo tom Dieck]],  p. 144 and Thm 6.7.9 in: _Algebraic topology_, European Mathematical Society, Zürich (2008) ([doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf))

[[!redirects n-connected continuous functions]]

[[!redirects n-connected map]]
[[!redirects n-connected maps]]

[[!redirects n-simply connected map]]
[[!redirects n-simply connected maps]]

[[!redirects n-simply connected continuous function]]
[[!redirects n-simply connected continuous functions]]

[[!redirects n-equivalence]]
[[!redirects n-equivalences]]


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#### $(\infty,1)$-Topos Theory
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# Contents
* table of contents
{: toc}

## Idea

An **$n$-connected object** is an object all whose [[homotopy group]]s _equal to or below_ degree $n$ are trivial.

More precisely, an object in an [[∞-stack]] [[(∞,1)-topos]] is $n$-connected if its [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]] equal to or below degree $n$ are trivial.

The complementary notion is that of an [[n-truncated object of an (∞,1)-category]].

The [[Whitehead tower]] construction produces $n$-connected objects.

## Definition
 {#Definition}

+-- {: .num_defn #Connectedness}
###### Definition

An [[object]] $X$ in an [[(∞,1)-topos]] $\mathbf{H}$ is called **$n$-connected** for $ -1 \leq n \in \mathbb{Z}$ if

1. the [[terminal object|terminal]] morphism $X \to *$ is an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]];

1. all [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]]
   equal to or below degree $n$ are trivial.

   $$
     \pi_k(X) = * \;\;\; \text{for} \; k \leq n
     \,.
   $$

A [[morphism]] $f : X \to Y$ in an $(\infty,1)$-topos is called **$n$-connected** if

1. it is an [[effective epimorphism in an (∞,1)-category]]

1. regarded as an object in the
   [[over quasi-category|over-(∞,1)-category]]
   $\mathbf{H}_{/Y}$ all [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]]
   equal to or below degree $n$ are trivial.

=--

This appears as [[Higher Topos Theory|HTT, def. 6.5.1.10]], but under the name "[[n-connective object|$(n+1)$-connective]]".  Another possible term is "$n$-simply connected"; see [[n-connected space]] for discussion.

One adopts the following convenient terminology.

* Every object is $(-2)$-connected.

* A $(-1)$-connected object is also called an [[inhabited object]].

* A 0-connected object is simply called a **connected object**.

Notice that [[effective epimorphism in an (∞,1)-category|effective epimorphisms]] are precisely the $(-1)$-connected morphisms. For more on this see _[[n-connected/n-truncated factorization system]]_.

## Properties
 {#Properties}

### General
 {#PropertiesGeneral}

+-- {: .num_prop #ConnectednessViaTruncationComodalness}
###### Proposition

An object $X$ is $n$-connected, def. \ref{Connectedness}, precisely if its [[n-truncated object in an (∞,1)-category|n-truncation]] $\tau_{\leq n} X$ is [[generalized the|the]] [[terminal object]] of $\mathbf{H}$ (hence precisely if it is $\tau_{\leq n}$-[[comodal type|comodal]]).

=--

This is [[Higher Topos Theory|HTT, prop. 6.5.1.12]].

\begin{lemma}
Every [[equivalence in a quasi-category|equivalence]] is $\infty$-connected.
\end{lemma}

This is [[Higher Topos Theory|HTT, prop. 6.5.1.16, item 2]].

+-- {: .num_remark}
###### Remark

In a general $(\infty,1)$-topos the converse is not true: not every $\infty$-connected morphisms needs to be an equivalence. It is true in a [[hypercomplete (∞,1)-topos]].
=--

+-- {: .num_prop}
###### Proposition

The class of $n$-connected morphisms is stable under [[pullback]] and [[pushout]].

If the pullback of a morphism along an [[effective epimorphism]] is $n$-connected, then so is the original morphism.

=--

This is [[Higher Topos Theory|HTT, prop. 6.5.1.16, item 6]].

### Recursive characterization
 {#RecursiveCharacterization}

+-- {: .num_prop #CharacterizationByDiagonal}
###### Proposition

A morphism $f : X \to Y$ is $n$-connected precisely if it is
an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]] and the [[diagonal]] morphism into the [[(∞,1)-pullback]]

$$
  \Delta_f : X \to X \times_Y X
$$

is $(n-1)$-connected.

=--

This appears as [[Higher Topos Theory|HTT, prop. 6.5.1.18]].

### Factorization system

+-- {: .num_prop}
###### Proposition

Let $\mathbf{H}$ be an [[(∞,1)-topos]]. For all $(-2) \leq n \leq \infty$ the [[class]] of $n$-connected morphisms in $\mathbf{H}$ forms the left class in a [[orthogonal factorization system in an (∞,1)-category]]. The right class is that of [[n-truncated]] morphisms in $\mathbf{H}$.

=--

See also [[n-connected/n-truncated factorization system]].

This appears as a remark in [[Higher Topos Theory|HTT, Example 5.2.8.16]]. A construction of the factorization in terms of a [[model category]] [[presentable (∞,1)-category|presentation]] is in ([Rezk, prop. 8.5](#Rezk)).

### Blakers-Massey theorem

* [[Blakers-Massey theorem]]

### The truncated / connected clock
 {#Clock}

In a [[hypercomplete (∞,1)-topos]] the $\infty$-connected morphisms are precisely the [[equivalence in an (∞,1)-category|equivalences]].

Therefore in such a context we have the following "clock" of notions of [[truncated object in an (infinity,1)-category]] / connected :

* any morphism = $(-2)$-connected

* [[effective epimorphism in an (∞,1)-category|effective epimorphism]] = $(-1)$-connected

* 0-connected, 1-connected, 2-connected, $\cdots$;

* $\infty$-connected = [[equivalence in an (∞,1)-category|equivalence]] = $(-2)$-truncated

* [[monomorphism in an (∞,1)-category|monomorphism]] = $(-1)$-truncated

* 0-truncated, 1-truncated, 2-truncated, $\cdots$

* $\infty$-truncated = any morphism

## Examples

### In $Top$
 {#InTop}

In the the [[(∞,1)-category]] $L_{whe}$[[Top]] we have that an object is $n$-connected precisely if it is an [[n-connected topological space]]:

* a $(-1)$-connected object is an [[inhabited set|inhabited space]].

* a $0$-connected object is a [[path-connected space]].

* a $1$-connected object is a [[simply connected space]].

* a $\infty$-connected object is a [[contractible space]].

More generally, a continuous function represents an $n$-connected morphism in $L_{whe} Top$ precisely if it is an [[n-connected continuous function]] ("[[n-equivalence]]").

### In $Grpd$
 {#ExamplesInGrpd}

+-- {: .num_prop}
###### Proposition

Let $f : X \to Y$ be a [[functor]] between [[groupoids]].
Regarded as a morphism in [[∞Grpd]] $f$ is 0-connected precisely
if it is an [[essentially surjective and full functor]].

=--

+-- {: .proof}
###### Proof

As discussed there, an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] in [[∞Grpd]] between 1-groupoids is precisely an
[[essentially surjective functor]].

So it remains to check that for an essentially surjective $f$, being
0-connected is equivalent to being full.

The [[homotopy pullback]] $X \times_Y X$ is given by the groupoid
whose objects are triples $(x_1, x_2 \in X, \alpha : f(x_1) \to f(x_2))$
and whose morphisms are corresponding tuples of morphisms in $X$ making the
evident square in $Y$ commute.

By prop. \ref{CharacterizationByDiagonal}
it is sufficient to check that the [[diagonal]] functor $X \to X \times_Y X$ is
(-1)-connected, hence, as before, essentially surjective, precisely if $f$ is full.

First assume that $f$ is full.
Then for $(x_1,x_2, \alpha) \in X \times_Y X$
any object, by [[full functor|fullness]] of $f$
there is a morphism $\hat \alpha : x_1 \to x_2$ in $X$,
such that $f(\hat \alpha) = \alpha$.

Accordingly we have a morphism
$(\hat \alpha,id) : (x_1, x_2) \to (x_2, x_2)$ in $X \times_Y X$

$$
  \array{
     f(x_1) &\stackrel{f(\hat \alpha)}{\to}& f(x_2)
     \\
     \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}}
     \\
     f(x_2) &\stackrel{id}{\to}& f(x_2)
  }
$$

to an object in the diagonal.

Conversely, assume that the diagonal is essentially surjective. Then
for every pair of objects $x_1, x_2 \in X$ such that there is a morphism
$\alpha : f(x_1) \to f(x_2)$ we are guaranteed morphisms
$h_1 : x_1 \to x_2$ and $h_2 : x_2 \to x_2$ such that

$$
  \array{
    f(x_1) &\stackrel{f(h_1)}{\to}& f(x_2)
    \\
    \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}}
    \\
    f(x_2) &\stackrel{f(h_2)}{\to}& f(x_2)
  }
  \,.
$$

Therefore $h_2^{-1}\circ h_1$ is a preimage of $\alpha$ under $f$,
and hence $f$ is full.

=--

See also [[(eso+full, faithful) factorization system]].

## Related concepts

* [[n-truncated object in an (infinity,1)-category]]

* [[n-connective object]]

* [[Eilenberg subcomplex]]

* [[connective spectrum]], [[connective cover]]

* [[Freudenthal suspension theorem]]

* [[Blakers-Massey theorem]]

## References

Section 6.5.1 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_ .

A discussion in terms of [[model category]] [[presentable (∞,1)-category|presentations]] is in section 8 of

* [[Charles Rezk]], _Toposes and homotopy toposes_ ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf))

[[!redirects (-2)-connected]]
[[!redirects (-1)-connected]]
[[!redirects 0-connected]]
[[!redirects 1-connected]]

[[!redirects 2-connected]]
[[!redirects 3-connected]]
[[!redirects 4-connected]]
[[!redirects 5-connected]]
[[!redirects ∞-connected]]

[[!redirects n-connected]]

[[!redirects n-connected object]]

[[!redirects n-connected objects]]
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[[!redirects n-simply connected objects]]

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[[!redirects n-connected object of an (infinity,1)-category]]
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[[!redirects n-connected objects of (infinity,1)-categories]]

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[[!redirects n-simply connected objects of (infinity,1)-categories]]

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[[!redirects n-connected morphism of an (infinity,1)-category]]
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[[!redirects n-connected morphisms of (infinity,1)-categories]]

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[[!redirects n-connected morphism of an infinity,1-category]]
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[[!redirects n-connected morphisms of infinity,1-categories]]

[[!redirects connected object in an infinity-topos]]
[[!redirects connected object of an infinity-topos]]
[[!redirects n-connected object of an infinity-topos]]
[[!redirects n-connected object of an infinity1-topos]]

[[!redirects connected object in an (infinity,1)-category]]
[[!redirects connected object in an (∞,1)-category]]
[[!redirects connected object in an (infinity,1)-topos]]
[[!redirects connected object in an (∞,1)-topos]]

[[!redirects 0-connected morphism]]
[[!redirects 0-connected morphisms]]

[[!redirects n-connected morphism]]
[[!redirects n-connected morphisms]]

[[!redirects connectivity]]


n-connected space

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#### Homotopy theory
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# $n$-(simply) connected spaces
* table of contents
{: toc}

## Idea

An _$n$-connected space_ is a generalisation of the pattern:

*  [[space]],
*  [[inhabited space]],
*  [[path-connected space]],
*  [[simply connected space]],
*  etc.

For the general concept see at _[[n-connected object of an (infinity,1)-topos]]_.
## Definition

A [[topological space]] $X$ is __$n$-connected__ or __$n$-simply connected__ if its [[homotopy groups]] are trivial up to degree $n$.

More explicitly, $X$ is __precisely $k$-connected__ if every [[continuous map]] to $X$ from the $k$-[[sphere]] extends to a continuous map to $X$ from the $k+1$-[[disk]].  Then $X$ is __$n$-(simply) connected__ if $X$ is precisely $k$-connected for $-1 \leq k \leq n$.

## Special cases

*  Any space is $(-2)$-simply connected.

*  A space is $(-1)$-simply connected precisely if it has an element; that is if it is [[inhabited set|inhabited]].

*  A space is $0$-simply connected precisely if it is [[path-connected space|path-connected]].

*  A space is $1$-simply connected precisely if it is [[simply connected space|simply connected]].

*  A space is $\infty$-simply connected precisely if it is [[contractible space|weakly contractible]].

## Terminology

The traditional terminology is '$n$-connected', but this violates the rule that '$1$-foo' should mean the same as 'foo'.  This can be fixed by saying '$n$-simply connected' instead, which also has the advantage of stressing that we are extending the change from connected to simply connected spaces.

## Properties

An $n$-connected topological space is precisely an [[n-connected object]] in the [[(∞,1)-topos]] [[∞Gpd]], presented by the [[model category]] [[Top]] of [[topological spaces]].

## Related concepts

* [[n-connected continuous function]]

[[!redirects n-connected space]]
[[!redirects n-connected spaces]]
[[!redirects n-simply connected space]]
[[!redirects n-simply connected spaces]]

[[!redirects n-connected topological space]]
[[!redirects n-connected topological spaces]]
[[!redirects n-simply connected topological space]]
[[!redirects n-simply connected topological spaces]]

[[!redirects n-connective space]]
[[!redirects n-connective spaces]]

[[!redirects n-unicoherent space]]
[[!redirects n-unicoherent spaces]]


n-connected type

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### Context
#### Type theory
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#### Homotopy theory
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[[!include homotopy - contents]]
=--
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=--

\tableofcontents

## Idea

In [[dependent type theory]], a type is **$n$-connected** if its [[n-truncation modality|n-truncation]] is a [[contractible type]].

In the [[categorical semantics]] of [[homotopy type theory]] in [[(infinity,1)-toposes|$\infty$-toposes]], $n$-connected types are interpreted as [[n-connected object of an (infinity,1)-topos|$n$-connected objects of the $\infty$-topos]].

## See also

* [[n-truncated type]]

* [[n-connected object]]

## References

* {#UFP13} [[Univalent Foundations Project]], §7.5 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

[[!redirects n-connected types]]


n-connective object

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#### Stable Homotopy theory
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
=--
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=--

#Contents#
* table of contents
{:toc}

## Idea

### In stable $\infty$-categories

#### Definition

In broad generality, given a [[stable (infinity,1)-category|stable $\infty$-category]] $\mathcal{A}$ equipped with a [[t-structure]] $\mathcal{A}_{\geq n}, \mathcal{A}_{\leq n} \hookrightarrow \mathcal{A}$, the objects in the [[full sub-(infinity,1)-category|full sub-$\infty$-category]] $\mathcal{A}_{\geq n}$ are the *$n$-connective objects*, for any $n \in \mathbb{Z}$.

Here typically one understands that a plain "connective" is short for "0-connective."

Accordingly, the [[coreflective subcategory|coreflections]] $\mathcal{A} \to \mathcal{A}_{\geq n}$ are called the *[[connective cover]]*-constructions.

Dually, the objects of $\mathcal{A}_{\leq n}$ may be called the "$n$-*co*connective objects". For non-negative $k \in \mathbb{N}$ the [[intersection]] $\mathcal{A}_{\geq 0} \cap \mathcal{A}_{\leq k}$ of [[sub-(infinity,1)-categories]] of objects which are both connective and $k$-coconnective  are equivalently the [[n-truncated object in an (infinity,1)-category|$k$-truncted]] connective object:

$$
 \tau_{\leq k} \mathcal{A}_{\geq 0}
 \;\simeq\;
 \mathcal{A}_{\geq 0} \cap \mathcal{A}_{\leq k}
  \,.
$$

([Lurie, Warning 1.2.1.9](#LurieHA))

#### Examples

* For the standard [[t-structure]] on an [[(infinity,1)-category of chain complexes|$\infty$-category of chain complexes]], the $n$-connective objects are the $n$-[[connective chain complexes]], namely those which are concentrated in degrees $\geq n$.

* For the standard [[t-structure]] on an [[(infinity,1)-category of spectra|$\infty$-category of spectra]], the $n$-connective objects are the $n$-[[connective spectra]], namely those whose [[stable homotopy groups]] are concentrated in degree $\geq n$.

### In $\infty$-toposes

In an [[(infinity,1)-topos|$\infty$-topos]] one would ---  following traditional in [[algebraic topology]] --- instead speak of the *[[n-connected object in an (infinity,1)-topos|$k$-connected]]* for $k \in \mathbb{N}$. If one insists on saying "connective" also in this case (as is the convention in [[Jacob Lurie|Lurie]]'s *[[Higher Topos Theory]]*) then there is a shift in degree: $n$-connected corresponds to $n+1$ connective. (See [there](n-connected+object+of+an+infinity1-topos#Connectedness) for more.)

## Related concepts

* [[t-structure]]

* [[connective chain complex]]

* [[connective spectrum]]

* [[connective spectrum object]]

## References

For general discussion in the context of [[stable (infinity,1)-categories|stable $\infty$-categories]] see the references at *[[t-structure]]*, such as

* {#LurieHA} [[Jacob Lurie]], Section 1.2.1 in: *[[Higher Algebra]]*

For the terminology "connective"/"coconnective" in this context, see for instance:

* Harry Smith, Def. 2.1 in: *Bounded t-structures on the category of perfect complexes over a Noetherian ring of finite Krull dimension*, Advances in Mathematics **399** (2022) 108241 [[doi:10.1016/j.aim.2022.108241](https://doi.org/10.1016/j.aim.2022.108241), [pdf](http://homepages.math.uic.edu/~hsmith36/Bounded%20t-structures%20on%20perfect%20complexes.pdf)]

* Emanuele Pavia, p. 4 of: *t-structures on ∞-categories* (2021) [[pdf](https://genoa-logic-group.github.io/itaca-workshop-2021/assets/slides/PaviaE.pdf)]

[[!redirects n-connective objects]]

[[!redirects connective object]]
[[!redirects connective objects]]

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[[!redirects 0-connective]]
[[!redirects 1-connective]]
[[!redirects 2-connective]]
[[!redirects 3-connective]]
[[!redirects ∞-connective]]


n-duality

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#### Duality
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#Contents#
* table of contents
{:toc}

## Idea

The fact that for a [[manifold]] $X$ of [[dimension]] $k$, then for any [[embedding]] $\iota \colon X\hookrightarrow \mathbb{R}^n$ (which exists by the [[Whitney embedding theorem]]) the [[Thom space]] $X^{\nu}$ of the [[normal bundle]] $\nu$ behaves like a dual to $X_+$ under [[smash product]] of [[pointed topological spaces]].

Under passing to [[suspension spectra]] this becomes [[Atiyah duality]] in [[stable homotopy theory]].

## Related concepts

* [[Atiyah duality]]

## Referenes

* [[Cary Malkiewich]], p.15 of _The stable homotopy category_, 2014 ([pdf](http://math.stanford.edu/~carym/stable.pdf))

For ex-spaces see around def. 1.4.2 of

* [[Kate Ponto]], _Fixed point theory and trace for bicategories_ ([pdf](http://arxiv.org/PS_cache/arxiv/pdf/0807/0807.1471v1.pdf))

For [[parametrized spectra]] see page xyz of

* [[Peter May]], J. Sigurdsson, _[[Parametrized Homotopy Theory]]_


n-epimorphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In an [[(∞,1)-topos]] a morphism $f \colon X \to Y$ is an _$n$-epimorphism_ for $n \in \mathbb{N}$ equivalently if

* in its [[n-image]] factorization

  $$
    f \colon X \to im_n(f) \to Y
  $$

  the second morphism is an [[equivalence in an (∞,1)-category]];

* it is an [[n-connected|(n-2)-connected]] object in the [[slice (∞,1)-topos]] $\mathbf{H}_{/Y}$.

## Properties

The $n$-epimorphisms in an [[(∞,1)-topos]] are the left half of the [[(n-epi, n-mono) factorization system|((n-2)-epi, (n-2)-mono) factorization system]] which factors every morphism through its [[n-image]].

## Examples

* The $\infty$-epimorphisms are precisely the [[equivalences]].

* The 1-epimorphism are the [[effective epimorphisms in an (∞,1)-category|effective epimorphisms]].

* Every morphism is a $0$-epimorphism.

* The 1-epimorphisms between [[0-truncated]] objects are precisely the ordinary [[epimorphisms]] in the underlying [[1-topos]].

## Related concepts

* [[Postnikov system]]

* [[n-types cover]]

## References

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

Disucssion in [[homotopy type theory]] is in

* [[Univalent Foundations Project]], section 7.6 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

[[!redirects n-epic]]

[[!redirects n-epimorphisms]]

[[!redirects 1-epimorphism]]
[[!redirects 1-epimorphisms]]


n-equivalence > history

[[!redirects history > n-equivalence]]
[[!redirects history > n-equivalence]]


n-excisive (∞,1)-functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Goodwillie calculus
+--{: .hide}
[[!include Goodwillie calculus - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

### (Co-)Cartesian cubical diagrams

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]].

+-- {: .num_defn #StronglyCoCartesian}
###### Definition

An $n$-[[cube]] in $\mathcal{C}$, hence an [[(∞,1)-functor]] $\Box^n \longrightarrow \mathcal{C}$, is called _strongly homotopy co-cartesian_ or just _strongly co-cartesian_, if all its 2-dimensional square faces are [[homotopy pushout]] [[diagrams]] in $\mathcal{C}$.

=--

+-- {: .num_defn #Cartesian}
###### Definition

An $n$-[[cube]] in $\mathcal{D}$, hence an [[(∞,1)-functor]] $\Box^n \longrightarrow \mathcal{D}$, is called _homotopy cartesian_ or just _cartesian_, if its "first" object exhibits a [[homotopy limit]]-[[cone]] over the remaining objects.

=--

(e.g. [Lurie, def. 6.1.1.2 with prop. 6.1.1.15](#HigherAlg))

### $n$-Excisive functors

+-- {: .num_defn}
###### Definition

An [[(∞,1)-functor]] $F \colon \mathcal{C} \longrightarrow \mathcal{D}$ is **$n$-excisive** for $n \in \mathbb{N}$ (or _polynomial_ of degree at most $n$) if whenever $X$ is a strongly cocartesian $(n + 1)$-[[cube]] in $\mathcal{C}$, def. \ref{StronglyCoCartesian}, then $F(X)$ is a cartesian cube in $\mathcal{D}$, def. \ref{Cartesian}.

A 1-excisive (∞,1)-functor is often just called _[[excisive (∞,1)-functor]]_ for short.

=--

An $(\infty,1)$-functor which is $n$-excisive for some $n \in \mathbb{N}$ is also called a __polynomial (∞,1)-functor__ (not to be confused with [[polynomial (∞,1)-functor|other concepts having the same name]]).  It has **degree $k$** when the smallest value of $n$ for which it is $n$-excisive is $k$.

+-- {: .num_remark}
###### Remark

This notion is comparable to how a [[polynomial]] of degree at most $n$ is determined by its values on $n + 1$ distinct points.

=--

## Properties

### $n$-Excisive approximation and reflection
 {#nExcisiveApproximation}

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]] and a [[terminal object in an (∞,1)-category|terminal object]], and let $\mathcal{D}$ be a [[Goodwillie-differentiable (∞,1)-category]].

+-- {: .num_defn #InclusionOfNExcisive}
###### Definition

For $n \in \mathbb{N}$

$$
  Exc^n(\mathcal{C}, \mathcal{D})
  \hookrightarrow
  Func(\mathcal{C}, \mathcal{D})
$$

for the [[full sub-(∞,1)-category]] of the [[(∞,1)-functor (∞,1)-category]] on those [[(∞,1)-functors]] which are $n$-excisive.

=--

+-- {: .num_prop #nExcisiveApproximations}
###### Proposition

The inclusion of def. \ref{InclusionOfNExcisive} is [[left exact (∞,1)-functor|lex]] [[reflective sub-(∞,1)-category|reflective]], hence the inclusion functor has a [[left adjoint|left]] [[adjoint (∞,1)-functor]]

$$
  P_n \;\colon\; Func(\mathcal{C}, \mathcal{D}) \longrightarrow
  Exc^n(\mathcal{C}, \mathcal{D})
$$

which moreover is [[left exact (infinity,1)-functor|left exact]] (preserves [[finite (∞,1)-limits]]).

=--

This is essentially the statement of ([Goodwillie 03, theorem 1.8](#Goodwillie03)). In the above form it appears explicitly as ([Lurie, theorem 6.1.1.10](#HigherAlg)). The construction of the reflector $P_n$ is in ([Lurie, constrution 6.1.1.27](#HigherAlg)).

For $n = 1$ this reflection is _[[spectrification]]_.

\begin{proposition}\label{nExcisiveFunctorsFormInfinityTopos}
**($n$-excisive functors form an $\infty$-topos)**
\linebreak

For $\mathcal{D} = \mathbf{H}$ an [[(∞,1)-topos]], then for all $n \in \mathbb{N}$ we have that

$$
  Exc^n(\mathcal{C}, \mathbf{H}) \in (\infty,1)Topos
$$

is an [[(∞,1)-topos]]. (For $n  \gt 1$ this is in general _not_ a  [[hypercomplete (∞,1)-topos]], even if $\mathbf{H}$ is.)

\end{proposition}

This observation is due to [[Charles Rezk]]. It appears as ([Lurie, remark 6.1.1.11](#HigherAlg)). See also at *[[Joyal locus]]*.

+-- {: .num_remark}
###### Remark

A [[site]] of definition of $Exc^n(\mathcal{C}, \mathbf{H}) \hookrightarrow PSh(\mathcal{C}^{op}, \mathbf{H})$ is the _[[Weiss topology]]_ on $\mathcal{C}^{op}$.

=--

+-- {: .num_remark}
###### Remark

As $n$ ranges, the tower of $n$-excisive approximations of an  $(\infty,1)$-functor, accordding to prop. \ref{nExcisiveApproximations}, forms a tower [[analogy|analogous]] to the the [[Taylor series]] of a [[smooth function]]. This is called the [[Goodwillie-Taylor tower]]

$$
  \cdots \to P_{n+1} F \to P_n F \to \cdots \to P_1 F \to P_0 F
  \,.
$$

If this converges to $F$, then $F$ is analogous to an [[analytic function]] and is called an _[[analytic (∞,1)-functor]]_.

=--

+-- {: .num_defn}
###### Definition

In the situation of def. \ref{nExcisiveApproximations}, the functors $F$ for which $P_{n-1}F \simeq \ast$  (hence the $P_{n-1}$ [[anti-modal types]]) are called $n$-[[reduced (∞,1)-functors]].

=--

(e.g. [Lurie, def. 6.1.2.1](#HigherAlg))

### Homogeneous pieces

A polynomial $\infty$-functor of degree $k$ --- that is, a $k$-excisive functor which is not $n$-excisive for any $n\lt k$ --- is a **homogeneous** polynomial if its approximation by an $(k-1)$ degree polynomial is trivial.

(...)

+-- {: .num_prop}
###### Proposition

Let $\mathcal{C}$ be an [[(∞,1)-category]] with [[finite (∞,1)-colimits]] and with [[terminal object in an (∞,1)-category|terninal object]]. Let $\mathcal{D}$ be a [[pointed category|pointed]] [[Goodwillie-differentiable (∞,1)-category]]. Write $\mathcal{C}^{\ast/}$ for the [[pointed objects]] in $\mathcal{C}$.

Then for all natural numbers $n \geq 1$ composition with the [[forgetful functor]] $\mathcal{C}^{\ast/} \to \mathcal{C}$ induces an [[equivalence of (∞,1)-categories]]

$$
  Homog^n(\mathcal{C},\mathcal{D}) \stackrel{\simeq}{\longrightarrow} Homog^n(\mathcal{C}^{\ast/}, \mathcal{D})
$$

=--

([Lurie, prop. 6.1.2.11](#HigherAlg))

## Examples

### Goodwillie $n$-jets
 {#GoodwillienJets}

Write $\infty Grpd_{fin}$ for the [[(∞,1)-category]] of [[finite homotopy types]], hence those freely generated by [[finite (∞,1)-colimits]] from the point. Write $\infty Grpd_{fin}^{\ast/}$ for the [[pointed object|pointed]] finite homotopy types.

+-- {: .num_example}
###### Example

For $\mathbf{H}$ an [[(∞,1)-topos]] we have that

* $Exc^0(\infty Grpd_{fin}^{\ast/},\mathbf{H}) \simeq \mathbf{H}$ is the collection of constant functors, hence the original [[(∞,1)-topos]] itself;

* $Exc^1(\infty Grpd_{fin}^{\ast/},\mathbf{H}) \simeq T\mathbf{H}$ is the collection of [[parameterized spectra]] in $\mathbf{H}$, hence the [[tangent (∞,1)-topos]] of $\mathbf{H}$.

Hence one might refer to the tower of toposes

$$
  \cdots \to J^n \mathbf{H} \to \cdots \to J^2 \mathbf{H} \to T \mathbf{H} \to \mathbf{H}
$$

with

$$
  J^n \mathbf{H} \coloneqq Exc^n(\infty Grpd^{\ast/}, \mathbf{H})
$$

the tower of "[[Goodwillie calculus|Goodwillie]] [[jet (∞,1)-categories]]" of $\mathbf{H}$.

=--

see ([Lurie, def. 1.4.2.8 and around p. 823](#HigherAlg))

## Related concepts

* [[excisive (∞,1)-functor]]

* [[Goodwillie calculus]]

  * [[Goodwillie-Taylor tower]]

  * [[analytic (∞,1)-functor]]

* [[polynomial (∞,1)-functor]]

* [[model structure for n-excisive functors]]

## References

The notion of $n$-excisive functors was introduced in

* [[Thomas Goodwillie]], _Calculus II, Analytic functors_, K-Theory 5 (1991/92), no. 4, 295-332

The [[Taylor tower]] formed by $n$-excisive functors was then studied in

* {#Goodwillie03} [[Thomas Goodwillie]], *Calculus III: Taylor Series*, Geom. Topol. **7** (2003) 645-711 [[arXiv:math/0310481](http://arxiv.org/abs/math/0310481), [doi:10.2140/gt.2003.7.645](https://doi.org/10.2140/gt.2003.7.645)]

See also

* {#Rezk08} [[Charles Rezk]], *A streamlined proof of Goodwillie's $n$-excisive approximation*, Algebr. Geom. Topol. **13** (2013) 1049-1051 [[arXiv:0812.1324](http://arxiv.org/abs/0812.1324), [doi:10.2140/agt.2013.13.1049](https://doi.org/10.2140/agt.2013.13.1049)]

A discussion in the general abstract context of [[(∞,1)-category theory]] is in

* {#HigherAlg} [[Jacob Lurie]], section 6.1.1 of _[[Higher Algebra]]_

A [[model structure for n-excisive functors]] is given in

* [[Georg Biedermann]], [[Boris Chorny]], Oliver Röndigs, _Calculus of functors and model categories_, Advances in Mathematics 214 (2007) 92-115 ([arXiv:math/0601221](http://arxiv.org/abs/math/0601221))

* [[Georg Biedermann]], Oliver Röndigs, _Calculus of functors and model categories II_ ([arXiv:1305.2834v2](http://arxiv.org/abs/1305.2834v2))

Relation to [[Mackey functors]] is discussed in

* [[Saul Glasman]], _Goodwillie calculus and Mackey functors_ ([arXiv:1610.03127](https://arxiv.org/abs/1610.03127))

[[!redirects n-excisive (∞,1)-functors]]

[[!redirects n-excisive (infinity,1)-functor]]
[[!redirects n-excisive (infinity,1)-functors]]

[[!redirects n-excisive functor]]
[[!redirects n-excisive functors]]

[[!redirects n-excisive approximation]]
[[!redirects n-excisive approximations]]


n-fibration

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An **$n$-fibration** is the version of a [[Grothendieck fibration]] appropriate for $n$-[[n-category|categories]].

The idea is that a functor $p:E\to B$ between $n$-categories is an $n$-fibration if the assignation $x\mapsto E_x = p^{-1}(x)$ of an object $x\in B$ to its fiber can be made into a (contravariant) functor from $B$ to the $(n+1)$-category $n Cat$.

## Definition

(This definition is schematic, and needs to be adapted to be made precise for any particular definition of $n$-category.)

Let $p:E\to B$ be a functor between (weak) $n$-categories.

+--{: .num_defn}
###### Definition
A morphism $\phi:b\to a$ in $E$ is **cartesian** (relative to $p$) if for any $c\in E$, the following square:
$$\array{E(c,b) & \overset{\phi\circ -}{\to} & E(c,a)\\
  ^p \downarrow && \downarrow ^p\\
  B(p c, p b) & \underset{p\phi\circ -}{\to} & B(p c, p a)}$$
(which commutes, up to equivalence, by functoriality of $p$) is a (weak) [[pullback]] of $(n-1)$-categories.
=--

+--{: .num_defn}
###### Definition
We say that $p:E\to B$ is an **$n$-fibration** (or just a **fibration**) if

1. For any object $a\in E$ and morphism $f:x\to p a$ in $B$, there exists a cartesian $\phi:b\to a$ and an [[equivalence]] $p\phi \simeq f$ in the [[over category|slice]] $n$-category $B/p a$,
1. For any objects $a,b\in E$, the functor $p:E(b,a) \to B(p b, p a)$ is an $(n-1)$-fibration, and
1. For any $a,b,c\in E$, the square
$$\array{E(b,c)\times E(a,b) &\overset{\circ}{\to} & E(a,c)\\
  ^p\downarrow && \downarrow^p\\
  B(p b, p c)\times B(p a, p b) & \underset{\circ}{\to} & B(p a, p c)}$$
is a morphism of $(n-1)$-fibrations.
=--

+--{: .num_defn}
###### Definition
If $p_1:E_1\to B_1$ and $p_2:E_2\to B_2$ are $n$-fibrations, a commutative square
$$\array{E_1 & \overset{h}{\to} & E_2\\
  p_1 \downarrow && \downarrow p_2\\
  B_1 & \underset{g}{\to} & B_2}$$
is a **morphism of $n$-fibrations** if

1. Whenever $\phi$ is cartesian for $p_1$, $h(\phi)$ is cartesian for $p_2$, and
1. For any $a,b\in E_1$, the square
$$\array{E_1(b,a) & \overset{h}{\to} & E_2(h b, h a)\\
  p_1 \downarrow && \downarrow p_2\\
  B_1(p_1 b, p_1 a)& \underset{g}{\to} & B_2(g p_1 b, g p_1 a)}$$
is a morphism of $(n-1)$-fibrations.
=--

## Remarks

* The definition is recursive in $n$, but if we unravel it, it makes perfect sense for $n=\omega$.  That is, saying that $f$ is a fibration requires some things about cartesian 1-cells, and also that its action on hom-categories be a fibration--which in turn requires some things about cartesian 2-cells, and also that *its* action on hom-categories be a fibration---which in turn which requires some things about cartesian 3-cells, and so on.  After $\omega$ steps of unraveling, we are left with a list of conditions on cartesian $n$-cells for every $n$.

  An equivalent, conciser way to say this is that we interpret the definition in the case $n=\omega$ as a [[coinductive definition]].

* When $n=1$, this reduces to a [[Street fibration]], a weakened version of a [[Grothendieck fibration]].  We recover Grothendieck's original notion by requiring that for any $a\in E$ and $f:x\to p a$ in $B$, there exists a cartesian $\phi:b\to a$ such that $p\phi$ and $f$ are _equal_.  This condition violates the [[principle of equivalence]] as stated, but not if it is rephrased to apply to [[displayed categories]] instead.

* When $n=2$, so that weak 2-categories are [[bicategories]], this notion of fibration can be found in [(Buckley)](#Buckley).  A strict version for [[strict 2-categories]] (though with one condition missing) was originally studied by [(Hermida)](#Hermida).

* In general, given any notion of (semi)strict $n$-category, we can expect to appropriately strictify the definition to make it correspond to stricter notions of pseudofunctor.

## Fibrations versus functors

If $p:E\to B$ is an $n$-fibration, we define a functor (or '$n$-pseudofunctor') from $B$ to $n Cat$ as follows.  (Like the above definition, this is only a schematic sketch.)

* Send $x\in B$ to the [[essential fiber]] $E_x$, whose objects are objects $a\in E$ equipped with a equivalence $p a \simeq x$.

* For a morphism $f:y\to x$ in $B$, define $f^*:E_x\to E_y$ by choosing, for each $a\in E_x$, a cartesian $\phi:b\to a$ over $f$ and defining $f^*(a)=b$.  The universal property of cartesian arrows makes $f^*$ a functor.

* For a 2-cell $\alpha:f\to g:y\to x$ in $B$, define a transformation $\alpha^*:g^*\to f^*$ as follows.  Given $a\in E_x$, we have a cartesian arrow $\phi:g^*a\to a$ over $g$.  Now choose a cartesian 2-cell $\mu:\psi\to \phi$ over $\alpha$ in $E(g^*a,a)$.  Since $p \psi = f$, $\psi$ factors essentially uniquely through the cartesian arrow $\chi:f^*a\to a$, giving a morphism $g^*a \to f^*a$; we define this to be the component of the transformation $\alpha^*$ at $a$.

* and so on...

Note that the functor we obtain is "totally contravariant:" it is contravariant on $k$-cells for all $1\le k\le n$.

Conversely, if we have a totally contravariant '$n$-pseudofunctor' from $B$ to $n Cat$, we define $p:E\to B$ by a generalization of the [[Grothendieck construction]] as follows:

* The objects of $E$ over $x\in B$ are those of $F x \in n Cat$.

* The morphisms of $E$ over $f:y\to x$ in $B$ from $b\in F y$ to $a\in F x$ are the morphisms from $b$ to $F_f(a)$ in $F y$.

* The 2-cells of $E$ over $\alpha:f\to g:y\to x$ in $B$ from $\psi : b \to F_f(a)$ to $\phi : b \to F_g(a)$ are the 2-cells in $F y$ from $\psi$ to the composite $b \xrightarrow{\phi} F_g(a) \xrightarrow{F_\alpha(a)} F_f(a)$.

* and so on...

One expects that in this way, the $(n+1)$-category of fibered $n$-categories over $B$ is equivalent to the $(n+1)$-category of totally contravariant functors $B\to n Cat$.  These constructions are known precisely only for $n=2$.

## Examples

* The prototypical example is that if $K$ is an $n$-category with finite limits (or at least pullbacks), then the $n$-category $Fib_K$ of *internal* $(n-1)$-fibrations in $K$ should admit an $n$-fibration $cod : Fib_K \to K$.  Of course this requires defining the notion of internal $(n-1)$-fibration in an $n$-category; this is usually done representably.  For $n=2$ this gives the notion of [[fibration in a 2-category]], and the fact that $cod : Fib_K \to K$ is a 2-fibration is in [(Hermida)](#Hermida).  For $n=1$ it is just the standard fact that the [[codomain fibration]] is a fibration, i.e. every morphism in a 1-category is an "internal 0-fibration".

## Related concepts

* [[Grothendieck fibration]], [[Street fibration]]

* [[fibration in a 2-category]]

* A notion of fibration of [[(∞,1)-category|(∞,1)-categories]] exists in terms of [[Cartesian fibrations]] of [[simplicial sets]]. (See also *[[fibration of quasi-categories]]*, *[[left fibration]]*, and *[[right fibration]]*.)

## References

The notion of 2-fibrations was introduced in:

* {#Hermida} [[Claudio Hermida]], *Some properties of $Fib$ as a fibred 2-category*, Journal of Pure and Applied Algebra **134** (1999) 83-109 [doi:10.1016/S0022-4049(97)00129-1, [ps.gz](http://maggie.cs.queensu.ca/chermida/papers/jpaa.ps.gz)]

In fact they also appeared earlier, in some form, in [[Gray-adjointness-for-2-categories|Gray's book]].

However, Hermida's definition was missing the stability of cartesian 2-cells under postcomposition, which is necessary for the "Grothendieck construction" turning a pseudofunctor into a fibration to have an inverse.  This was rectified, and the definition generalized to bicategories, in

* {#Bakovic} [[Igor Baković]], *Fibrations of bicategories* (2010) [[pdf](http://www.irb.hr/korisnici/ibakovic/groth2fib.pdf), [[Bakovic-FibrationsOfBicategories.pdf:file]]]

* {#Buckley} [[Mitchell Buckley]], *Fibred 2-categories and bicategories*, Journal of Pure and Applied Algebra **218** 6 (2014) 1034-1074 [[arXiv:1212.6283](https://arxiv.org/abs/1212.6283), [doi:10.1016/j.jpaa.2013.11.002](https://doi.org/10.1016/j.jpaa.2013.11.002)]

A definition for strict $n$-categories due to Hermida is unpublished, but it is used and presented in another joint work with [Marta Bunge](http://www.math.mcgill.ca/bunge), presented at CATS07 at Calais:

* [[Marta Bunge]], "Intrinsic $n$-stack completions over a topos," slides [here](https://www.researchgate.net/publication/230792798_Intrinsic_n-stack_completions_over_a_topos)

 $n$-pseudofunctors may be viewed (and defined) as [[anafunctor]]s.  For $n$-groupoids such an approach to $n$-pseudofunctors has been studied in

* [D. Bourn](http://www-lmpa.univ-littoral.fr/~bourn),
Pseudo functors and non abelian weak equivalences, in "Categorical algebra and its applications", Springer LNM 1348 (1988), 55--70.

[[!redirects fibered n category]]
[[!redirects fibered n-category]]
[[!redirects strict fibered n-category]]
[[!redirects fibered strict n-category]]

[[!redirects fibred n category]]
[[!redirects fibred n-category]]
[[!redirects strict fibred n-category]]
[[!redirects fibred strict n-category]]

[[!redirects 2-fibration]]
[[!redirects 2-fibrations]]

[[!redirects fibered 2-category]]
[[!redirects fibered 2-categories]]

[[!redirects fibred 2-category]]
[[!redirects fibred 2-categories]]

[[!redirects fibered bicategory]]
[[!redirects fibered bicategories]]

[[!redirects fibred bicategory]]
[[!redirects fibred bicategories]]

[[!redirects fibration of n-categories]]
[[!redirects fibrations of n-categories]]

[[!redirects fibration of 2-categories]]
[[!redirects fibrations of 2-categories]]

[[!redirects fibration of bicategories]]
[[!redirects fibrations of bicategories]]


n-fold 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
{:toc}

## Idea

The notion of **$n$-fold category** or **multiple category** is what is obtained by iterating the process of forming [[internal categories]] $n$-times, starting with sets: an $0$-fold category is just an object of the ambient category (say a [[set]]) and then inductively an $n+1$-fold category is a [[internal category]] in the category of $n$-fold categories.

If the ambient category is instead an [[(∞,1)-category]] such as [[∞Grpd]], then an $n$-fold category is an "[[n-fold Segal space]]".  If a "completeness" condition is added (analogous to "globularity" of an n-fold category), one obtaines an [[n-fold complete Segal space]], which is a model for $(\infty,n)$-categories.

## Definition

### $n$-fold categories

A **$0$-fold category** is a [[set]].  A (strict) **$n$-fold category** for $n \gt 0$ is an [[internal category]] in the category of $(n-1)$-fold categories.  An $n$-fold category is also known as an **$n$-tuple category**.

In particular:

* A $1$-fold category is precisely a [[category]].
* A $2$-fold category is precisely a [[double category]] (introduced by [[Charles Ehresmann]] in 1963).
* A $3$-fold category is precisely a [[triple category]].

## Advantages

A key advantage of $n$-fold categories is the ease of expressing multiple compositions, and so  the idea of "algebraic inverse to subdivision". This is important because subdivision is a key tool in many local-to-global problems in mathematics and science, and these themselves are an important class of problems.

Thus subdividing an $n$-cube amounts to dividing the cube into small cubes by hyperplanes parallel to the faces. For a $2$-fold category, we can define a _composable array_ $(a_{ij})$ of $2$-dimensional elements (called squares) to be such that any $a_{ij}$ is composable with its immediate neighbours. In such case, the associative and interchange laws imply that the composition $[a_{ij}]$ is well defined. This  process is easily extended to $n$-fold categories, using  elements say $a_{(r)}$ where $(r)$ is multi-index, and  is applied  widely in the JPAA papers by Brown and Higgins listed below. It seems much more difficult to express these ideas in the globular or simplicial contexts.

### $n$-fold groupoids

Analogously, a **$0$-fold groupoid** is again a set, and an **$n$-fold groupoid** is an [[internal groupoid]] in $(n-1)$-fold groupoids; in particular, a $1$-fold groupoid is a [[groupoid]].

Analogous to how a [[group]] is a groupoid with a single object, one can consider $(n+1)$-fold groupoids for which all morphisms in one of the $(n+1)$ directions are endomorphisms. These are the _[[cat-n-groups]]_.

More generally, an **$(n,r)$-fold groupoid** is an $r$-fold category in $(n-r)$-fold groupoids; compare $(n,r)$-[[(n,r)-category|category]].

Note also that a category  object in the category of groups is actually a groupoid object.

## Properties
 {#Properties}

### Cartesian closure
 {#CartesianClosure}

The category of $n$-fold categories is a [[cartesian closed category]]. By [[induction]] from the statement at _[Internal category - Cartesian closure](internal+category#CartesianClosure)_.

### Homotopy types

Even though an $n$-fold category is a _strict_ version of an [[n-category]] in that all $n$ composition operations are strictly unital and associative and strictly commute with each other,  still $n$-fold _groupoids_ model all [[homotopy n-types]]. See [[homotopy hypothesis]].

### Relation to strict globular $\omega$-categories

By a theorem by Al-Agl, Brown and Steiner, [[strict omega-category|strict omega-categories]] are equivalent to those $\infty$-fold categories that satisfy a couple of restrictive properties (something like that all 1-categories of $n$-cells for all $n$ are the same and that all the "thin" identity elements exist, called "connections"): these are the "cubical $\omega$-categories with connections". Because it is relatively straightforward to define a monoidal closed category in the cubical theory, using the formula $I^m \otimes I^n = I^{m+n}$, this leads to a monoidal closed structure for strict globular $\omega$ categories.

This is a category version of a corresponding groupoid theorem of Brown and Higgins which follows from the two papers listed below.

[[Jean-Louis Loday]] introduced in the paper listed below the category of what he called $n$-cat groups, but are now called cat$^n$-groups as they are exactly $n$-fold groupoids internal to the category of groups. He showed that these objects model weak, pointed homotopy $n$-types,  see [[homotopy hypothesis]]. The paper by Brown and Loday shows that these structures can be used, via a van Kampen type theorem,  for explicit computations in homotopy theory, and this is further developed in the paper by Ellis and Steiner. This paper   relates the theory to that of crossed modules, $n$-ad homotopy groups, and the important $n$-ad connectivity theorem, which is related to results on homotopical excision.

## Related concepts

* [[double groupoid]]

* [[double bicategory]]

* [[cat-n-group]]

* [[crossed n-cube]]

* [[n-fold complete Segal space]]

* [[(n × k)-category]]

## References ##

$n$-fold categories in general were introduced in Définition 15 of

* {#Ehresmann63} [[Charles Ehresmann]], _Catégories structurées_, Annales scientifiques de l'École Normale Supérieure. Vol. 80. No. 4. Elsevier, 1963 ([EUDML:81794](https://eudml.org/doc/urn:eudml:doc:81794))

as _*n*-uple categories_.

See also:

* {#BastianiEhresmann74} [[Andrée Bastiani]], [[Charles Ehresmann]], pages 272-273 of  _Multiple functors. I. Limits relative to double categories_, Cah. Top. Géom. Différ. Catég. 15 (1974) 215–292

* [[Andrée Bastiani]], [[Charles Ehresmann]], _Multiple functors. II. The monoidal closed category of multiple categories_, Cahiers de topologie et géométrie différentielle 19.3 (1978): 295-333.

* [[Andrée Bastiani]], [[Charles Ehresmann]], _Multiple functors. III. The cartesian closed category $Cat_n$_, Cahiers de topologie et géométrie différentielle 19.4 (1978): 387-443.

* {#BastianiEhresmann79} [[Andrée Bastiani]] and [[Charles Ehresmann]]. _Multiple functors IV. Monoidal closed structures on $Cat_n$_, Cahiers de topologie et géométrie différentielle, Volume 20 (1979) no. 1, pp. 59-104. ([link](http://www.numdam.org/item/CTGDC_1979__20_1_59_0/))

A thorough theory of multiple categories appears in:

* {#Grandis19} [[Marco Grandis]], _Higher dimensional categories: from double to multiple categories_, [World Scientific]((https://doi.org/10.1142/11406)), 2019

Other references:

* [[Ronnie Brown]] and P.J. Higgins, The equivalence of $\infty$-groupoids and crossed  complexes,  Cah. Top. G\'eom. Diff.  22 (1981) 371--386.

* [[Ronnie Brown]] and P.J. Higgins. On the algebra of cubes, J. Pure
Appl.  Algebra, 21 (1981) 233-260.

* G.J. Ellis, and R.J. Steiner. Higher-dimensional crossed modules and the homotopy groups of $(n+1)$-ads, J. Pure Appl. Algebra, 46  {1987} 117--136.

* J.-L. Loday. Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra, 24 (1982) 179--202.

* [[Ronnie Brown]] and J.-L. Loday. Van Kampen theorems for diagrams of spaces. Topology 26 (1987) 311--335. With an appendix by M. Zisman.

*  [[Ronnie Brown]],  and J.-L. Loday. Homotopical excision, and Hurewicz theorems for $n$-cubes of spaces. Proc. London Math. Soc. (3) 54  (1987) 176--192.

* F.A.  Al-Agl, [[Ronnie Brown]] and R.J. Steiner, Multiple categories: the
equivalence between a globular and cubical approach, Advances in
Mathematics, 170 (2002) 71--118.

* S.  Paoli.  Internal categorical structures in homotopical algebra. In  Towards higher categories, 85–103,
IMA Vol. Math. Appl., 152, Springer, New York, 2010.

* T.  M. Fiore and  S. Paoli.  A Thomason model structure on the category of small $n$-fold categories. Algebr. Geom. Topol. 10 (2010) 1933–-2008.

* [[Marco Grandis]], [[Robert Paré]], _An introduction to multiple categories (On weak and lax multiple categories, I) _, Cahiers de Topologie et Géométrie Différentielle Catégoriques, [(pdf)](http://cahierstgdc.com/wp-content/uploads/2017/11/CTGDC_57_2_Grandis-Pare-I.pdf)

* [[Marco Grandis]], [[Robert Paré]], _Limits in multiple categories (On weak and lax multiple categories, II)_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, [(pdf)](http://cahierstgdc.com/wp-content/uploads/2017/11/Grandis-Pare-57-3.pdf)

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n-fold complete Segal space

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## Idea
_$n$-fold complete Segal spaces_ are a model for [[(∞,n)-category|(∞,n)-categories]], i.e. the [[homotopy theory|homotopical]] version of [[n-category|n-categories]].

We can view [[strict n-category|strict n-categories]] as [[n-fold category|n-fold categories]] where part of the structure is trivial; for example, strict 2-categories can be described as double categories where the only vertical morphisms are identities.
$n$-fold Segal spaces similarly result from viewing $(\infty,n)$-categories as a special class of $n$-fold internal $(\infty,1)$-categories in [[∞Grpd|∞-groupoids]].

If $\mathcal{C}$ is an $(\infty,1)$-category, then $(\infty,1)$-categories [[category object in an (infinity,1)-category|internal]] to $\mathcal{C}$ can be defined as certain simplicial objects in $\mathcal{C}$ (namely those satisfying the "Segal condition"). Thus $n$-fold internal $(\infty,1)$-categories in $\infty$-groupoids correspond to a class of $n$-simplicial $\infty$-groupoids, and  $n$-fold Segal spaces are defined by additionally specifying certain constancy conditions.

To describe the correct homotopy theory of $(\infty,n)$-categories we also want to regard the fully faithful and essentially surjective morphisms between $n$-fold Segal spaces as equivalences. It turns out that, just as in the case of [[Segal spaces]], the localization at these maps can be accomplished by restricting to a full subcategory of _complete_ objects.

## Definition

### $n$-fold Segal objects
If $\mathcal{C}$ is an $(\infty,1)$-category with pullbacks, we say that a simplicial object $X_\bullet : \Delta^{op} \to \mathcal{C}$ satisfies the [[Segal condition]] if
the squares
$$
  \array{
     X_{m+n} &\to& X_{n}
     \\
     \downarrow && \downarrow
     \\
     X_{m} &\to& X_{0}
  }
$$
are all pullbacks. Such _Segal objects_ give the $(\infty,1)$-categorical version of [[internal category|internal categories]] as _algebraic_ structures. (I.e. we have not inverted a class of fully faithful and essentially surjective morphisms.)

If $Seg(\mathcal{C})$ denotes the full subcategory of $Fun(\Delta^{op}, \mathcal{C})$ spanned by the Segal objects, then this is again an $(\infty,1)$-category with pullbacks, so we can iterated the definition to obtain a full subcategory $Seg^{n}(\mathcal{C})$ of $Fun(\Delta^{n,op}, \mathcal{C})$ of _Segal $\Delta^{n}$-objects_ in $\mathcal{C}$.

We can now inductively define $n$-fold Segal objects by imposing constancy conditions:
An _$n$-fold Segal object_ in $\mathcal{C}$ is a Segal $\Delta^{n}$-object $X$ such that

1. The $(n-1)$-simplicial object $X_0 : \Delta^{n-1,op} \to \mathcal{C}$ is constant
1. The $(n-1)$-simplicial objects $X_i$ are $(n-1)$-fold Segal objects for all $i$.

When $\mathcal{C}$ is the $(\infty,1)$-category of spaces (or $\infty$-groupoids) we refer to $n$-fold Segal objects as _$n$-fold Segal spaces_.

### Complete $n$-fold Segal spaces

We now define fully faithful and essentially surjective morphisms between $n$-fold Segal inductively in terms of the corresponding notions for [[Segal spaces]]:

+-- {: .num_defn}
###### Definition

A morphism $X \to Y$ between $n$-fold Segal spaces is _fully faithful and essentially surjective_ if:

1. $X_{\bullet,0,\ldots,0} \to Y_{\bullet,0,\ldots,0}$ is a fully faithful and essentially surjective morphism of Segal spaces
1. $X_{1,\bullet,\ldots,\bullet} \to Y_{1,\bullet,\ldots,\bullet}$ is a fully faithful and essentially surjective morphism of $(n-1)$-fold Segal spaces

=--

+-- {: .num_defn}
###### Definition

An $n$-fold Segal space $X$ is _complete_ if:

1. The Segal space $X_{\bullet,0,\ldots,0}$ is complete.
1. The $(n-1)$-fold Segal space $X_{1,\bullet,\ldots,\bullet}$ is complete.

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+-- {: .num_remark}
###### Remark

There are several equivalent ways to reformulate these inductive definitions. For example, a morphism $f : X \to Y$ is fully faithful and essentially surjective if and only if:

1. $X_{\bullet,0,\ldots,0} \to Y_{\bullet,0,\ldots,0}$ is essentially surjective.
1. For all objects $x,x' \in X_{0,\ldots,0}$ the induced map on $(n-1)$-fold Segal spaces of morphisms $X(x,x') \to Y(fx,fx')$ is fully faithful and essentially surjective.

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+-- {: .num_theorem}
###### Theorem

The complete $n$-fold Segal spaces are precisely the $n$-fold Segal spaces that are local with respect to the fully faithful and essentially surjective morphisms. Thus the localization of the $(\infty,1)$-category of $n$-fold Segal spaces at this class of morphisms is equivalent to the full subcategory of complete $n$-fold Segal spaces.

=--

This was first proved in Barwick's [thesis](#BarwickThesis), generalizing Rezk's proof in the case $n=1$. Later, [Lurie](#Lurie09) extended the notion of complete Segal objects to more general contexts than spaces, which allows an inductive definition of complete $n$-fold Segal spaces as complete Segal objects in complete $(n-1)$-fold Segal spaces.
The theorem for $n$-fold Segal spaces then follows by inductively applying the generalization of Rezk's theorem (for the case $n=1$) to this setting.

### Definition via the model category of simplicial sets

(...)

## Related concepts

* [[Theta-spaces]], [[Segal n-categories]]

* [[higher Segal space]]

* [[semi-Segal space]]

* [[weakly globular n-fold category]]

## References

The definition originates in the thesis

* {#BarwickThesis} [[Clark Barwick]], _$(\infty,n)$-$Cat$ as a closed model category_ PhD (2005)

which however remains unpublished. It appears in print in section 12 of

* [[Clark Barwick]], [[Chris Schommer-Pries]], _On the Unicity of the Homotopy Theory of Higher Categories_ ([arXiv:1112.0040](http://arxiv.org/abs/1112.0040), [slides](http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/))
 {#BarwickSchommerPries}

The basic idea was being popularized and put to use in

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

A detailed discussion in the general context of [[internal categories in an (∞,1)-category]] is in section 1 of

* {#Lurie09} [[Jacob Lurie]], _$(\infty,2)$-Categories and the Goodwillie Calculus I_ ([arXiv:0905.0462](http://arxiv.org/abs/0905.0462))

A [[Quillen adjunction]] relating [[n-complicial sets|$n$-complicial sets]] to [[n-fold complete Segal spaces|$n$-fold complete Segal spaces]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories*, Advances in Mathematics **421** (2023) 108980 [[doi:10.1016/j.aim.2023.108980](https://doi.org/10.1016/j.aim.2023.108980)]

For related references see at _[[(∞,n)-category]]_ .

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[[!redirects globular n-fold complete Segal spaces]]


n-functor

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# Higher functors
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## Idea

An __$n$-functor__ is simply a [[functor]] between $n$-[[n-category|categories]].  Similarly, an __$\infty$-functor__ is a functor between $\infty$-[[infinity-category|categories]].

Of course, as the definition of $n$-category gets more complicated as $n$ increases, so does the appropriate definition of functor.  This explains why one says '$n$-functor' instead of simply 'functor' all along.  On the other hand, anything that goes between $n$-categories, if it deserves to be called anything like 'functor' at all, will be an $n$-functor, so the prefix is not really necessary.

An $n$-[[n-natural transformation|natural transformation]] goes between $n$-functors, and there are things to go between those as well, etc.  The most general concept is an $n$-$k$-[[transfor]].

## Special cases

*  [[implication]] (a $(-1)$-[[(-1)-functor|functor]])
*  [[function]] (a $0$-[[0-functor|functor]])
*  [[functor]] (a $1$-[[1-functor|functor]])
*  [[2-functor]], including [[pseudo functor]]
* [[(∞,1)-functor]]
* [[(∞,n)-functor]]

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[[!redirects n-functors]]

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[[!redirects omega-functors]]


n-globe > history

< [[n-globe]]
[[!redirects n-globe -- history]]


N-graded module

[[!redirects N-graded vector spaces]]

[[!redirects N-graded modules]]
[[!redirects N-graded prevector space]]
[[!redirects N-graded prevector spaces]]

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## Idea ##

A mathematical structure used to define [[geometric algebras]].

## Definition ##

Given a [[commutative ring]] $R$ (usually a [[field]] $F$), an **$\mathbb{N}$-graded $R$-module** is an $R$-[[module]] $V$ (usually a [[vector space]] $V$) with a binary function $\langle - \rangle_{(-)}: V \times \mathbb{N} \to V$ called the **grade projection operator** such that

* for all $v:V$, $v = \sum_{n:\mathbb{N}} \langle v \rangle_n$

* for all $a:R$, $b:R$, $v:V$, $w:V$, and $n:\mathbb{N}$, $\langle a v + b w \rangle_n = a \langle v \rangle_n + b \langle w \rangle_n$

* for all $v:V$ and $n:\mathbb{N}$, $\langle \langle v \rangle_n \rangle_n = \langle v \rangle_n$

* for all $v:V$, $m:\mathbb{N}$, and $n:\mathbb{N}$, $(m \neq n) \Rightarrow (\langle \langle v \rangle_m \rangle_n = 0)$

Terms of $V$ are called **multivectors**.

For a natural number $n:\mathbb{N}$, the [[image]] of $\langle - \rangle_n$ under $V$ is called the **space of $n$-vectors** and is denoted as $\langle V \rangle_n$.

$$\langle V \rangle_n \coloneqq \mathrm{im}(\langle - \rangle_n)$$

The terms of $\langle V \rangle_n$ are called **$n$-vectors**.

We define the **n-truncation operator** $\vert - \vert_{(-)}: V \times \mathbb{N}  \to V$

$$\vert v \vert_n = \sum_{m = 0}^n \langle v \rangle_m$$

For a natural number $n:\mathbb{N}$, the [[image]] of $\vert - \vert_n$ under $V$ is called the **space of $n$-multivectors** and is denoted as $\vert V \vert_n$.

$$\vert V \vert_n \coloneqq \mathrm{im}(\vert - \vert_n)$$

The terms of $\vert V \vert_n$ are called **$n$-multivectors** or **$n$-truncated multivectors**. The **$n$-multivectors** are exactly those multivectors whose grade projections for all natural numbers $i \gt n$ are all zero, where $\vert V \vert_i$ is the trivial $R$-module for all natural numbers $i \gt n$.

## Examples ##

* Every [[polynomial ring]] is an $\mathbb{N}$-graded module, where the $n$-vectors are homogeneous polynomials of degree $n$, and the $n$-multivectors are general polynomials of degree $n$.

* Every [[geometric algebra]] is an $\mathbb{N}$-graded module.

* Every [[exterior algebra]] is an $\mathbb{N}$-graded module.

## See also ##

* [[commutative ring]]

* [[module]]

* [[graded vector space]]

* [[geometric algebra]]

* [[non-associative geometric algebra]]

## References ##

* {#AragonEtAl97} G. Aragón, J.L. Aragón, M.A. Rodríguez (1997), Clifford Algebras and Geometric Algebra, _Advances in Applied Clifford Algebras_ Vol. 7 No. 2, pg 91–102, doi:10.1007/BF03041220, S2CID:120860757


n-group

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## Definition

An __$n$-group__ is a [[group object]] [[internalization|internal to]] $(n-1)$-[[n-groupoid|groupoids]].

If it is [[delooping|deloopable]],
an $n$-group $G$ is the [[hom-object]] $G = Aut_{\mathbf{B}G}({*})$ of an [[n-groupoid]] $\mathbf{B}G$ with a single [[object]] ${*}$.

If $\mathbf{B}G$ is a [[strict omega-groupoid|strict n-groupoid]], then the corresponding $n$-group is called a __strict $n$-group__. Strict $n$-groups are equivalent to [[crossed complex]]es of groups, of length $n$.

Under the [[homotopy hypothesis]] $n$-groups correspond to [[pointed space|pointed]] [[connected space|connected]] [[homotopy n-type]]s.

## Examples

* A $1$-group is simply a [[group]]; see also [[2-group]] and [[3-group]].

* For $n \lt 1$, there is a single $n$-group, the [[point]].

* For arbitrary $n$, there is a [[circle n-group]].

* In [[string theory]], we have the [[string 2-group]] and the [[fivebrane 6-group]].

## Related concepts

See also

* [[group]]

* [[2-group]]

  * [[strict 2-group]]

  * [[crossed module]]

* [[3-group]]

* [[cat-n-group]]

* [[k-tuply groupal n-groupoid]]

* [[∞-group]]

## References

The [[homotopy theory]] of [[k-tuply groupal n-groupoids]] is discussed in

* [[A.R. Garzón]], J.G. Miranda, _Serre homotopy theory in subcategories of simplicial groups_, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (doi:10.1016/S0022-4049(98)00143-1)

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n-groupoid

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## Idea

An $n$-groupoid is an

* [[n-category]] in which all [[k-morphism]]s are [[equivalence]]s;

* [[(n,r)-category|(n,0)-category]];

* $n$-[[truncated]] [[∞-groupoid]] (a [[homotopy n-type]]).

## Definitions

In terms of a known notion of [[(n,r)-category]], we can define an __$n$-groupoid__ explicitly as an $\infty$-category such that:

* every $j$-morphism (at any level) is an [[equivalence]];
* every parallel pair of $j$-morphisms is equivalent, for $j\gt n$.

Or we define an __$n$-groupoid__ abstractly as an [[n-truncated object of an (infinity,1)-category|n-truncated object in]] the [[(∞,1)-category]] [[∞Grpd]].

The $n$-groupoids form an [[(n,1)-category|(n+1,1)-category]], [[nGrpd]].

## Models

### As Kan complexes
 {#AsKanComplexes}

A general model for [[∞-groupoids]] is provided by [[Kan complexes]] (the [[fibrant objects]] in the [[classical model structure on simplicial sets]] which [[presentable (infinity,1)-category|presents]] [[∞Grpd]]).

In this context an  $n$-groupoid in the general sense is modeled by a Kan complex all of whose [[homotopy groups]] vanish in degree $k \gt n$. In this generality one also speaks of a *[[homotopy n-type|homotopy $n$-type]]* or an *[[n-truncated object of an (infinity,1)-category|$n$-truncated object]]* of [[∞Grpd]].

Every such $n$-type is [[equivalence in an (infinity,1)-category|equivalent]] to a "small" model, an $(n+1)$-[[coskeletal]] Kan complex (see [there](simplicial+skeleton#Truncation)): one in which every $k$-sphere $\partial \Delta^{k+1}$ for $k \geq n+1$ has a _unique_ filler.

Yet a bit smaller than this are Kan complexes that is an $n$-[[hypergroupoid]], where in addition to these sphere-fillers also the [[horn]] fillers in degree $n+1$ are unique.

(For [[1-groupoids]]/[1-hypergroupoids](hypergroupoid#OneHypergroupoids)  this situation is further spelled out [here](simplicial+skeleton#CoskeletalityOfSimplicialNervesOfCategories)).

## Related entries

[[!include homotopy n-types - table]]

* [[homotopy n-type]] / [[n-groupoid]] / [[n-truncated object in an (∞,1)-category]]

* [[model structure for homotopy n-types]]

* [[n-truncation modality]]

## References

Discussion of [[homotopy n-types|homotopy $n$-types]]/[[n-truncated object in an (infinity,1)-category|$n$-truncated objects]] in [[homotopy type theory]]:

* [[Univalent Foundations Project]], §7 of: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* [[Nicolai Kraus]], *Truncation levels in Homotopy Type Theory*, Nottingham (2015) [[pdf](https://eprints.nottingham.ac.uk/28986/1/thesis.pdf), [eprints:28986](https://eprints.nottingham.ac.uk/28986)]

* [[Felix Cherubini]], [[Egbert Rijke]], Thm. 3.10 of: *Modal Descent*, Mathematical Structures in Computer Science , **31** 4 (2021) 363-391  [[doi:10.1017/S0960129520000201](https://doi.org/10.1017/S0960129520000201), [arXiv:2003.09713](https://arxiv.org/abs/2003.09713)]

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n-homogeneous (∞,1)-functor

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## Definition

In the context of [[Goodwillie calculus]], an [[(∞,1)-functor]] is called _$n$-homogeneous_ if it is both [[n-excisive functor|n-excisive]] and [[n-reduced (∞,1)-functor|n-reduced]]. (see e.g. [Lurie, def. 6.1.2.1](#HigherAlg)).

In the [[analogy]] with ordinary [[analysis]], these functors correspond to [[functions]] which are homogeneous [[polynomials]], whence the name.

## References

* {#HigherAlg} [[Jacob Lurie]], section 6.1.1 of _[[Higher Algebra]]_

[[!redirects n-homogeneous (∞,1)-functors]]

[[!redirects n-homogeneous (infnity,1)-functor]]
[[!redirects n-homogeneous (infnity,1)-functors]]

[[!redirects homogeneous (∞,1)-functor]]
[[!redirects homogeneous (∞,1)-functors]]

[[!redirects homogeneous (infinity,1)-functor]]
[[!redirects homogeneous (infinity,1)-functors]]


n-image

[[!redirects infinity-image]]

+-- {: .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 _[[image]]_ from [[category theory]] to [[(∞,1)-category theory]].

## Definition

### By epi/mono factorization in an $(\infty,1)$-topos
 {#ViaEpiMonoFactorization}

Let $\mathbf{H}$ be an [[(∞,1)-topos]]. Then by the discussion at _[[(n-connected, n-truncated) factorization system]]_ there is a [[orthogonal factorization system in an (∞,1)-category|orthogonal factorization system]]

$$
  (Epi(\mathbf{H}), Mono(\mathbf{H})) \subset Mor(\mathbf{H}) \times Mor(\mathbf{H})
$$

whose left class consists of the [[n-connected object in an (∞,1)-topos|(-1)-connected]] morphisms, the [[effective epimorphism in an (∞,1)-category|(∞,1)-effective epimorphisms]], while the right class consists of the [[n-truncated object in an (∞,1)-category|(-1)-truncated morphisms]] morphisms, hence the [[monomorphism in an (∞,1)-category|(∞,1)-monomorphisms]].

So given a morphism $f : X \to Y$ in $\mathbf{H}$ with epi-mono factorization

$$
  f : X \to im_1(f) \hookrightarrow Y
  \,,
$$

we may call $im_1(f) \hookrightarrow Y$ the **image** of $f$.

### As the ∞-colimit of the kernel ∞-groupoid
 {#ViaColimitOfCechNerve}

In a sufficiently well-behaved 1-[[category]], the [[coimage|(co)image]] of a morphism $f \colon X \to Y$ may be defined as the [[coequalizer]] of its [[kernel pair]], hence by the fact that

$$
  X \times_Y X \stackrel{\longrightarrow}{\longrightarrow} X \stackrel{}{\longrightarrow} im(f)
$$

is a [[colimit|colimiting]] [[cocone]] under the [[parallel morphism]] [[diagram]].

In an [[(∞,1)-topos]] the 1-image is the [[(∞,1)-colimit]] not just of these two morphisms, but of the full "$\infty$-kernel", namely of the full [[Cech nerve]] [[simplicial object]]

$$
  \cdots
   \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}}
  X \times_Y X \times_Y X
   \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
  X \times_Y X \stackrel{\longrightarrow}{\longrightarrow} X \stackrel{}{\longrightarrow} im(f)
  \,.
$$

(Here all degeneracy maps are notionally suppressed.)

To see that this gives the same notion of image as given by the epi-mono factorization as discussed [above](#ViaEpiMonoFactorization), let $f \colon X \stackrel{f}{\longrightarrow} im(f) \hookrightarrow Y$ be such a factorization. Then using (by the discussion at _[truncated morphism -- Recursive characterization](n-truncated+object+of+an+%28infinity%2C1%29-category#RecursiveDefinition)_) that the [[(∞,1)-pullback]] of a [[monomorphism in an (∞,1)-category|monomorphism]] is its domain, we find a [[pasting diagram]] of [[(∞,1)-pullback]] squares of the form

$$
  \array{
    X \times_{im(f)} X &\to & X &\stackrel{\simeq}{\to} & X
    \\
    \downarrow && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f}}
    \\
    X &\stackrel{f}{\to}& im(f) &\stackrel{\simeq}{\to}& im(f)
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} && \downarrow
    \\
    X &\stackrel{f}{\to}& im(f) &\hookrightarrow& Y
  }
  \,,
$$

which shows, by the [[pasting law]], that $X \times_{Y} X \simeq X \times_{im(f)} X$ and hence that the [[Cech nerve]] of $X \stackrel{f}{\to} Y$ is equivalently that of the effective epimorphism $X \stackrel{f}{\to} im(f)$. Now, by one of the [[Giraud-Rezk-Lurie axioms]] satisfied by [[(∞,1)-toposes]], the [[(∞,1)-colimit]] over the [[Cech nerve]] of an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] is that morphism itself. Therefore the 1-image defined this way coincides with the one defined by the epi/mono factorization.

### Tower of $n$-images

More generally for $f \colon X \to Y$ a morphism we have a [[tower]] of $n$-images

$$
  X \simeq im_\infty(f) \to \cdots \to im_2(f) \to im_1(f) \to im_0(f) \simeq Y
$$

factoring $f$, such that for each $n \in \mathbb{N}$ the factorization $X \to im_{n+2}(f) \to Y$ is the [[(n-connected, n-truncated) factorization system]] of $f$.

This is the relative [[Postnikov system]] of $f$.

## Properties

### Syntax in homotopy type theory
 {#SyntaxInHomotopyTypeTheory}

Let the ambient [[(∞,1)-category]] be an [[(∞,1)-topos]] $\mathbf{H}$. Then its [[internal language]] is [[homotopy type theory]].
We discuss the [[syntax]] of 1-images in this theory.

+-- {: .num_prop #SyntaxOfInfinityImage}
###### Proposition

If
$$
  a \colon A
  \;\vdash\;
  f(a) \,\colon\, B
$$
is a [[term]] whose [[categorical semantics]] is a [[morphism]] $A \xrightarrow{f} B$ in $\mathbf{H}$, then the 1-image of that morphism, when regarded as an object of $\mathbf{H}/B$, is the [[categorical semantics]] of the [[dependent type]]

$$
  (b:B)
  \;\;
  \vdash
  \;\;
    \Big[
      (a \colon A)
      \times
      \big(b = f(a) \big)
    \Big]_0
    \,\colon\,
    Type
    \,,
$$
=--

Here

* $(a \colon A) \times C_a$ denotes the [[dependent pair type]],

* "$=$" denotes the [[identification type]],

* $[-]_0$ denotes the [[bracket type]] (which is constructible in homotopy type theory either as a [[higher inductive type]] as described at *[[n-truncation modality]]*m or using the [[univalence axiom]] and a resizing rule to obtain a [[subobject classifier]]).

+-- {: .proof}
###### Proof

Let $\mathbf{M}$ be a suitable [[model category]] [[presentable (∞,1)-category|presenting]] $\mathbf{H}$.  Then by the rules for [[categorical semantics]] of [[identity types]] and [[substitution]], the interpretation of

$$
  (b \colon B)
  ,\,
  (a \colon A)
  \;\;\vdash\;\;
  \big(
    b = f(a)
  \big)
  \,:\, Type
$$

in $\mathbf{M}$ a is the following [[pullback]] $\tilde A$ (see _[[homotopy pullback]]_ for more details):

$$
  \array{
    \tilde A &\to& B^{I}
    \\
    \downarrow && \downarrow
    \\
    A \times B &\stackrel{(f,id_B)}{\to}& B \times B
  }.
$$

Here all objects now denote [[fibrant object]] representatives of the given objects in $\mathbf{H}$, and the right-hand morphism is the [[fibration]] out of a [[path space object]] for $B$. By the [[factorization lemma]] the composite $\tilde A \to A \times B \to B$ here is a [[fibration]] [[resolution]] of the original $A \stackrel{f}{\to} B$ and $\tilde A \to A \times B$ is a fibration resolution of $A \stackrel{(id_A,f)}{\to} A \times B$. Regarded in the [[slice category]] $\mathbf{M}/(A \times B)$, this now interprets the syntax $(b = f(a))$ as an $(A \times B)$-[[dependent type]].

Now the interpretation of the sum $\sum_{a:A}$ is simply that we forget the map to $A$ (or equivalently compose with the projection $A\times B\to B$), regarding $\tilde A$ as an object of $\mathbf{M}/B$.  Of course, this is just a fibration resolution of $f$ itself.

Finally, the interpretation of the [[bracket type]] of this is precisely the [[n-truncated object in an (infinity,1)-category|(-1)-truncation]] of this morphism, which by the discussion there is its 1-image $im_1(\tilde A \to B)$, regarded as a dependent type over $B$.  Thus, it is precisely the the 1-image of $f$.
=--

By additionally forgetting the remaining map to $B$, we obtain:

+-- {: .num_cor #NonDependentSyntax}
###### Corollary
In the above situation, the 1-image of $f$, regarded as an object of $\mathbf{H}$ itself, is the semantics of the non-dependent type
$$
  \vdash \;
  \Big(\Big(
    \sum_{b:B}\Big[\sum_{a:A} (b = f(a))\Big]
  \Big)
  : Type\Big).
$$
=--

+-- {: .num_remark }
###### Remark
The bracket type of a [[dependent sum]] is the [[propositions as some types]] version of the [[existential quantifier]], so we can write the dependent type in Prop. \ref{SyntaxOfInfinityImage} as

$$
  (b:B) \; \vdash \;
  \left(\exists a:A . (b = f(a))
  \;:\; hProp\right).
$$

The dependent sum *of* an [[h-proposition]] is then the propositions-as-some-types version of the [[comprehension rule]], $\{b \in B | \phi(b)\}$, so the non-dependent type in Cor. \ref{NonDependentSyntax} may be written as

$$
  \left\{
    b \in B \,|\, \exists a \in A . (b = f(a))
  \right\}
$$

which is manifestly the naive definition of [[image]].
=--

### Compatibility with limits

+-- {: .num_prop #nImagePreservesProducts}
###### Proposition

In an [[(∞,1)-topos]] $\mathbf{H}$, the $n$-image of a product is the product of $n$-images:

$$
  im_n\left(X_1 \times X_2 \stackrel{(f,g)}{\longrightarrow}) X_2 \times Y_2\right)
  \simeq
  \left(
     im_n(f) \times im_n(g)
     \longrightarrow
     X_2 \times Y_2
  \right)
  \,.
$$

=--

+-- {: .proof}
###### Proof

The morphism  $(f,g)$ is the product of $(f,id)$ with $(g,id)$ in the [[slice (∞,1)-topos]] over $X_2 \times Y_2$, namely there is a [[homotopy fiber product]] in $\mathbf{H}$ of the form

$$
  \array{
     && X_1 \times Y_1
     \\
     & {}^{\mathllap{(id,g)}}\swarrow && \searrow^{\mathrlap{(f,id)}}
     \\
     X_1 \times Y_2 && \swArrow_{\simeq} && X_2 \times Y_1
     \\
     & {}_{\mathllap{(f,id)}}\searrow && \swarrow_{\mathrlap{(id,g)}}
     \\
     && X_2 \times Y_2
  }
$$

But by [this proposition](n-truncated+object+of+an+infinity-category#nTruncationInToposPreservesFiniteProducts), $n$-truncation in the slice $\infty$-topos preserves products, so that

$$
  \begin{aligned}
     im_n(f,g)
     & =
     \tau_n (f,g)
     \\
     & \simeq \tau_n ((f,id)\times_{X_2 \times Y_1} (id,g))
     \\
     &\simeq (\tau_n(f,id)) \times_{X_2 \times Y_2} (\tau_n(id,g))
     \\
     & \simeq (im_n f, id)\times_{X_2 \times Y_2} (id, im_n g)
     \\
     & \simeq (im_n f, im_n g)
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

The $n$-image of the [[looping]] of a morphism $f \colon X \to Y$ of [[pointed objects]] is the looping of its $(n+1)$-image

$$
  im_n (\Omega(f))
  \simeq
  \Omega(im_{n+1}(f))
  \,.
$$

i.e. we have the following diagram, where the columns are [[homotopy fiber sequences]]

$$
  \array{
    \Omega f \colon
     & \Omega X &\longrightarrow& im_n(\Omega f)
     &\longrightarrow& \Omega Y
    \\
    & \downarrow && \downarrow && \downarrow
    \\
    id_\ast \colon & \ast &\longrightarrow & im_{n+1}(id_\ast) \simeq \ast &\longrightarrow& \ast
    \\
    & \downarrow && \downarrow && \downarrow
    \\
    f\colon & X &\stackrel{}{\longrightarrow}& im_{n+1}(f) &\stackrel{}{\longrightarrow}& Y
  }
$$

=--

## Examples

### Automorphisms

Let $V \in \mathbf{H}$ be a $\kappa$-[[compact object in an (∞,1)-category|small object]], and let

$$
  * \stackrel{\vdash V}{\to} Obj_\kappa
$$

the corresponding morphism to the [[object classifier]]. Then the 1-image of this is $\mathbf{B}\mathbf{Aut}(V)$, the [[delooping]] of the internal [[automorphism ∞-group]] of $V$.

### Of functors between groupoids
 {#NImagesOf1FunctorsBetweenGroupoids}

The simplest nontivial case of higher images after the ordinary case of images of functions of sets is images of [[functors]] between [[groupoids]] hence betwee [[1-truncated]] objects $X, Y \in Grpd \hookrightarrow$ [[∞Grpd]].

+-- {: .num_prop #FactorizationOf1FunctorsBetween1Groupoids}
###### Proposition

For $f \colon X \to Y$ a [[functor]] between [[groupoids]], its image factorization is

$$
  f \colon X \to im_2(f) \to im_1(f) \to Y
  \,,
$$

where (up to [[equivalence of groupoids]])

* $im_1(f) \to Y$ is the [[full subcategory|full subgroupoid]] of $Y$ on those objects $y$ such that there is an object $x \in X$ with $f(x) \simeq y$;

* $im_2(f)$ is the groupoid whose objecs are those of $X$ and whose morphisms are equivalence classes of morphisms in $X$ where $\alpha,\beta \in Mor(X)$ are equivalent if they have the same domain and codomain in $X$ and if $f(\alpha) = f(\beta)$

  * $im_2(f)\to im_1(f)$ is the identity on objects and the canonical inclusion on sets of morphisms;

  * $X \to im_2(f)$ is the identity on objects and the defining [[quotient]] map on sets of morphisms.

=--

+-- {: .proof}
###### Proof

Evidently $im_1(f) \to Y$ is a [[fibration]] ([[isofibration]]) and so the [[homotopy fiber]] over every point is given by the 1-categorical [[pullback]]

$$
  \array{
    F_y &\to& im_1(y)
    \\
    \downarrow && \downarrow
    \\
    * &\stackrel{y}{\to}& Y
  }
  \,.
$$

If there is no $x \in X$ such that $f(x) \simeq y$ then fiber $F_y$ is [[empty set]]. If there is such $x$ then the fiber is the groupoid with that one object and all morphisms in $X$ on that object which are mapped to the identity morphism, which by construction is only the identity morphism itself, hence the fiber is the point. Hence indeed the homotopy fibers of $im_1(f) \to Y$ are [[(-1)-truncated]] objects and the map from homotopy fiber of $f$ to those of $im_1(f)$ is their (-1)-truncation.

Next, the [[homotopy fibers]] of $im_2(f) \to Y$ over a point $y \in Y$ are the groupoids whose objects are pairs $(x, (f(x) \to y))$ and whose morphisms are pairs

$$
  \left(
    \array{
      x_1 &\stackrel{[\alpha]}{\to}& x_2
      \\
      f(x_1) &\stackrel{f(\alpha)}{\to}& f(x_2)
      \\
      \downarrow && \downarrow
      \\
      y &=& y
    }
  \right)
  \,.
$$

Notice first that the above is [[0-truncated]]: an automorphism of $(x,(f(x) \to y))$ is of the form $x \stackrel{[\alpha]}{to} x$ such that $f(\alpha) = id_{f(x)}$ and so there is precisely one such, namely the equivalence class of $id_x$.

Notice second that the homotopy fibers of $f$ itself have the same form, only that $\alpha \colon x_1 \to x_2$ appears itself, not as its equivalence class. Also if two objects in the homotopy fiber of $f$ are connected by a morphism, then by construction so they are in the homotopy fiber of $im_2(f) \to Y$ and hence the latter is indeed the [[0-truncation]] of the former.

=--

+-- {: .num_remark }
###### Remark

The factroization $f \colon X \to im_2(f) \to im_1(f) \to Y$ of prop. \ref{FactorizationOf1FunctorsBetween1Groupoids} exhibits a _[[ternary factorization system]]_ in [[Grpd]].

=--

+-- {: .num_remark }
###### Remark

This situation can also be considered from the perspective of the formalization of [[stuff, structure, property]]. See there at _[A factorization system](stuff%2C+structure%2C+property#AFactorizationSystem)_.

=--

## Related concepts

* [[image]], [[coimage]]

* [[homotopy image]]

* [[Postnikov system]]

* [[k-ary factorization system]]

## References

Discussion of $n$-image factorization in [[homotopy type theory]] is in

* [[Univalent Foundations Project]], section 7.6 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_

A construction of $n$-image factorizations in [[homotopy type theory]] using only [[homotopy pushouts]] and specifically [[join of topological spaces|joins]] (instead of more general [[higher inductive types]]) is described in

* {#Rijke17} [[Egbert Rijke]], _The join construction_ ([arXiv:1701.07538](https://arxiv.org/abs/1701.07538))

[[!redirects ∞-image]]
[[!redirects ∞-images]]
[[!redirects infinity-images]]

[[!redirects n-image]]
[[!redirects n-images]]

[[!redirects n-image]]
[[!redirects n-images]]

[[!redirects 0-image]]
[[!redirects 0-images]]
[[!redirects 1-image]]
[[!redirects 1-images]]
[[!redirects 2-image]]
[[!redirects 2-images]]
[[!redirects 3-image]]
[[!redirects 3-images]]


n-Lie algebra

> This entry is about the concept that became famous with the [[BLG model]] while it is NOT about what in [[homotopy theory]] are known as [[Lie n-algebras]] ([[homotopy theory|homotopy-theoretic]] [[Lie algebras]]), NOR [[n-algebras]] ([[categorification|categorifies]] [[associative algebras]]). Some authors also say _Filippov algebra_.

***

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

# $n$-Lie algebras #
* table of contents
{: toc}

## Idea

An **$n$-Lie algebra** is defined to be an algebraic structure which

* looks formally like the special case of an $L_\infty$-[[L-infinity-algebra|algebra]] for which only the $n$-ary bracket $D_n$ is non-vanishing (see there);

* _but_ without necessarily the grading underlying an $L_\infty$-algebra, and in particular without the requirement that $D_n$ be of homogeneous degree $-1$ in any grading.

Therefore, any "$n$-Lie algebras" that appear in the literature are **not** examples of _[[Lie n-algebras]]_, hence of  _[[L-∞ algebras]]_.
(So in particular $n$-Lie algebras in this sense in general don't integrate to [[Lie infinity-groupoids]] via the usual [[Lie theory]]. )

Instead, at least certain "3-Lie algebras" can be understood as encoding structure in [[Lie 2-algebras]] equipped with a binary [[invariant polynomial]] ([Saemann-Ritter 13, section 2.5](#SaemannRitter13)).

## Related concepts

* [[higher Poisson bracket]]

  * [[Nambu bracket]]

* [[membrane matrix model]]

* [[BLG model]]

## References

### Original

A discussion of $n$-Lie algebras (without the $L_\infty$-grading) is in

* V. T. Filippov, _$n$-Lie algebras_, Sib. Math. Zh. No 6 126--140 (195)

* V. T. Filippov, _On the $n$-Lie algebra of Jacobians_, Sibirsk. Mat. Zh., 1998, Volume 39, Number 3, Pages 660–669 ([English translation](http://link.springer.com/article/10.1007%2FBF02673915))

* A. S. Dzhumadil'daev, _Wronskians as $n$-Lie multiplications_ ([arXiv:math/0202043](http://arxiv.org/abs/math/0202043))

Similar (but different) discussion is in

* [[Phil Hanlon]], [[Michelle Wachs]], _On Lie $k$-Algebras_, Advances in Mathematics Volume 113, Issue 2, July 1995, Pages 206–236 ([doi:10.1006/aima.1995.1038](https://doi.org/10.1006/aima.1995.1038))

* [[José de Azcárraga]], J. C. Perez Bueno, _Higher-order simple Lie algebras_, Commun. Math. Phys. **184** (1997) 669-681 [[arXiv:hep-th/9605213](http://arxiv.org/abs/hep-th/9605213), [doi:10.1007/s002200050079](https://doi.org/10.1007/s002200050079)]

* [[José de Azcárraga]], [[José M. Izquierdo]], *$n$-Ary algebras: a review with applications*, J. Phys. A: Math. Theor. **43** (2010) 293001 [[doi:10.1088/1751-8113/43/29/293001](https://doi.org/10.1088/1751-8113/43/29/293001)]

### Re-inventions

The notion of $n$-Lie algebras, for $n=3$, was re-invented, in the form of the _[[M-brane 3-algebra]]_ by string physicists in the [[BLG model]]

* [[Jonathan Bagger]], [[Neil Lambert]], _Gauge Symmetry and Supersymmetry of Multiple M2-Branes_ ([arXiv](http://arxiv.org/abs/0711.0955))

which sparked a tremendous amount of activity.

See the blog entry

* [Lie 3-Algebras on the Membrane](http://golem.ph.utexas.edu/category/2008/11/linfinity_algebras_on_the_memb.html)

for further details and links. And see this blog discussion

* [3-Algebra and Chern-Simons terms](http://golem.ph.utexas.edu/category/2009/03/twisted_differential_string_an.html#c022696)

for discussion about the relation to proper $L_\infty$-algebraic formalism.

Further re-inventions of the concept of $n$-Lie algebras in this context are appearing. For instance in

* Tamar Friedman, _Orbifold singularities, the LATKe, and Yang-Mills with Matter_ ([arXiv](http://arxiv.org/abs/0806.0024))

### Relation to representations of metric Lie algebras
 {#ReferencesAsMetricLienAlgebras}

The full generalized axioms on the [[M2-brane 3-algebra]] and first insights into their relation to [[Lie algebra representations]] of [[metric Lie algebras]] is due to

* {#CherkisSaemann08} [[Sergey Cherkis]], [[Christian Saemann]], _Multiple M2-branes and Generalized 3-Lie algebras_, Phys. Rev. D78:066019, 2008 ([arXiv:0807.0808](https://arxiv.org/abs/0807.0808))

The full identification of M2-brane 3-algebras with dualizable [[Lie algebra representations]] over [[metric Lie algebras]] is due to

* {#MFFMER08} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], _On the Lie-algebraic origin of metric 3-algebras_, Commun. Math. Phys. 290:871-902, 2009 ([arXiv:0809.1086](http://arxiv.org/abs/0809.1086))

See also:

* Sam Palmer, [[Christian Saemann]], section 2 of _M-brane Models from Non-Abelian Gerbes_, JHEP 1207:010, 2012 ([arXiv:1203.5757](http://arxiv.org/abs/1203.5757))

* {#SaemannRitter13} [[Patricia Ritter]], [[Christian Saemann]], section 2.5 of _Lie 2-algebra models_, JHEP 04 (2014) 066 ([arXiv:1308.4892](http://arxiv.org/abs/1308.4892))

* {#Saemann16} [[Christian Saemann]], appendix A of _Lectures on Higher Structures in M-Theory_ ([arXiv:1609.09815](https://arxiv.org/abs/1609.09815))

* [[José Figueroa-O'Farrill]], _Triple systems and Lie superalgebras_ in _M2-branes, ADE and Lie superalgebras_, talk at IPMU 2009 ([pdf](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf), [[FigueroaOFarrillM2Branes02.pdf:file]])

[[!redirects n-Lie algebras]]

[[!redirects 3-Lie algebra]]
[[!redirects 3-Lie algebras]]

[[!redirects Filippov algebra]]
[[!redirects Filippov algebras]]


n-localic (infinity,1)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[(∞,1)-topos]] is **$n$-localic** if

* regarded as a [[little topos]] it behaves like a generalized [[n-groupoid]];

* it behaves like the [[(∞,1)-category of (∞,1)-sheaves]] over an [[(∞,1)-site]] that is a [[(n,1)-category]].

More precisely: if [[(∞,1)-geometric morphisms]] into it are fixed by their restriction to the underlying [[(n,1)-toposes]] of [[n-truncated object of an (infinity,1)-category|(n-1)-truncated objects]].

To the tower of [[(n,1)-toposes]] of [[n-truncated object of an (infinity,1)-category|(n-1)-truncated objects]]

$$
  \cdots \to
  \tau_{\leq 3-1} \mathcal{X}
    \longrightarrow
   \tau_{\leq 2-1} \mathcal{X}
     \longrightarrow
   \tau_{\leq 1-1} \mathcal{X}
     \longrightarrow
    \tau_{\leq 0-1} \mathcal{X}
  \to *
$$

of a given [[(∞,1)-topos]] $\mathcal{X}$ corresponds a tower of $n$-localic toposes $\mathcal{X}_n$ such that $\tau_{\leq n -1} \mathcal{X} \simeq \tau_{\leq n-1} \mathcal{X}_n$. We may think of the $n$-localic $\mathcal{X}_n$ as being $n$th stage in the [[Postnikov tower]] decomposition of $\mathcal{X}$.

A 0-localic $(1,1)$-topos is a [[localic topos]] from ordinary [[topos theory]].

## Definition

We write [[(∞,1)Topos]] for the [[(∞,1)-category]] of [[(∞,1)-toposes]] and [[(∞,1)-geometric morphisms]] between them.

For $\mathcal{X}$ an [[(∞,1)-topos]] we denote by

$$
  \tau_{\leq n-1} \mathcal{X} \hookrightarrow \mathcal{X}
$$

the [[(n,1)-topos]] of $(n-1)$-[[truncated]] objects of $\mathcal{X}$.

We write $(n,1)Topos$ for the [[(n,1)-category|(n+1,1)-category]] of [[(n,1)-topos]]es and $(n,1)$-geometric morphisms between them.

+-- {: .num_defn}
###### Definition
**($n$-localic $(\infty,1)$-topos)**

An [[(∞,1)-topos]] $\mathcal{X}$ is **$n$-localic** if for any other
$(\infty,1)$-topos $\mathcal{Y}$ the canonical morphism

$$
  (\infty,1)Topos(\mathcal{Y},\mathcal{X}) \to
   (n,1)Topos(\tau_{\leq n-1} \mathcal{Y}, \tau_{\leq n-1}\mathcal{X})
$$

is an [[equivalence of (∞,1)-categories]] (of [[∞-groupoids]]).

More generally,

a [[(n,1)-topos|(k,1)-topos]] $\mathcal{X}$ is **$n$-localic** for $0 \leq n \leq k \leq \infty$ if for any other
$(k,1)$-topos $\mathcal{Y}$ the canonical morphism

$$
  (k,1)Topos(\mathcal{Y},\mathcal{X}) \to
   (n,1)Topos(\tau_{\leq n-1} \mathcal{Y}, \tau_{\leq n-1}\mathcal{X})
$$

is an [[equivalence of (∞,1)-categories]] (of [[∞-groupoids]]).

=--

This is ([[Higher Topos Theory|HTT, def. 6.4.5.8]]).

+-- {: .num_remark}
###### Remark

This implies that an $n$-localic $(\infty,1)$-topos is also $(n+1)$-localic and generally $k$-localic for all $k \geq n$.

=--

## Examples
  {#Examples}

+-- {: .num_prop}
###### Proposition

The [[(∞,1)-category of (∞,1)-sheaves]] over an [[(∞,1)-site]] $C$ with finite limits which is an [[(n,1)-category]] is $n$-localic.

=--

This is ([[Higher Topos Theory|HTT, lemma 6.4.5.6]]).

+-- {: .num_remark}
###### Remark

For $n = 0$ this implies the familiar statement from ordinary [[topos theory]]: a [[category of sheaves]] over a [[posite]]=[[(0,1)-site]] is a [[localic topos]] (= 0-localic $(1,1)$-topos).

=--

This is ([LurieStructured, lemma 2.3.16](#JurieStructured)).

+-- {: .num_prop}
###### Proposition

For $n \in \mathbb{N}$ and $\mathcal{X}$ an $n$-localic $(\infty,1)$-topos, the [[over-(∞,1)-topos]] $\mathcal{X}/U$ is $n$-localic precisely if the object $U$ is $n$-[[truncated]].

=--

This is ([StrSp, lemma 2.3.14](#JurieStructured)).

+-- {: .num_prop}
###### Proposition

For $\mathcal{X}$ an $n$-localic $(\infty,1)$-topos let $U \in \mathcal{X}$ be an [[object]]. Then the following are equivalent

1. the restriction of the [[inverse image]] $U^* : \mathcal{X} \to \mathcal{X}/U$ (of the [[etale geometric morphism ]] from the [[over-(∞,1)-topos]]) to $(n-1)$-[[truncated]] objects is an [[equivalence of (∞,1)-categories]];

1. the object $U$ is $n$-[[connected]].

=--

This is ([StrSp, lemma 2.3.14](#JurieStructured)).

## Properties

+-- {: .num_prop}
###### Proposition

Every [[(n,1)-topos]] $\mathcal{Y}$ is the [[(n,1)-category]] of $(n-1)$-[[truncated]] objects in an $n$-localic $(\infty,1)$-topos $\mathcal{X}_n$

$$
  \tau_{n-1} X_n  \stackrel{\simeq}{\to} \mathcal{Y}
  \,.
$$

=--

This is ([[Higher Topos Theory|HTT, prop. 6.4.5.7]]).

Let $\mathcal{G}$ be a [[geometry (for structured (∞,1)-toposes)]].

+-- {: .num_prop}
###### Proposition

If $\mathcal{G}$ is an [[(∞,n)-category]] then a $n$-localic $\mathcal{G}$-[[structured (∞,1)-topos]] is an $n$-[[truncated]] object in the [[(∞,1)-category]] $Topos(\mathcal{G})$.

=--

This is [StrSp, lemma 2.6.17](#LurieStructured)

## Related concepts

* [[localic topos]]

* [[n-localic 2-topos]]

* [[n-types cover]]

## References

The general noion is the topic of section 6.4.5 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

Remarks on the application of $n$-localic $(\infty,1)$-toposes in [[higher geometry]] are in

* [[Jacob Lurie]], _[[Structured Spaces]]_
{#JurieStructured}

[[!redirects localic (∞,1)-topos]]
[[!redirects localic (infinity,1)-topos]]
[[!redirects n-localic (∞,1)-topos]]