n-localic 2-topos

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## Definition

Let $K$ be a [[Grothendieck 2-topos]].  We say that $K$ is **$n$-truncated** if it has a small [[eso-generator]] consisting of $(n-1)$-[[truncated object|truncated objects]].  It is easy to see that if a coproduct of $(n-1)$-truncated objects is $(n-1)$-truncated (as is the case for all $n\ge 1$), then this is equivalent to saying that $K$ has _enough $(n-1)$-truncated objects_ (i.e. every object admits an eso from an $(n-1)$-truncated one).  In particular:

* $K$ is always **2-truncated**.
* $K$ is **(2,1)-truncated** if it has [[core|enough groupoids]].
* $K$ is **1-truncated** if it has enough discretes.
* $K$ is **(0,1)-truncated** if the subterminal objects are eso-generating.
* $K$ is **(-1)-truncated** if the terminal object is an eso-generator.

## $n$-sites and $n$-sheaves

By the [[2-Giraud theorem]], small eso-generating sets of objects correspond to small [[2-site|2-sites]] of definition for $K$.  Thus, if we define an $n$-site to be a 2-site which is an $n$-category (where $n\le 2$ as usual), we have:

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###### Theorem
A Grothendieck 2-topos is $n$-truncated iff it is equivalent to the 2-category of 2-sheaves on some $n$-site.
=--

Note that a 1-site is the same as the usual notion of [[site]], and a [[(0,1)-site]] is sometimes called a _[[posite]]_. In particular, any [[frame]] is a (0,1)-site with its [[canonical coverage]] (the covering families are given by [[unions]]).

Particular cases include:

* $K$ is 1-truncated iff it is equivalent to the 2-category of 2-sheaves (stacks) on an ordinary small (1-)site, and therefore to the 2-category of stacks for the canonical coverage on some Grothendieck 1-topos.

* $K$ is (0,1)-truncated iff it is equivalent to the 2-category of stacks on a posite, and therefore also to the 2-category of stacks on some [[nlab:locale|locale]].  We call such a $K$ _localic_.

* If $K$ is (-1)-truncated, then it is in particular localic, and its terminal object is a (strong) generator.  It is not hard to see that this is equivalent to saying that the corresponding locale $X$ is a sublocale of the terminal locale $1$.  Thus, just as [[nlab:(-1)-category|(-1)-categories]] are sub*sets* of $1$, (-1)-toposes are sub*locales* of $1$.  If $Cat$ has classical logic, this implies that either $X\cong 0$ or $X\cong 1$; and hence that either $K\simeq 1$ or $K\simeq Cat$.  However, constructively there may be many other sublocales of $1$.

* It would be nice if the only (-2)-truncated Grothendieck 2-topos were $Cat$.  However, I don't see a way to make this happen except by fiat.

## Grothendieck $n$-toposes

Now, if $C$ is an $n$-site, it is also reasonable to consider **$n$-sheaves** on $C$, by which we mean 2-sheaves taking values in $(n-1)$-categories.  Thus, a 1-sheaf on a 1-site is precisely the usual notion of [[nlab:sheaf|sheaf]] on a site.  And a (0,1)-sheaf on a (0,1)-site is easily seen to be a lower set that is an "ideal" for the coverage.

We define a **Grothendieck $n$-topos** to be an $n$-category equivalent to the $n$-category of $n$-sheaves on an $n$-site.  The case $n=1$ gives classical [[nlab:Grothendieck topos|Grothendieck toposes]]; the case $n=(0,1)$ gives [[nlab:locale|locales]].  Note the distinction between a Grothendieck $n$-topos and an $n$-truncated Grothendieck 2-topos.  The relationship is that

1. The 2-category of 2-sheaves for the canonical coverage on a Grothendieck $n$-topos is an $n$-truncated Grothendieck 2-topos, and
1. Any $n$-truncated Grothendieck 2-topos arises in this way from the Grothendieck $n$-topos which is its full subcategory of $(n-1)$-truncated objects.

This relationship is completely analogous to the classical relationship between locales and localic toposes.  In fact, if $Gr n Top$ denotes the $(n+1)$-category of Grothendieck $n$-toposes (that is, $n$-categories of $n$-sheaves on an $n$-site), we have inclusions
$$\array{
  &&&&&& Gr(1,2)Top\\
  &&&&& \nearrow && \searrow\\
  Gr (-1) Top &\to&
  Gr(0,1)Top &\to &
  Gr1Top &&&&
  Gr2Top\\
  &&&&& \searrow && \nearrow\\
  &&&&&& Gr(2,1)Top
}$$
where the inclusion from $Gr n Top$ to $Gr (n+1) Top$ is given by taking the $(n+1)$-category of $(n+1)$-sheaves for the canonical coverage.  (See [[2-geometric morphism]] for the morphisms in these categories.)

## Related concepts

* [[n-localic (∞,1)-topos]]

[[!include flavors of higher toposes -- list]]

## References

The material on this page is taken from

* [[Mike Shulman]], _[[michaelshulman:truncated 2-topos]]_

[[!redirects n-site]]
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n-mesh

[[!redirects meshes]]
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### Context
#### Higher category theory
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## Idea

Meshes are towers of [[stratified space#constructible_strat_bun|constructible]] [[stratified space#stratified_bundle|stratified bundles]] whose fibers are points or [[framing|framed]] stratified 1-manifolds. Meshes play a central role in the local description of directed (namely, [[directed space#framed_spaces|framed]]) regular [[cell complex|cell]] and [dual-cell](https://en.wikipedia.org/wiki/Poincar%C3%A9_duality#Dual_cell_structures) complexes.

## Definition

Meshes are certain [[stratified space|stratified spaces]] with additional [[structure]] (namely, structure that records spatial [[directed space|directions]] via a notion of [[directed space#framed_spaces|framings]]). In the following definition, framings can be understood in the [[framing|ordinary sense]].

\begin{defn} A _1-mesh_ is a framed [[contractible]] $k$-[[manifold]] $M$, $k \leq 1$, together with a [[stratification]] $f$ on $M$ whose strata are open $l$-disks, $l \leq 1$.
\end{defn}

Contractability implies that any 1-mesh framed embeds in standard framed $\mathbb{R}$: we call such embeddings _mesh-trivializations_.

To prepare for the definition of a bundle of 1-meshes, recall, the [[stratified space#fundamental_categories|fundamental $\infty$-category]] functor $\mathcal{E} : \mathcal{S}\mathit{trat} \to \mathbf{Cat}_\infty$ maps [[stratified space#conical_strat|conical stratifications]] to their fundamental [[∞-categories]]: for defining the latter, we work with [[stratified space#exit_entrance_conv|entrance paths]] (which is the opposite convention to working with [[stratified space#exit_entrance_conv|exit paths]]).

\begin{rmk} For simplicity, below one can replace the [[(infinity,1)-functor|$\infty$-functor]] $\mathcal{E}$ with the ordinary fundamental *[[poset]]* [[functor]] $\mathsf{E}$. This yields an equivalent definition of $n$-meshes (even though the definition of $n$-mesh *bundles* will differ).
\end{rmk}

The following definition abstractly characterizes bundles of 1-meshes (see Rmk. \ref{rmk_mesh_bundles_concrete} for an elaboration). We assume all stratifications to be conical.

\begin{defn} A _1-mesh bundle_ $p : (M,f) \to (B,g)$ is a [[stratified space#stratified_bundle|stratified bundle]] with 1-mesh fibers $(M_x, f_x)$, for $x \in B$, and a fiberwise mesh-trivializing bundle embedding $p \hookrightarrow (\pi_B : B \times \mathbb{R} \to B)$ into the trivial $\mathbb{R}$-bundle over $B$, such that

1. $\mathcal{E}(p)$ is an [[Conduché functor|exponentiable fibration]],
2. $\mathcal{E}(p_{(0)})$ is an [[Grothendieck fibration#opfibrations_and_bifibrations|opfibration]],
3. $\mathcal{E}(p_{(1)})$ is a [[Grothendieck fibration|fibration]],

where $p_{(i)}$ denotes the restriction of $p$ to the union of strata in its domain whose non-empty intersections with fibers are all of dimension $i$.
\end{defn}

\begin{rmk} \label{rmk_mesh_bundles_concrete} A concrete way to think about 1-mesh bundles is explained in [Dorn-Douglas 2021, Ch. 4](#DornDouglas21): a 1-mesh bundle $p$ is a stratified bundle of 1-meshes that can be trivialized in the trivial framed line bundle, such that the image of the trivialization has continuous upper and lower fiberwise bounds, and such that entrance paths $b \to b'$ in the base, with $p(x) = b$ for $x$ a point in a $0$-stratum in the fiber over $b$, have unique lifts to entrance paths $x \to x'$.
\end{rmk}

\begin{defn} An _$n$-mesh bundle_ $M$ over a base stratification $(B,g)$ is a tower of 1-mesh bundles
$$
    (M_n, f_n) \to (M_{n-1}, f_{n-1}) \to ... \to (M_1, f_1) \to (M_0,f_0) = (B,g)
$$
If $(B,g) = \ast$ is the trivial stratification then the $n$-mesh bundle is simply called an _$n$-mesh_ $M$.
\end{defn}

\begin{rmk} There are several equivalent phrasings and variations of the definitions of 1- and $n$-mesh bundles. For instance, the assumption of contractability in the definition of 1-meshes can be weakened.
\end{rmk}

\begin{terminology} \label{closed_and_open_meshes} An $n$-mesh $M$ is call _closed_ if $M_n$ is [[compact]], and _open_ if $M_n \cong \mathbb{R}^n$.
\end{terminology}

\begin{defn} An _$n$-mesh map_ $F : M \to N$ is a stratified map of towers that fiberwise preserves the framing. (In fact, this makes mesh maps framed maps in the sense of [[directed topological space#framed_spaces|framed directed spaces]].)
\end{defn}

## Properties

### Cell structures

Closed meshes are regular cell complexes (cf. [Wikipedia entry](https://en.wikipedia.org/wiki/CW_complex#Regular_CW_complexes)). More generally, meshes $M$ are [[stratified space#fundamental_categories|stratified $0$-types]] (meaning the fundamental $\infty$-categories $\mathcal{E}(M_i,f_i)$ are $0$-[[truncated]] as $\infty$-categories, i.e. equivalent to [[posets]]).

### Classification

The fundamental categories $\mathcal{E}(M) = \mathcal{E}(f_n \to f_{n-1} \to ... \to f_0)$ of meshes are certain towers of poset maps 'with structure': this gives rise to a combinatorial notion called [[trusses]]. In fact, $n$-meshes up to framed stratified homeomorphism are classified by $n$-trusses (see [[n-truss#combinatorial_meshes|at trusses]] for details), and $n$-mesh bundles up to bundle isomorphism are classified by $n$-truss bundles (and thus share the same [[n-truss#truss_bundle_classifications|classifying category]]).

### Duality

As a consequence of [[n-truss#duality|truss duality]] and of the classification of meshes by trusses, we obtain a involutive [[dualization]] endofunctor on the category of meshes. This functor can be understood as a geometric dualization operation in the sense of [[Poincare duality]]. It maps open to closed meshes and vice versa (see Trm. \ref{closed_and_open_meshes}).

## Related concepts

* [[n-truss|n-trusses]]

* [[manifold diagrams]]

* [[framed combinatorial topology]]

## References

* {#DornDouglas22} [[Christoph Dorn]] and [[Christopher Douglas]], _Manifold diagrams and tame tangles_, 2022 ([arXiv](https://arxiv.org/abs/2208.13758), [latest](https://cxdorn.github.io/manifold-diagram-paper/))

* {#DornDouglas21} [[Christoph Dorn]] and [[Christopher Douglas]], _Framed combinatorial topology_, 2021 ([arXiv](https://arxiv.org/abs/2112.14700), [latest](https://cxdorn.github.io/fct-book/))

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n-monomorphism

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## Definition

For $n \in \mathbb{N}$
a [[morphism]] $f \colon X \to Y$ in an [[(infinity,1)-category]] is an **$n$-monomorphism** equivalently if

* in its [[n-image]] factorization

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

  the first morphism is an [[equivalence in an (infinity,1)-category]]

* it is an [[n-connected|(n-2)-truncated morphism]];

### In homotopy type theory

In [[homotopy type theory]], a [[function]] $f \colon X \to Y$ is an $n$-monomorphism if its $n$-image factorization is via an [[equivalence in homotopy type theory]].

More explicitly, given a natural number $n:\mathbb{N}$ and types $A$ and $B$, a function $f:A \to B$ is a __$n$-monomorphism__ if for all terms $b:B$ the [[fiber]] of $f$ over $b$ has an [[homotopy level]] of $n$.

$$isMonic(n, f) \coloneqq \prod_{b:B} hasHLevel(n, fiber(f, b))$$

A [[equivalence]] is a $0$-monomorphism. $1$-monomorphisms are typically just called __monomorphisms__ or __[[embeddings]]__.

The type of all $n$-monomorphisms with domain $A$ and codomain $B$ is defined as
$$Monos(n, A, B): \sum_{f:A \to B} isMonic(n, f)$$

## Dual concept

The dual concept is that of _[[n-epimorphism]]_.

## Examples

* $0$-monomorphism are precisely the [[equivalences]].

* Every morphism is an $\infty$-monomorphism.

* 1-monomorphisms are often just called [[monomorphisms in an (∞,1)-category]]. The 1-monomorphisms into a fixed object are called the [[subobject in an (∞,1)-category|subobjects]] of that object.

* A 1-monomorphism between [[0-truncated]] objects is precisely an ordinary [[monomorphism]] in the underlying [[1-topos]].

## Related concepts

* [[Postnikov system]]

* [[(n-connected, n-truncated) factorization system]]

* [[subtype]]

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n-plectic form

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## Idea

The notion of _$n$-plectic form_ is a generalization of the notion of [[symplectic form]] to [[differential forms]] of more than two arguments. This is considered in [[higher symplectic geometry]], specifically: in [[n-plectic geometry]]/[[multisymplectic geometry]].

## Definition

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###### Definition

For $X$ a [[smooth manifold]] and $n \in \mathbb{N}$, $n \geq 1$, a [[differential form]] $\omega$ on $X$ is **$n$-plectic**
if

1. it is an $(n+1)$-form, $\omega \in \Omega^{n+1}(X)$;

1. it is closed: $d_{dR} \omega = 0$;

1. it is non-degenerate in that the [[contraction]] map

   $$
     \iota_{(-)}\omega \;\colon\; \Gamma(T X) \to \Omega^{n}(X)
   $$

   has trivial [[kernel]].

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###### Remark

An 1-plectic form is equivalently a [[symplectic form]].

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###### Remark

If the last condition is dropped, then by analogy with [[presymplectic forms]] one may speak of a _pre-$n$-plectic_ form. Of course this is just a closed $(n+1)$-form, but as in the presymplectic case, the plectic-terminology indicates that one wants to regard it as input datum for [[higher geometric quantization]].

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###### Remark

This definition has an evident generalization to the case where also $X$ is allowed to be a "higher" generalization of a smooth manifold, namely a [[smooth ∞-groupoid]] or [[L-∞ algebroid]]. See [[higher symplectic geometry]] for more on this case.

=--

## References

See the references at _[[n-plectic geometry]]_ and at _[[multisymplectic geometry]]_.

For instance definition 2.1 in

* [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis ([arXiv](http://arxiv.org/abs/1106.4068)).

See also

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _Higher geometric prequantum theory_ ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))

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n-plectic geometry

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## Idea

$n$-plectic geometry is a generalization of [[symplectic geometry]] to [[higher category theory]].

It is closely related to [[multisymplectic geometry]] and [[n-symplectic manifold]]s.

## Definition

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###### Definition

For $n \in \mathbb{N}$, an **[[n-plectic vector space]]**
is a [[vector space]] $V$ (over the [[real numbers]]) equipped with an $(n+1)$-linear skew function

$$
  \omega : \wedge^{n+1} V \to \mathbb{R}
$$

such that regarded as a function

$$
  V \to \wedge^n V^*
$$

is has trivial [[kernel]].

=--

Let $X$ be a [[smooth manifold]], $\omega \in \Omega^{n+1}(X)$ a [[differential form]].

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

We say $(X,\omega)$ is a **$n$-plectic manifold** if

* $\omega$ is closed: $d \omega = 0$;

* for all $x \in X$
  the map

  $$
    \hat \omega : T_x X \to \Lambda^n T_x^* X
  $$

  given by contraction of vectors with forms

  $$
    v \mapsto \iota_v \omega
  $$

  is [[injective]].

=--

See also the definition at _[[multisymplectic geometry]]_.

## Examples

* $n = 1$ -- this is the case of an ordinary [[symplectic manifold]] this appears in [[Hamiltonian mechanics]];

* $n = 2$, this appears naturally in 1+1 dimensional [[quantum field theory]].

1. For $X$ orientable, take $\omega$ the [[volume form]]. This is $(dim(X)-1)$-plectic.

1. $\wedge^n T^* X \to X$

1. $G$ a [[compact space|compact]] [[simple group|simple]] [[Lie group]],

   let $\nu : (x,y,z) \mapsto \langle x, [y,z]\rangle$ be the canonical
   [[Lie algebra cohomology|Lie algebra 3-cocycle]]
   and extend it left-invariantly
   to a 3-form $\omega_\nu$ on $G$.  Then $(G,\omega_\nu)$ is 2-plectic.

## Poisson $L_\infty$-algebras
 {#PoissonLInfinityAlgebras}

To an ordinary [[symplectic manifold]] is associated its [[Poisson algebra]]. Underlying this is a [[Lie algebra]], whose Lie bracket is the _Poisson bracket_.

We discuss here how to an $n$-plectic manifold for $n \gt 1$ there is correspondingly assoociated not a Lie algebra, but a [[Lie n-algebra]]: the **[[Poisson bracket Lie n-algebra]]**. It is natural to call this a **Poisson Lie $n$-algebra** (see for instance at _[[Poisson Lie 2-algebra]]_).

(Not to be confused with the Lie algebra of a [[Poisson Lie group]], which is a Lie group that itself is equipped with a compatible [[Poisson manifold]] structure. It is a bit unfortunate that there is no better established term for the Lie algebra underlying a Poisson algebra apart from "Poisson bracket".)

Consider the ordinary case in $n=1$ for how a [[Poisson algebra]] is obtained from a [[symplectic manifold]] $(X, \omega)$.

Here

$$
  \hat \omega : T_x X \to T^*_x X
$$

is an [[isomorphism]].

Given $f \in C^\infty(X)$, $\exists ! \nu_f \in \Gamma(T X)$ such that  $d f = - \omega(v_f, -)$

Define $\{f,g\} := \omega(v_f, v_g)$. Then $(C^\infty(X,), \{-,-\})$ is a [[Poisson algebra]].

We can generalize this to $n$-plectic geometry.

Let $(X,\omega)$ be $n$-plectic for $n \gt 1$.

Observe that then $\hat \omega : T_x X \to \wedge^n T_x X$ is no longer an [[isomorphism]] in general.

**Definition**

Say

$$
  \alpha \in \Omega^{n-1}(X)
$$

is **Hamiltonian** precisely if

$$
  \exists v_\alpha \in \Gamma(T X)
$$

such that

$$
  d \alpha = - \omega(v_\alpha, -)
  \,.
$$

This makes $v_\alpha$ uniquely defined.

Denote the collection of [[Hamiltonian forms]] by $\Omega^{n-1}_{Hamilt}(X)$.

Define a bracket

$$
  \{-,-\} : \Omega^{n-1}_{Hamilt}(X)^{\otimes_2} \to \Omega^{n-1}_{Hamilt}(X)
$$

by

$$
  \{\alpha, \beta\} = - \omega(v_\alpha, v_\beta, -, \cdots, -)
  \,.
$$

This satisfies

1. k

   $$
    d \{\alpha, \beta\} = - \omega([v_\alpha, v_\beta], -, \cdots, -)
    \,.
   $$

1. $\{-,-\}$ is skew-symmetric;

1. $\{\alpha_1, \{\alpha_2, \alpha_3\}\}$ + cyclic permutations
   $d \omega(v_{\alpha_1}, v_{\alpha_2}, v_{\alpha_3}, -, \cdots)$.

So the Jacobi dientity fails up to an exact term. This will yield the structure of an [[L-infinity algebra]].

**Proposition**

Given an $n$-plectic manifold $(X,\omega)$ we get a [[Lie n-algebra]] structure on the complex

$$
  C^\infty(X) \stackrel{d_{dR}}{\to} \Omega^1(X) \stackrel{d_{dR}}{\to}
  \to \cdots \to
  \Omega^{n-1}_{Hamilt}(X)
$$

(where the rightmost term is taken to be in degree 0).

where

* the unary bracket is $d_{dR}$;

* the $k$-ary bracket is

  $$
    [\alpha_1, \cdots, \alpha_k]
    =
    \left\{
      \array{
        \pm \omega(v_{\alpha_1}, \cdots, v_{\alpha_k}) & if \forall i : \alpha_i \in \Omega^{n-1}_{Hamilt}(X)
        \\
        0 & otherwise
      }
    \right.
  $$

This is the **[[Poisson bracket Lie n-algebra]]**.

This appears as ([Rogers 11, theorem 3.14](#Rogers11)).

For $n = 1$ this recovers the definition of the [[Lie algebra]] underlying a [[Poisson algebra]].

## Prequantization

### Review of the symplectic situation

Recall for $n=1$ the mechanism of [[geometric quantization]] of a [[symplectic manifold]].

Given a 2-form $\omega$ and the corresponding complex [[line bundle]] $P$, consider the [[Atiyah Lie algebroid]] sequence

$$
  ad P \to T P/U(1) \to T X
$$

The smooth [[section]]s of  $T P/U(1) \to X$ are the $U(1)$ invariant [[vector field]]s on the total space of $P$.

Using a [[connection on a bundle|connection]] $\nabla$ on $P$ we may write such a section as

$$
  s(v) + f \partial_t
$$

for $v \in \Gamma(T X)$ a vector field downstairs, $s(v)$ a horizontal lift with respect to the given connection and $f \in C^\infty(X)$.

Locally on a suitable patch $U \subset X$ we have that $s(V)|_U = v|_U + \iota_v \theta_i|_U$ .

We say that $\tilde v = s(v) + f \partial_t$ **preserves the splitting** iff $\forall u \in \Gamma(X)$ we have

$$
  [\tilde v, s(u)] = s([v,u])
  \,.
$$

One finds that this is the case precisely if

$$
  d f = - \iota_v \omega
  \,.
$$

This gives a [[homomorphism]] of [[Lie algebra]]s

$$
  C^\infty(X) \to \Gamma(T P / U(1))
$$

$$
  f \mapsto s(v_f) + f \partial_t
  \,.
$$

### 2-plectic geometry and Courant algebroids

We consder now [[geometric quantization|prequantization]] of [[2-plectic manifolds]].

Let $(X,\omega)$ be a 2-plectic manifold such that the [[de Rham cohomology]] class $[\omega]/2 \pi i$ is in the image of [[integral cohomology]] (Has integral periods.)

We can form a cocycle in [[Deligne cohomology]] from this, encoding a [[bundle gerbe]] with connection.

On a [[cover]] $\{U_i \to X\}$ of $X$ this is given in terms of [[Cech cohomology]] by data

* $(g_{i j k} : U_{i j k} \to U(1)) \in C^\infty(U_{i j k}, U(1))$

* $A_{i j} \in \Omega^1(U_{i j})$;

* $B_i \in \Omega^2(U_i)$

satisfying a cocycle condition.

Now recall that an exact [[Courant algebroid]] is given by the following data:

* a [[vector bundle]] $E \to X$;

* an _anchor_ morphism $\rho : E \to T X$ to the [[tangent bundle]];

* an _inner product_ $\langle -,-\rangle$ on the fibers of $E$;

* a bracket $[-,-]$ on the sections of $E$.

Satisfying some conditions.

The fact that the [[Courant algebroid]] is _exact_ means that

$$
  0 \to T^* X \to E \to T X \to 0
$$

is an [[exact sequence]].

The **standard Courant algebroid** is the example where

* $E = T X \oplus T^* X$;

* $\langle v_1 + \alpha_1, v_2 + \alpha_2\rangle = \alpha_2(v_1) + \alpha_1(v_2)$;

* the bracket is the skew-symmetrization of  the Dorfman bracket

  $$
    (v_1 + \alpha_1, v_2 + \alpha_2) = [v_1, v_2] - \mathbb{L}_{v_1}\alpha_2
    - (d \alpha_1)(v_2,-)
  $$

Now with respect to the above Deligne cocycle, build a Courant algebroid as follows:

* on each patch $U_i$ is is the standard Courant algebroid
  $E_i := T U_i \oplus T^* U_i$;

* glued together on double intersections using the $d A_{i j}$

This gives an exact Courant algebroid $E \to X$ as well as a splitting $s : T X \to E$ given by the $\{B_i\}$.

The bracket on this $E$ is given by the skew-symmetrization of

$$
  [ [ s(v_1)  \alpha_1, s(v_2) + \alpha_2 ] ] = s([v_1, v_2])
  +
  \mathcal{L}_{v_1} \alpha_2 - (d \alpha_2)(v_2, -) -
  \omega(v_1, v_2, \cdots)
  \,.
$$

Here a section $e = s(v) + ... $ preserves the splitting precisely if

for all $u \in \Gamma(T X)$ we have

$$
  [ [ e, s(u)] ]_D = s([v,u])
$$

exactly if $\alpha$ is Hamiltonian and $v = v_\alpha$.

**Theorem**

Recall that to every [[Courant algebroid]] $E$ is associated a [[Lie 2-algebra]] $L_\infty(E)$.

Then: we have an embedding of [[L-infinity algebra]]s

$$
  \phi : L_\infty(X,\omega) \to L_\infty(E)
$$

given by $\phi(\alpha) = s(v_\alpha) + \alpha$.

## Related entries

* [[geometry of physics -- prequantum geometry]]

## Properties

### Central extensions under geometric quantization

[[!include geometric quantization extensions - table]]

## Related concepts

* [[2-plectic geometry]]

* [[symplectic infinity-groupoid]]

* [[higher geometric quantization]]

[[!include Isbell duality - table]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References
 {#References}

### General
 {#ReferencesGeneral}

The observation that the would-be Poisson bracket induced by a higher degree closed form extends to the [[Poisson bracket Lie n-algebra]] is due to

* {#Rogers10} [[Chris Rogers]], _$L_\infty$ algebras from multisymplectic geometry_ , Letters in Mathematical Physics  **100** 1 (2012) 29-50  [[arXiv:1005.2230](http://arxiv.org/abs/1005.2230), [journal](http://link.springer.com/article/10.1007%2Fs11005-011-0493-x)]

* {#Rogers11} [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))

with first discussion of application to [[prequantization]] in

* [[Chris Rogers]], _2-plectic geometry, Courant algebroids, and categorified prequantization_ ,  [arXiv:1009.2975](http://arxiv.org/abs/1009.2975).

* [[Chris Rogers]], _Higher geometric quantization_, talk at _Higher Structures 2011_ in Göttingen ([pdf slides](http://www.crcg.de/wiki/Higher_geometric_quantization))

Discussion in the more general context of [[higher differential geometry]]/[[extended prequantum field theory]] is in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]],

  _[[schreiber:Higher geometric prequantum theory]]_,

  _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_

See also the references at [[multisymplectic geometry]] and [[n-symplectic manifold]].

A [[higher differential geometry]]-generalization of [[contact geometry]] in line with [[multisymplectic geometry]]/$n$-plectic geometry is discussed in

* Luca Vitagliano, _L-infinity Algebras From Multicontact Geometry_ ([arXiv.1311.2751](http://arxiv.org/abs/1311.2751))

### Applications

Some more references on application, on top of those mentioned in the articles [above](#ReferencesGeneral).

A survey of some (potential) applications of 2-plectic geometry in [[string theory]] and [[M2-brane]] models is in section 2 of

* [[Christian Saemann]], [[Richard Szabo]], _Groupoid quantization of loop spaces_ ([arXiv:1203.5921](http://arxiv.org/abs/1203.5921))

and in

* [[Christian Saemann]], [[Richard Szabo]], _Quantization of 2-Plectic Manifolds_ ([arXiv:1106.1890](http://arxiv.org/abs/1106.1890))

### Examples

On $n$-plectic formulation of [[gravity]], [[Maxwell theory]] and [[Einstein-Maxwell theory]]:

* [[Petr Hořava]], _On a covariant Hamilton-Jacobi framework for the Einstein-Maxwell theory_ Classical and Quantum Gravity **8** 11 (1991) 2069 [[doi:10.1088/0264-9381/8/11/016](https://iopscience.iop.org/article/10.1088/0264-9381/8/11/016)]

* [[Dmitri Vey]], _$n$-Plectic Maxwell Theory_ [[arxiv:1303.2192](https://arxiv.org/abs/1303.2192)]

* [[Dmitri Vey]], *Multisymplectic formulation of vielbein gravity. De Donder-Weyl formulation, Hamiltonian $(n-1)$-forms*,  Classical and Quantum Gravity **32** 9 (2015) [[arxiv:1404.3546](https://arxiv.org/abs/1404.3546), [doi:10.1088/0264-9381/32/9/095005](https://iopscience.iop.org/article/10.1088/0264-9381/32/9/095005)[

For [[teleparallel gravity]]

* [[Igor V. Kanatchikov]]. *The De Donder-Weyl Hamiltonian formulation of TEGR and its quantization* (2023) [[arXiv:2308.10052](https://arxiv.org/abs/2308.10052)]

[[!redirects n-plectic manifold]]
[[!redirects n-plectic manifolds]]

[[!redirects Hamiltonian 1-form]]
[[!redirects Hamiltonian 1-forms]]
[[!redirects Hamiltonian n-form]]
[[!redirects Hamiltonian n-forms]]

[[!redirects n-plectic structure]]
[[!redirects n-plectic structures]]

[[!redirects 3-plectic geometry]]

[[!redirects pre-n-plectic manifold]]
[[!redirects pre-n-plectic manifolds]]


n-plectic vector space

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the notion of [[symplectic vector space]] from [[symplectic geometry]] to [[n-plectic geometry]].

## Definition

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

For $n \in \mathbb{N}$, an **[[n-plectic vector space]]**
is a [[vector space]] $V$ (over the [[real numbers]]) equipped with an $(n+1)$-linear skew function

$$
  \omega : \wedge^{n+1} V \to \mathbb{R}
$$

such that regarded as a function

$$
  V \to \wedge^n V^*
$$

is has trivial [[kernel]].

=--

## Related concepts

* [[n-plectic manifold]]

* [[Heisenberg Lie n-algebra]]

[[!redirects n-plectic vector spaces]]


n-point function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[relativistic field theory|relativistic]] [[quantum field theory]] an [[observable]] that evaluates the [[time-ordered products]] of the point-evaluation observables of the basic [[field (physics)|fields]] at $n$ points of [[spacetime]] in some [[state on a star-algebra]] is called an _$n$-point function_, typically denoted

$$
  \left\langle
    :
    \mathbf{\Phi}(x_1)
    \mathbf{\Phi}(x_2)
    \cdots
    \mathbf{\Phi}(x_n)
    :
  \right\rangle
$$

For details on what this means see at _[[geometry of physics -- perturbative quantum field theory]]_ chapter _[7. Observables](geometry+of+physics+--+perturbative+quantum+field+theory#Observables)_ and chapter [15. Interacting quantum fields](geometry+of+physics+--+perturbative+quantum+field+theory#QuantumObservables)

Specifically the 2-point functions are also known as _[[Feynman propagators]]_, see Section _[9. Propagators](geometry+of+physics+--+perturbative+quantum+field+theory#Propagators)_.

In [[Euclidean field theory]] (say under [[Wick rotation]]) $n$-point functions are also called _[[correlators]]_, but in fact both terms are often used interchangeably.

Traditionally $n$-point functions are thought of as [[distributions]] [[distribution of several variables|of several variables]]. In [[relativistic field theory]] these have [[singularities]] on the "relative [[light cones]]", hence whenever two points $x_i$ are [[lightlike]].

On the other hand, in [[Euclidean field theories]] the $n$-point functions/correlators are distributions with singularities only on the [[fat diagonal]], hence when at least two of their arguments coincide. This means that in [[Euclidean field theory]] $n$-point functions/correlators [[restriction of distributions|restrict]] to smooth (non-singular) [[differential forms]] on [[configuration spaces of points]]. For more on this perspective see at _[[correlators as differential forms on configuration spaces of points]]_.

## Related concepts

* [[Feynman amplitude]]

## References

Discussion in [[AdS-CFT]]:

* Adam Bzowski, Paul McFadden, Kostas Skenderis, *A handbook of holographic 4-point functions* [[arXiv:2207.02872](https://arxiv.org/abs/2207.02872)]

[[!redirects n-point functions]]

[[!redirects 2-point function]]
[[!redirects 3-point function]]


n-poset

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

# Contents
* table of contents
{: toc}

## Idea

An $n$-poset is any of several concepts that generalize [[partial order|posets]] in [[higher category theory]]. In fact, $n$-posets are the same as $(n-1,\infty)$-[[(n,r)-category|categories]].

## Definition

Fix a meaning of $(\infty,\infty)$-[[infinity-category|category]], however weak or strict you wish. Then an __$n$-poset__ is a $(n-1)$-[[truncation]] of an $(\infty,\infty)$-category. As all $j$-morphisms for $j \geq n$ are equivalent in an $(n-1)$-truncated $(\infty,r)$-category,  all [[parallel morphisms|parallel]] pairs of $j$-morphisms are [[equivalence|equivalent]] for $j \geq n$ by definition, which means that $(n-1,\infty)$-categories are the same as $(n-1,n)$-categories. Thus, up to equivalence, there is no point in mentioning anything beyond $n$-morphisms, not even whether two given parallel $n$-morphisms are equivalent. This definition makes sense as low as $n = -1$; the statement that parallel $(-1)$-morphisms are equivalent simply means that there exists an object (a $0$-morphism).

## Special cases

* The concept of $(-1)$-[[(-1)-poset|poset]] is trivial.

* A $0$-[[0-poset|poset]] is a [[truth value]].

* A $1$-[[1-poset|poset]] or [[(0,1)-category]] is simply a [[partial order|poset]].

  Because, by the definition of $(0,1)$-category, we have that any two $1$-morphisms with the same source and target are equivalent. Hence there is, up to equivalence, at most one morphism for every ordered pair of objects. The rest of the axioms say that this is all the information there is in a $(0,1)$-category. Therefore, by the discussion at _[poset -- As a category with extra properties](http://ncatlab.org/nlab/show/partial+order#AsACategoryWithExtraProperties)_, a $(0,1)$-category is a poset.  (See also [[thin category]].)

* A $2$-[[2-poset|poset]] or [[(1,2)-category]] is a [[locally posetal 2-category]].

* In general, an $n$-poset is an $n$-[[n-category|category]] in which all parallel pairs of $n$-morphisms are equal.

* An $\infty$-poset is the same thing as an $(\infty,\infty)$-[[infinity-category|category]].

In the light of the general definition, one must interpret 'is' up to [[equivalence of categories]].  The last statement also depends on how strict your definition of $(\infty,\infty)$-category or $n$-category is; it is actually simpler to define $n$-posets from scratch as given above than to define them in terms of $n$-categories.

## Basic theorems

The $(\infty,\infty)$-category of ([[small category|small]]) $n$-posets, as a [[full subcategory|full sub-(∞,∞)-category]] of the $(\infty,\infty)$-category of $(\infty,\infty)$-categories, is an $(n+1)$-poset.  That is, $n$-posets form an $(n+1)$-poset.  This is well known for small values of $n$.

[[!redirects n-poset]]
[[!redirects n-posets]]
[[!redirects (n-1,n)-category]]
[[!redirects (n-1,n)-categories]]
[[!redirects (n-1,infinity)-category]]
[[!redirects (n-1,infinity)-categories]]
[[!redirects (n-1,∞)-categories]]
[[!redirects (n-1,∞)-categories]]


n-quasicategory

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

#Contents#
* table of content
{:toc}

## Idea

Where a [[quasicategory]] is a model for the notion of [[(infinity,1)-category]] so an _$n$-quasicategory_ is a model for the notion of [[(infinity,n)-category]] for all $n \in \mathbb{N}$.

A definition in terms of [[cellular sets]] had originally been suggested by [[André Joyal]]. This has then been made to work in terms of a [[model structure on cellular sets]] (see there for more details) in ([Ara 12](#Ara12)).

## Related concepts

* [[Theta-space]]

## References

* [[Dimitri Ara]], _Higher quasi-categories vs higher Rezk spaces_ ([arXiv:1206.4354](http://arxiv.org/abs/1206.4354))
 {#Ara12}

A survey is in

* [[Dimitri Ara]] _On higher quasicategories_, 2013 ([pdf](http://home.sandiego.edu/~shulman/cmshighercategories2013/Ara.pdf))

[[!redirects n-quasicategories]]
[[!redirects n-quasicategories]]


n-reduced (∞,1)-functor

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

#Contents#
* table of content
{:toc}

## Definition

In the context of [[Goodwillie calculus]], an [[(∞,1)-functor]] is called $n$-reduced for $n \in \mathbb{N}$, $n \gt 0$, if its [(n-1)-excisive approximation](n-excisive+functor#nExcisiveApproximation) is trivial, $P_{n-1} F \simeq \ast$ (hence if it is a $P_{n-1}$ [[anti-modal type]]).

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

Hence a functor is 1-reduced (or just _reduced_, for short), if $F(\ast) \simeq \ast$.

A functor that is both [[n-excisive functor|n-excisive]] and $n$-reduced is called an  _[[n-homogeneous (∞,1)-functor]]_.

## References

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

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

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

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

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

[[!redirects reduced functor]]
[[!redirects reduced functors]]


n-topos

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

#Contents#
* table of contents
{:toc}

## Idea

An _$n$-topos_ is an [[n-category]] analog of a [[topos]].

An $n$-topos that is an [[(n,1)-category]], hence where all [[k-morphism]]s for $k \geq 2$ are [[equivalence in an (infinity,1)-category|equivalences]] is called an [[(n,1)-topos]]. See there for more.

## Examples

For every $n$, The canonical $(n+1)$-topos is [[nCat]], the [[n-category|(n+1)-category]] of [[n-categories]].

## Related concepts

[[!include flavors of higher toposes -- list]]

## References

* [[Michael Shulman]], _[[michaelshulman:n-topos for large n]]_ .

[[!redirects n-toposes]]
[[!redirects n-topoi]]


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

# Contents
* table of contents
{: toc}

## Idea

An **$n$-truncated [[∞-groupoid]]** is an [[n-groupoid]].

An **$n$-truncated [[topological space]]** is a [[homotopy n-type]]: all [[homotopy groups]] above degree $n$ are trivial.

An **$n$-truncated object** in a general [[(∞,1)-category]] is an object such that all [[(∞,1)-categorical hom-space|hom-∞-groupoids]] into it are $n$-truncated.

If an object in an [[(∞,1)-topos]] is $k$-truncated for any (possibly large) $k$, then it is $n$-truncated precisely if all its [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]] above degree $n$ are trivial.

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

## Definition

### In terms of truncations
 {#InTermsOfTruncations}

+-- {: .num_defn #nTruncatedInfinityGroupoid}
###### Definition
**($n$-truncated $\infty$-groupoid)**

An [[∞-groupoid]] $A \in \infty Grpd$ is **$n$-truncated** for $n \in \mathbb{N}$ if it is an [[n-groupoid]]:

Precisely: in the model of [[∞-groupoid | ∞-groupoids]] given by [[Kan complex | Kan complexes]] $A$ is **$n$-truncated** if the [[simplicial homotopy group | simplicial homotopy groups]] $\pi_k(A,x)$ are trivial for all $x$ and all $k \gt n$.

=--

It makes sense for the following to adopt the convention that $A$ is called.

* _$(-1)$-truncated_ if it is empty or contractible -- this is a [[(-1)-groupoid]].

* $(-2)$-truncated if it is non-empty and contractible -- this is a [[(-2)-groupoid]].

(following [[Higher Topos Theory|HTT, p. 6]]).

To generalize this, let now $C$ be an arbitrary [[(∞,1)-category]]. For $X,A$ objects in $C$ write $C(X,A) \in $ [[∞Grpd]] for the [[(∞,1)-categorical hom-space]] (if $C$ is given as a [[simplicially enriched category]] then this is just the [[SSet]]-[[hom-object]] which is guaranteed to be a [[Kan complex]]).

Using this, it is useful to reformulate the above as follows slightly:

\begin{lemma}
An [[∞-groupoid]] $A$ is $n$-truncated precisely when for all [[∞-groupoids]] $X$ the [[(infinity,1)-categorical hom-space|hom-$\infty$-groupoid]] $\infty Grpd(X,A)$ is $n$-truncated.
\end{lemma}

In other category-theoretic terms this says that [[(n,k)-transformation|(∞,k)-transformation]] between $X$ and $A$, whose components are [[k-morphisms]] in $A$, cannot be nontrivial for $k \gt n$ if there are no nontrivial [[k-morphisms]] with $k \gt n$ in $A$.

Using this fact we can transport the notion of $n$-truncation to any [[(∞,1)-category]] by testing it on [[hom-infinity-groupoid|hom-$\infty$-groupoids]]:

+-- {: .num_defn #nTruncatedObject}
###### Definition
**($n$-truncated object in an $(\infty,1)$-category)**

Given an [[(∞,1)-category]] $\mathcal{C}$,
an [[object]] $A \in \mathcal{C}$ is called **$n$-truncated**, for $n \in \mathbb{N}$, if for all objects $X \in \mathcal{C}$ the [[hom-∞-groupoid]] $\mathcal{C}(X,A)$ is $n$-truncated according to Def. \ref{nTruncatedInfinityGroupoid}.

=--

This is [[Higher Topos Theory|HTT, def. 5.5.6.1]].

Some terminology:

* A 0-truncated object is also called **discrete**. Notice that this is _categorically discrete_ as in [[discrete category]], not discrete in the sense of [[discrete topological space]]. An object in an [[(∞,1)-topos]] is discrete in this sense if, regarded as an [[∞-groupoid]] with extra structure, it has only trivial morphisms.

* By the above convention on (-2)-truncated $\infty$-groupoids, it is only the [[terminal object | terminal objects]] of $C$ that are (-2)-truncated.

* Similarly, the (-1)-truncated objects are the [[subterminal objects]].

+-- {: .num_defn #nTruncatedMorphism}
###### Definition
**($n$-truncated morphism in an $(\infty,1)$-category)**

A [[morphism]] $f : X \to Y$ of [[∞-groupoids]] is $n$-truncated if all of its [[homotopy fibers]] are $n$-truncated by def. \ref{nTruncatedInfinityGroupoid}.

A morphism $f : X \to Y$ in an [[(∞,1)-category]] $C$ is $n$-truncated if for all $W \in C$ the postcomposition morphism

$$
  C(W,f) : C(W,X) \to C(W,Y)
$$

is $n$-truncated in [[∞Grpd]].

=--

This is [[Higher Topos Theory|HTT, def. 5.5.6.8]].

By the [characterization of homotopy fiber of functor categories](http://ncatlab.org/nlab/show/fiber+sequence#OfFuncCats)
this is equivalent to saying that $f$ is $k$-truncated when it is so regarded as an
object of the [[over quasi-category|over (∞,1)-category]] $C_{/Y}$. (See also [[Higher Topos Theory|HTT, rem. 5.5.6.12]].)

Unwinding the definitions and applying the [[long exact sequence of homotopy groups]], we have:

+-- {: .num_prop }
###### Proposition
For $n \geq -1$, a morphism $f : X \to Y$ of ∞-groupoids is $n$-truncated
iff, for every point $x \in X$, we have $\pi_{n+1}(X, x) \to \pi_{n+1}(Y, f(x))$ is a [[monomorphism]] and for every $k \geq n+2$, $\pi_k(X, x) \to \pi_k(Y, f(x))$ is an [[isomorphism]].

$f$ is $(-2)$-trunacated iff it is a [[weak homotopy equivalence]].
=--

### In terms of categorical homotopy groups

At least if the ambient [[(∞,1)-category]] is even an [[∞-stack]] [[(∞,1)-topos]] there is an alternative, more intrinsic, characterization of $n$-truncation in terms of [[categorical homotopy groups in an (∞,1)-topos]]:

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

Suppose that an object $X$ in an [[∞-stack]] [[(∞,1)-topos]] is $k$-truncated for _some_ $k \in \mathbb{N}$ (possibly very large).

Then for any $n \in \mathbb{N}$ this $X$ is $n$-truncated precisely if all the [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]] above degree $n$ are trivial.

=--

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

This is [[Higher Topos Theory|HTT, prop 6.5.1.7]].

=--

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

Notice that this expected statement does require the assumption that $X$ is $k$-truncated for some $k$. Without any _a priori_ truncation assumption on $X$, there is no comparable statement about the relation to categorical homotopy groups. See [[Higher Topos Theory|HTT, remark 6.5.1.8]].

=--

## Properties

### Recursive definition
 {#RecursiveDefinition}

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

In an $(\infty,1)$-category $C$ with [[finite limits]], a morphism
$f : X \to Y$ is $k$-truncated (for $k \geq -1$) precisely if the
[[diagonal]] morphism $X \to X \times_Y X$ is $(k-1)$-truncated.

=--

This is [[Higher Topos Theory|HTT, lemma 5.5.6.15]].

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

By definition $f$ is $k$-truncated if for each object
$d \in C$ we have that $C(d,f)$ is $k$-truncated in
[[∞Grpd]]. Since the [[hom-functor | hom-functors]] $C(d,-)$ preserve
[[(∞,1)-limits]], we have
in particular that $X \to X \times_Y X$ in $C$
is $k$-truncated
if $C(d,X) \to C(d,X) \times_{C(d,Y)} C(d,X)$ is
$k$-truncated for all $d$ in [[∞Grpd]]. Therefore it is
sufficient to prove the statement for
morphisms in $C =$ [[∞Grpd]].

So let now $f : X \to Y$ be a morphism of [[∞-groupoid | ∞-groupoids]].
We may find a fibration $\bar \phi : \bar X \to \bar Y$
between [[Kan complex|Kan complexes]]  in [[sSet]] that models $f : X \to Y$
in the standard [[model structure on simplicial sets]], and by the
standard rules for [[homotopy pullback | homotopy pullbacks]] it follows that the object
$X \times_Y X$ in $\infty$-Grpd is then modeled by the ordinary
[[pullback]] $\bar X \times_{\bar Y} \bar X$ in [[sSet]]. And the
homotopy fibers of $f$ over $y \in Y$ are then given by the ordinary fibers
$\bar X_y$ of $\bar f$ in $sSet$.

This way, the statement is reduced to the following fact: a [[Kan complex]]
$\bar X_y$ is $k$-truncated precisely if the [[homotopy fiber | homotopy fibers]] of
$\bar X_y \to \bar X_y \times \bar X_y$ are $(k-1)$-truncated.

We now write $X$ for $\bar X_y$ for simplicity. To see the last statement,
let $(a,b) : * \to X \times X$ and compute the [[homotopy pullback]]

$$
  \array{
     Q &\to& *
     \\
     \downarrow && \downarrow^{\mathrlap{(a,b)}}
     \\
     X &\to& X \times X
  }
$$

as usual by replacing the right vertical morphism with the fibration
$(X \times X)^I \times_{X \times X} (a,b) \to X \times X$
and then forming the ordinary pullback. This shows that
$Q$ is equivalent to the space of paths $P_{a,b}X$ in $X$ from $a$ to $b$.
(Use that gluing of [[path space object | path space objects]] at endpoints of
paths produces a new
path space; see, for instance, section 4 of [[BrownAHT]]).

If $X$ is [[connected]], then choosing any path $a \to b$
gives an isomorphism from the homotopy groups of $P_{a,b} X$
to those of the [[loop space]] $\Omega_a X$. These latter are indeed those
of $X$, shifted down in degree by one (as described, for instance, at
[[fiber sequence]]).

If $X$ is not connected, we can easily reduce to the case that it is.

=--

### General
 {#PropertiesGeneral}

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

For $C$ an $(\infty,1)$-category and $k \geq -2$, the full [[sub-(∞,1)-category]] $\tau_{\leq k} C$ is stable under all [[limit in a quasi-category|$\infty$-limits]] in $C$.

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.6.5]].

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

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

=--

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

### Model category presentations
 {#ModelCategoryPresentations}

There are [[model structures for homotopy n-types]] that [[presentable (∞,1)-category]] present the full [[sub-(∞,1)-categories]] of $n$-truncated objects in some ambient $(\infty,1)$-category. See there for more details.

## Truncation

### Definition

Under mild conditions there is for each $n$ a universal way to send an arbitrary object $A$ to its $n$-truncation $\tau_{\leq n} A$. This is a general version of [[decategorification]] where [[k-morphism|n-morphisms]] are identified if they are connected by an invertible $(n+1)$-morphism.

For $C$ an [[(∞,1)-category]] and $n \geq -2$ in $\mathbb{Z}$ write $\tau_{\leq n} C $ for the [[full subcategory]] of $C$ on its $n$-truncated objects.

So for instance for $C = $ [[∞Grpd]] we have $\tau_{\leq n} \infty Grpd = n Grpd$.

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

If $C$ is an [[(∞,1)-category]] that is [[presentable (∞,1)-category|presentable]] then the canonical inclusion [[(∞,1)-functor]]

$$
  \tau_{\leq n} C \hookrightarrow C
$$

has an [[accessible (infinity,1)-functor|accessible]] [[left adjoint]]

$$
  \tau_{\leq n} : C \to \tau_{\leq n} C
  \,.
$$

=--

This is [[Higher Topos Theory|HTT 5.5.6.18]].

Indeed, as the notation suggests, $\tau_{\leq n} C$ is the [[essential image]] of $C$ under $\tau_{\leq n}$. The image  $\tau_{\leq n} A$ of an object $A$ under this operation is the **$n$-truncation** of $A$.

So $n$-truncated objects form a [[reflective sub-(∞,1)-category]]

$$
  \tau_{\leq n} C
  \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow}
  C
  \,.
$$

### Properties {#Properties}

#### General {#GeneralPropsTruncation}

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

For any small $\infty$-category $C$,
$\tau_{\leq n} PSh(C, \infty Grpd) \simeq PSh(C, n Grpd) $,
and truncation acts pointwise.

=--

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

If $P$ is an $n$-truncated ∞-presheaf, then
$P(c) \simeq PSh(C)(C(-, c), P)$ is $n$-truncated; thus
$P$ takes values in $n Grpd$.

Conversely, if $P$ takes values in $n Grpd$, then
the fact every presheaf is a colimit of representables
implies $hom(Q, P)$ is a limit of $n$-truncated spaces and is thus
$n$-truncated.

Given this identification of the subcategory of $n$-truncated objects,
we can see that the truncation-inclusion adjunction between
$n Grpd$ and $\infty Grpd$ induces an adjunction whose right
adjoint is the inclusion $PSh(C, n Grpd) \to
PSh(C, \infty Grpd)$

=--

\begin{corollary}
**(left exact $\infty$-functors preserve truncation)**
\linebreak
A [[left exact functor|left exact]] [[(infinity,1)-functor|$\infty$-functor]]
$F \colon C \to D$
between $(\infty,1)$-categories with [[finite (infinity,1)-limit|finite $\infty$-limits]] sends
$k$-truncated objects/morphisms to $k$-truncated objects/morphisms.
\end{corollary}
([[Higher Topos Theory|HTT, prop. 5.5.6.16]])
\begin{proof}
This follows from the above recursive characterization of
$k$-truncated morphisms by the $(k-1)$-truncation of their
diagonal, which is preserved by the finite limit preserving $F$.
\end{proof}

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

A left exact presentable $(\infty,1)$-functor $F : C \to D$  between
[[locally presentable (∞,1)-categories]] $C$ and $D$ commutes with
truncation:

$$
  F \circ \tau^C_{\leq k} \simeq \tau^D_{\leq k} \circ F
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, prop. 5.5.6.28]].

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

By the above lemma, $F$ restricts to a functor on the truncations. So we need to show that the diagram

$$
  \array{
    C &\stackrel{F}{\to}& D
    \\
    {}^{\mathllap{\tau_{\leq k}}}\downarrow
    & (?) &
    \downarrow^{\mathrlap{\tau_{\leq k}}}
    \\
    \tau_{\leq k } C &\stackrel{F}{\to}&
    \tau_{\leq k} D
  }
$$

in [[(∞,1)Cat]] can be filled by a 2-cell. To see this, notice that the
[[adjoint (∞,1)-functor]] of both composite morphisms exists
(because that of $F$ exists by the [[adjoint (∞,1)-functor theorem]]
and bcause adjoints of composites are composites of adjoints)
and since the bottom morphism is just the restriction of the top morphism
and the right adjoints of the vertical morphisms are full inclusions
this adjoint diagram

$$
  \array{
    C &\stackrel{G}{\leftarrow}& D
    \\
    \uparrow
    & &
    \uparrow
    \\
    \tau_{\leq k } C &\stackrel{G}{\leftarrow}&
    \tau_{\leq k} D
  }
  \,.
$$

evidently commutes since it just expresses this restriction.

=--

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

If $C$ is an [[(∞,1)-topos]], then truncation $\tau_{\leq n} : C \to C$ preserves [[finite (∞,1)-limit|finite]] [[products]].

=--

This appears as [[Higher Topos Theory|HTT, lemma 6.5.1.2]].

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

First notice that the statement is true for $C = $ [[∞Grpd]]. For instance we can use the example [In ∞Grpd and Top](#InInfGrpd), model [[∞-groupoid|∞-groupoids]] by [[Kan complex|Kan complexes]] and notice that then $\tau_{\leq n}$ is given by the truncation functor $tr_{n+1} : sSet \to [\Delta^{op}_{\leq n+1}, Set]$. This is also a [[right adjoint]] and as such preserves in particular products in $sSet$, which are $(\infty,1)$-products in $\infty Grpd$.

From that we deduce that the statement is true for  $C$ any [[(∞,1)-category of (∞,1)-presheaves]] $C = PSh_{(\infty,1)}(K) = Func_{(\infty,1)}(K^{op}, \infty Grpd)$ because all relevant operations there are objectwise those in $\infty Grpd$.

So far, this shows even that on presheaf $(\infty,1)$-toposes, all products (not necessarily finite) are preserved by truncation.

A general [[(∞,1)-topos]] $C$ is (by definition) a left exact [[reflective sub-(∞,1)-category]] of a presheaf $(\infty,1)$-topos,

$$
  C \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}}
  PSh_{(\infty,1)}(K)
  \,.
$$

Let $\prod_{j} i(X_j)$ be the product of the objects in question
taken in $PSh(K)$. By the above there, we have an equivalence

$$
  \tau_{\leq k} \prod_j i(X_j)
  \stackrel{\simeq}{\to}
  \prod_j \tau_{\leq} i(X_j)
  \,.
$$

Now applying $L$ to this equivalence and using now that $L$ preserves the
_finite_ product, this gives an equivalence

$$
  L \tau_{\leq k} \prod_j i(X_j)
  \stackrel{\simeq}{\to}
  L \prod_j \tau_{\leq k} i(X_j)
  \simeq
  \prod_j L \tau_{\leq k} i(X_j)
$$

in $C$. The claim follows now with the above result that $L \circ \tau_{\leq n} \simeq \tau_{\leq n} \circ L$.

=--

#### Postnikov tower

By the fact that the truncation functor $\tau_{\leq n}$ is a [[left adjoint]], one obtains canonical morphisms

$$
  \tau_{\leq n}A  \to A
$$

as the [[adjunct]] of the [[identity]] on $A$, and then by iteration also canonical morphisms

$$
  \tau_{\leq (n+1)} A \to \tau_{\leq n} A
  \,.
$$

For any $A \in C$, the sequence

$$
  \cdots
   \to \tau_{\leq 2}A \to \tau_{\leq 1} A \to \tau_{\leq 0} A
$$

is the [[Postnikov tower in an (∞,1)-category]] of $A$. See there for more details.

#### Homotopy type theory syntax
 {#TruncationHomotopyTypeTheorySyntax}

Discussion of $n$-truncation of [[types]] in [[homotopy type theory]] via [[higher inductive types]] is in ([Brunerie](#Brunerie)). This sends a type to an [[h-level]] $(n+2)$-type. The $(-1)$-truncation in the context is forming the [[bracket type]] [[hProp]].

See at _[[n-truncation modality]]_.

### Relation to homotopy groups {#RelationToHomotopyGroups}

In an $(\infty,1)$-topos $C$ there is a notion of categorical [[homotopy groups in an (∞,1)-topos]]. For the $(\infty,1)$-topos [[∞Grpd]] given by the model of [[Kan complex|Kan complexes]] this coincides with the notion of [[simplicial homotopy groups]]:

+-- {: .num_lemma}
###### Observation

An object $A$ in the [[(∞,1)-topos]] [[∞Grpd]] is $n$-truncated precisely if its [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]] $\pi_k(A)$ vanish for all $k \gt n$.

=--

This simple relation between $n$-truncation and categorical homotopy groups is almost, but not exactly, true in an arbitrary [[(∞,1)-topos]].

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

Let $\mathbf{H}$ be an [[(∞,1)-topos]] and $A \in \mathbf{H}$ an $n$-truncated object.

Then

1. for $k \gt n$ we have for the [[homotopy groups in an (∞,1)-topos|categorical homotopy groups]] $\pi_k(A) = *$;

1. if (for $n \geq 0$) $\pi_n(A) = *$, then $X$ is in fact $(n-1)$-truncated.

=--

This implies

+-- {: .num_corollary}
###### Corollary

If $A \in \mathbf{H}$ is truncated at all (for any value), then it is $n$-truncated precisely if all categorical homotopy groups vanish $\pi_k(A) = *$ for $k \gt n$.

=--

**Notice.** If $A$, on the other hand, is not truncated at all, then all its homotopy groups may be trivial, and $A$ may still not be equivalent to the [[terminal object]]. This means that [[Whitehead's theorem]] may fail in a general [[(∞,1)-topos]] for untruncated objects. It holds, however, in [[hypercomplete (∞,1)-topos|hypercomplete (∞,1)-toposes]].

## Examples
 {#Examples}

### Truncated morphisms

#### General

A morphism $f : X \to 0$ is

* (-2)-truncated precisely if it is an
  [[equivalence in a quasi-category|equivalence]];

* (-1)-truncated precisely if it is a
  [[monomorphism in an (infinity,1)-category|monomorphism]].

#### Between groupoids
 {#TruncatedMorphismsBetweenGroupoids}

For morphisms between 1-[[groupoids]],
the notion of $n$-truncation for low $n$ reproduces standard concepts from
ordinary [[category theory]].

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

A [[functor]] $f : X \to Y$ between [[groupoids]], is $n$-truncated precisely when regarded as a morphism in [[∞Grpd]] it is

* for $n = -2$ -- an [[equivalence of categories]];

* for $n = -1 $ -- a [[full and faithful functor]];

* for $n = 0$ -- a [[faithful functor]].

=--

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

Notice that $f$ being [[faithful functor|faithful]] means precisely that
it induces a [[monomorphism]] on the first [[homotopy groups]].

For $x : * \to X$ any point and $F_{f(x)}$ the corresponding [[homotopy fiber]] of $f$,
the [[long exact sequence of homotopy groups]] gives that $\pi_1(F)$
is the [[kernel]] of an [[injective]] map

$$
  \cdots \to \pi_1(F) \to \pi_1(X) \hookrightarrow \pi_1(Y,f(x)) \to \cdots
  \,,
$$

hence $\pi_1(F_{y}) = *$ for all points $y$ in the essential image of $f$.
For $y$ not in the essential image we have $F_y \simeq \emptyset$. In either case, it follows that $F$ is 0-truncated.

By def. \ref{nTruncatedMorphism} this is the defining condition for $f$ to be 0-truncated.

=--

#### Between stacks
  {#TruncatedMorphismsBetweenStacks}

Let $C$ be a [[site]] and write
$Sh_{(2,1)}(C) \hookrightarrow Sh_{(\infty,1)}(C)$ for the [[(2,1)-topos]] of  [[stacks]]/[[(2,1)-sheaves]] inside the [[(∞,1)-sheaf (∞,1)-topos]] of all [[∞-stacks]]/[[(∞,1)-sheaves]].

Write $L_W [C^{op}, Grpd]$ for the [[simplicial localization]] of groupoid valued presheaves in $C$
and write $[C^{op}, sSet]_{proj,loc}$ for the local projective [[model structure on simplicial presheaves]] that [[presentable (∞,1)-category|presents]] $Sh_{(\infty,1)}(C)$.

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

Let $f : X \to Y$ be a morphism of stacks that has a presentation by a degreewise [[faithful functor]] that, under the [[nerve]], goes between fibrant [[simplicial presheaves]].

Then $f$ is 0-truncated as a morphism in $Sh_{(\infty,1)}(C)$.

=--

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

We need to check that for any $\infty$-stack $A$, the morphism
$Sh_\infty(A,f)$ is 0-truncated in [[∞Grpd]]. We may choose a cofibrant
model for $A$ in $[C^{op}, sSet]_{proj,loc}$ and by assumption that
$X$ and $Y$ is fibrant we have that the ordinary hom of simplicial presheaves
$[C^{op}, sSet](A, f) $ is the correct [[derived hom space]] morphism.
This is itself (the [[nerve]] of) a [[faithful functor]],
hence the statement follows with
prop. \ref{TruncatedFunctorsOfGroupoids}.

=--

### In $\infty Grpd$ and in $Top$ {#InInfGrpd}

An object in [[∞Grpd]] is $n$-truncated precisely if it is an [[n-groupoid]]. To some extent, this is so by definition. Equivalently, an object in [[Top]] is $n$-truncated if it is (in the equivalence class of) a [[homotopy n-type]].

So we have for $n \in \mathbb{N}$ a [[reflective sub-(∞,1)-category]]

$$
  n Grpd \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow}
  \infty Grpd
  \,.
$$

+-- {: .num_lemma}
###### Observation

If we model the [[(∞,1)-category]] [[∞Grpd]] as the [[Kan complex]]-[[enriched category]]/fibrant [[simplicial category]] $KanCplx \subset $ [[sSet]] of [[Kan complex|Kan complexes]], then the truncation adjunction

$$
  (\tau_{\leq n } \dashv i)
  :
  n Grpd \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow}
  \infty Grpd
  \,.
$$

is modeled by the [[simplicial skeleton|simplicial coskeleton]] [[sSet]]-enriched adjunction

$$
  (tr_{n+1} \dashv cosk_{n+1})
  :
  KanCplx_{n+1} \stackrel{\overset{tr_{n+1}}{\leftarrow}}{\underset{cosk_{n+1}}{\to}}
  KanCplx
  \,,
$$

where $KanCplx_{n+1}$ is the subcategory of $[\Delta^{op}_{\leq n+1}, Set]$ on those truncated simplicial sets that are truncations of Kan complexes, regarded as a Kan-complex-enriched category by the embedding via $cosk_{n+1}$.

=--

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

Notice that every [[Kan complex]] $X$ which is $n$-truncated is homotopy equivalent to one in the image of $cosk_{n+1}$, namely to $cosk_{n+1} tr_{n+1} X$, because by one of the [properties](http://ncatlab.org/nlab/show/simplicial+skeleton#Truncation) of $cosk_{n+1}$ we have that the unit

$$
  X \to cosk_{n+1} tr_{n+1} X
$$

induces isomorphisms on homotopy groups $\pi_k$ for $k \leq n$.

This shows that $KanCplx_{n+1}$ is indeed a full [[sub-(∞,1)-category]] of $KanCplx$ on $n$-truncated objects

Moreover, by the fact discussed at <a href="http://ncatlab.org/nlab/show/adjoint+(infinity%2C1)-functor#SimplicialAndDerived">Simplicial and derived adjunctions</a> at [[adjoint (∞,1)-functor]] we have that the [[sSet]]-enriched adjunction $(tr_{n+1} \dashv cosk_{n+1})$ on $KanCplx$ indeed presents a pair of [[adjoint (∞,1)-functor|adjoint (∞,1)-functors]] on [[∞Grpd]]. So $tr_{n+1} : KanCplx \to KanCplx$ indeed presents the [[left adjoint]] $\tau_{\leq} : \infty Grpd \to n Grpd$ to the inclusion $n Grpd \hookrightarrow \infty Grpd$.

=--

### Diagonals
 {#DiagonalExamples}

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

In ordinary [[category theory]] we have that a morphism is a
[[monomorphism]] (as discussed there), precisely if its [[diagonal]] is an [[isomorphism]]. Embedded into [[(∞,1)-category]], this becomes
the special case of prop. \ref{DiagonalCharacterization}
for $n = 0$: a morphism is (-1)-truncated
(hence a [[monomorphism in an (∞,1)-category]]), precisely
if its diagonal is (-2)-truncated (hence an [[equivalence in an (∞,1)-category]]).

=--

+-- {: .num_example}

###### Example

Let $X$ be an object that is $n$-truncated. This means that
$X \to *$ is an $n$-truncated morphism. So by
prop. \ref{DiagonalCharacterization} the [[diagonal]] on
that object

$$
  \Delta : X \to X \times X
$$

is an $(n-1)$-truncated morphism, and precisely if it is $(n-1)$-truncated is $X$ $n$-truncated.

In particular, the diagonal is a [[monomorphism in an (∞,1)-category]],
hence (-1)-truncated, precisely if $X$ is $0$-truncated (an [[h-set]]).

=--

## Related concepts

* [[n-truncation modality]]

* **$n$-truncated morphism** / [[n-connected]] morphism

* [[n-groupoid]], [[homotopy n-type]]

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

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

## References

### In $\infty$-category theory

Discussion in [[(infinity,1)-category theory|$(\infty,1)$-category]]:

* [[Jacob Lurie]], Section 5.5.6 of: _[[Higher Topos Theory]]_

Discussion of [[categorical homotopy groups in an (∞,1)-topos]] is in
section 6.5.1 there.

Discussion in terms of [[model category]] [[presentable (∞,1)-category|presentations]]:

* {#Rezk} [[Charles Rezk]], Section 7 of: _Toposes and homotopy toposes_, 2010 ([pdf](http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf), [[Rezk_HomotopyToposes.pdf:file]])

A classical article that amplifies the expression of [[Postnikov towers]] in [[sSet]] (hence in [[Infinity-Grpd|$Grpd_{\infty}$]]) in terms of [[coskeleta]]:

* [[William Dwyer]], [[Dan Kan]], _An obstruction theory for diagrams of simplicial sets_ ([pdf](http://www.nd.edu/~wgd/Dvi/ObstructionTheoryForDiagrams.pdf))

Discussion in the context of [[modal type theory|modal]] [[homotopy type theory]]:

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

* {#Brunerie} [[Guillaume Brunerie]], _Truncations and truncated higher inductive types_ ([web](http://homotopytypetheory.org/2012/09/16/truncations-and-truncated-higher-inductive-types/))

More discussion of the internal perspective:

* [[Nima Rasekh]], _An Elementary Approach to Truncations_ ([arXiv:1812.10527](https://arxiv.org/abs/1812.10527))

### In homotopy type theory

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-truncated structured (infinity,1)-topos

[[!redirects n-truuncated strcutured (infinity,1)-topos]]
A [[structured (infinity,1)-topos]] $\mathcal{O}_{\mathcal{X}} : \mathcal{G} \to \mathcal{X}$ is called **$n$-truncated** if the image of each object of the [[geometry (for structured (infinity,1)-toposes)]] $\mathcal{G}$ is an [[n-truncated object of an (infinity,1)-topos|n-truncated object of]] $\mathcal{X}$.

See [[derived scheme]] and [[derived Deligne-Mumford stack]] for discussion of examples of $n$-truncated structured $(\infty,1)$-toposes.


n-truncation modality

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

# Contents
* table of contents
{: toc}

## Idea

For all $n \in \{-2, -1, 0,1,2,3, \cdots\}$, [[truncated object in an (infinity,1)-category|$n$-truncation]] is a [[modality]] (in [[homotopy type theory]]).

## Definition

The following applies to [[dependent type theory]] with [[suspension types]] and [[localizations]], such as to standard [[homotopy type theory]].

Recall:

\begin{definition}\label{SphereTypes}
**([[sphere type|$n$-sphere type]] $S^n$)**

* $S^{-1}$ is the [[empty type]],

* $S^0$ (the [[0-sphere]]) is the [[boolean domain]] [type](boolean+domain#InTypeTheory)

* $S^{n+1} \,\coloneqq\, \Sigma S^n$ is the [[suspension type]] of the sphere type of dimension lower.

\end{definition}

### As nullification of the $n+1$-sphere

The $n$-truncation modality may also be defined to be the [[localization of a type at a family of functions|localization at the unique function]] from the $(n + 1)$-dimensional sphere type (Def. \ref{SphereTypes}) to the [[unit type]] $S^{n + 1} \to \mathbb{1}$

$$\left[A\right]_n \coloneqq L_{S^{n + 1}}(A)$$

By definition, the type of functions $(\mathbb{1} \to \left[A\right]_n) \to (S^{n + 1} \to \left[A\right]_n)$ is an [[equivalence of types]].

### Explicit inference rules
 {#ExplicitInferenceRules}

The following are the [[inference rules]] for the $n$-truncation $[X]_n$ of a given $X \,\colon\, Type$ regarded as a [[higher inductive type]] according to [UFP13, §7.3, p. 223](#UFP13) (the [[diagram]] indicates the [[categorical semantics]], for orientation):

\begin{tikzcd}
  X
  \ar[
    d,
    "{ \eta }"
  ]
  \ar[
    rr,
    equals
  ]
  &[-130pt]
  &[-10pt]
  X
  \ar[
    d,
    "{ \eta_D }"
  ]
  &[-130pt]
  \\
  {
    [X]_n
  }
  \ar[ddd, equals]
  \ar[
    rr,
    dashed,
    shift left=2pt,
    "{
       \mathrm{ind}_{
         (D,\, \eta_D,\, \mathrm{hub}_D,\, \mathrm{hmt}_D)
    } }"{description}
  ]
  &[-70pt]
  &[-10pt]
  D
  \ar[dddll]
  \\
  &[-70pt]
  &[-10pt]
  &[-70pt]
  \mathrm{Map}\big({S^{n+1}},\,{D}\big)
  \ar[
    ul,
    "{ \mathrm{hub}_D}"{sloped},
    "{\ }"{name=t2}
  ]
  &[+70pt]
  \\[+10pt]
  &&
  \mathrm{Map}\big({S^{n+1}},\,{D}\big) \times S^{n+1}
  \ar[
    ur,
    "{ \mathrm{pr}_1 }"{description}
  ]
  \ar[dddll]
  \ar[
    uu,
    bend left=50,
    gray,
    "{ \mathrm{ev} }"{pos=.6},
    "{\ }"{pos=.7, swap, name=s2}
  ]
  \\[-80pt]
  {
    [X]_n
  }
  \\
  &
  \mathrm{Map}\big({S^{n+1}},\,{[X]_n}\big)
  \ar[
    ul,
    "{ \mathrm{hub}}"{sloped},
    "{\ }"{name=t}
  ]
  \ar[
    from=uuurr,
    shorten <=-6pt,
    crossing over
  ]
  \\[+10pt]
  \mathrm{Map}\big({S^{n+1}},\,{[X]_n}\big) \times S^{n+1}
  \ar[
    ur,
    "{ \mathrm{pr}_1 }"{description}
  ]
  \ar[
    uu,
    bend left=40,
    "{ \mathrm{ev} }"{pos=.6},
    "{\ }"{pos=.7, swap, name=s}
  ]
  \ar[
    from=s,
    to=t,
    Rightarrow,
    "{ \mathrm{hmt} }"{yshift=1pt,sloped},
  ]
  \ar[
    from=s2,
    to=t2,
    Rightarrow,
    gray,
    "{ \mathrm{hmt}_D }"{yshift=1pt,sloped},
  ]
\end{tikzcd}

\linebreak

**[[type formation rule]]:**

$$
  \frac{
     X \,\colon\, Type
    \mathclap{\phantom{\vert_{\vert}}}
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    [X]_n \,\colon\, Type
  }
$$

\linebreak

**[[term introduction rule]]:**

$$
  \frac{
    x \,\colon\, X
    \mathclap{\phantom{\vert_{\vert}}}
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    \eta_X(x) \,\colon\, [X]_n
  }
  \;\;\;\;\;\;\;\;\;
  \frac{
    \phi \,\colon\, S^{n+1} \to [X]_n
    \mathclap{\phantom{\vert_{\vert}}}
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    hub_X^{\phi} \,\colon\, [X]_n
  }
  \;\;\;\;\;\;\;\;\;
  \frac{
    \phi \,\colon\, S^{n+1} \to [X]_n
    \;;
    \;\;\;
    s \,\colon\, S^{n+1}
    \mathclap{\phantom{\vert_{\vert}}}
  }{
    \mathclap{\phantom{\vert^{\vert}}}
    hmt_X^{\phi}(s)
     \,\colon\,
    Id_{[X]_n}\big(
      \phi(s)
      ,\,
      hub_X^{\phi}
    \big)
  }
$$

\linebreak

{#TermElimination} **[[term elimination rules]]:**

$$
  \frac{
    \begin{array}{l}
      \overline{x} \,\colon\, [X]_n
      \;\vdash\;
      D(\overline{x}) \,\colon\, Type \;;
      \\
      x \,\colon\, X
      \;\vdash\;
      \eta_D(x) \,\colon\, D\big( \eta_X(x) \big)
      \;;
      \\
      \phi \,\colon\, S^{n+1} \to [X]_n
      \,,
      \;
      \phi_D
        \,\colon\,
      \underset{s \,\colon\,S^{n+1}}{\prod}
      \,
      D\big( \phi(s) \big)
      \;\;\vdash\;\;
      hub^{\phi_D}_D
       \,\colon\,
      D\big( hub_X^\phi \big)
      \;;
      \\
      \phi \,\colon\, S^{n+1} \to [X]_n
      \,,
      \;
      \phi_D
        \,\colon\,
      \underset{s \,\colon\,S^{n+1}}{\prod}
      \,
      D\big( \phi(s) \big)
      \,,
      \;
      s \,\colon\, S^{n+1}
      \;\;\vdash\;\;
      hmt_D^{\phi_D}(s)
        \,\colon\,
      Id_{D\big(hub_X^{\phi}\big)}\Big(
        \big(hmt^{\phi}_X(s)\big)_\ast
        \phi_D(s)
        ,\,
        hub_D^{\phi_D}
      \Big)
      \mathclap{\phantom{\vert^{\vert^{\vert}}}}
    \end{array}
  }{
    \mathclap{\phantom{\vert^{\vert_{\vert}}}}
    ind_{\big(D,\,\eta_D,\, hub_D,\, hmt_D \big)}
    \,\colon\,
    \underset{\overline{x}\colon [X]_n}{\prod}
    \,
    D(\overline{x})
  }
$$

\linebreak

**[[computation rule]]:**

$$
  ind_{\big(D,\,\eta_D,\, hub_D,\, hmt_D \big)}
  \big(
    \eta(x)
  \big)
  \;=\;
  \eta_D(x)
$$

\linebreak

## Properties

### In low degree

$(-2)$-truncation is the [[unit type]] modality ([[constant map|constant]] on the unit type).

$(-1)$-truncation is given by forming _[[bracket types]]_, turning types into genuine [[propositions]].

[[classical logic|Classically]], this is the same as the [[double negation modality]]; in general, the bracket type ${\|A\|_{-1}}$ only entails the double negation $\neg(\neg{A})$:
there is a canonical [[function]]

$$
  {\|A\|_{-1}} \longrightarrow \neg(\neg{A})
$$

and this is a [[1-epimorphism]] precisely if the [[law of excluded middle]] holds.

## Related concepts

* [[decategorification]], [[core]]

* [[cone type]], [[bracket type]], [[set truncation]]

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

## References
 {#References}

Construction of $n$-truncation as a one-step [[higher inductive type]] in [[homotopy type theory]]:

* {#UFP13} [[Univalent Foundations Project]], §6.9 & §7.3 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)]

Alternative construction of $n$-truncation as an iterated [[pushout type]] is (somewhat implicit) in:

* {#Rijke19b} [[Egbert Rijke]], Section 7.2 (culminating in Thm. 7.2.10) of: _Classifying Types_ [[arXiv:1906.09435](https://arxiv.org/abs/1906.09435)]

Discussion of $n$-truncation as a [[modality]]:

* {#RSS20} [[Egbert Rijke]], [[Mike Shulman]], [[Bas Spitters]], Exp. 1.6 in: *Modalities in homotopy type theory*,  Logical Methods in Computer Science, **16** 1 (2020)  [[arXiv:1706.07526](https://arxiv.org/abs/1706.07526), doi:10.23638/LMCS-16(1:2)2020]

and in addition via [[lifting properties]] against [[n-spheres]]:

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

Earlier discussion (and in view of [[homotopy levels]]):

* [[Nicolai Kraus]], Def. 2.3.8 in: *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)]

Precursor discussion of the material that became [UFP (2013, §7.3)](#UFP13):

* [[Peter LeFanu Lumsdaine]], *Reducing all HIT's to 1-HIT's* (May 2012) [[blog posy](https://homotopytypetheory.org/2012/05/07/reducing-all-hits-to-1-hits/)]

* [[Guillaume Brunerie]], *Truncations and higher inductive types* (September 2012) [[blog post](http://homotopytypetheory.org/2012/09/16/truncations-and-truncated-higher-inductive-types/)]

and precursur discussion of the material that became [RSS (2020)](#RSS20):

* {#Shulman12} [[Mike Shulman]], _Higher modalities_, talk at [[UF-IAS-2012]] (October 2012)  [[pdf](https://ncatlab.org/ufias2012/files/modalitt.pdf)]

* [[Mike Shulman]], *All modalities are Higher Inductive Types* (November 2012) [[blog post](http://homotopytypetheory.org/2012/11/19/all-modalities-are-hits/)]

Considering the combination of $n$-truncation modality and [[shape modality]]:

* [[David Jaz Myers]], Def. 3.0.1 in: *Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory* ([arXiv:2205.15887](https://arxiv.org/abs/2205.15887))

[[!redirects n-truncation modality]]
[[!redirects n-truncation modalities]]

[[!redirects truncation modality]]
[[!redirects truncation modalities]]

[[!redirects n-truncated type]]
[[!redirects n-truncated types]]