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\begin{document}

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\section*{n-truncated object of an (infinity,1)-category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{modalities_closure_and_reflection}{}\paragraph*{{Modalities, Closure and Reflection}}\label{modalities_closure_and_reflection}

[[!include modalities - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_truncations}{In terms of truncations}\dotfill \pageref*{in_terms_of_truncations} \linebreak
\noindent\hyperlink{in_terms_of_categorical_homotopy_groups}{In terms of categorical homotopy groups}\dotfill \pageref*{in_terms_of_categorical_homotopy_groups} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RecursiveDefinition}{Recursive definition}\dotfill \pageref*{RecursiveDefinition} \linebreak
\noindent\hyperlink{PropertiesGeneral}{General}\dotfill \pageref*{PropertiesGeneral} \linebreak
\noindent\hyperlink{ModelCategoryPresentations}{Model category presentations}\dotfill \pageref*{ModelCategoryPresentations} \linebreak
\noindent\hyperlink{truncation}{Truncation}\dotfill \pageref*{truncation} \linebreak
\noindent\hyperlink{definition_5}{Definition}\dotfill \pageref*{definition_5} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{GeneralPropsTruncation}{General}\dotfill \pageref*{GeneralPropsTruncation} \linebreak
\noindent\hyperlink{postnikov_tower}{Postnikov tower}\dotfill \pageref*{postnikov_tower} \linebreak
\noindent\hyperlink{TruncationHomotopyTypeTheorySyntax}{Homotopy type theory syntax}\dotfill \pageref*{TruncationHomotopyTypeTheorySyntax} \linebreak
\noindent\hyperlink{RelationToHomotopyGroups}{Relation to homotopy groups}\dotfill \pageref*{RelationToHomotopyGroups} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{truncated_morphisms}{Truncated morphisms}\dotfill \pageref*{truncated_morphisms} \linebreak
\noindent\hyperlink{general_3}{General}\dotfill \pageref*{general_3} \linebreak
\noindent\hyperlink{TruncatedMorphismsBetweenGroupoids}{Between groupoids}\dotfill \pageref*{TruncatedMorphismsBetweenGroupoids} \linebreak
\noindent\hyperlink{TruncatedMorphismsBetweenStacks}{Between stacks}\dotfill \pageref*{TruncatedMorphismsBetweenStacks} \linebreak
\noindent\hyperlink{InInfGrpd}{In $\infty Grpd$ and in $Top$}\dotfill \pageref*{InInfGrpd} \linebreak
\noindent\hyperlink{DiagonalExamples}{Diagonals}\dotfill \pageref*{DiagonalExamples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An \textbf{$n$-truncated [[∞-groupoid]]} is an [[n-groupoid]].

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

An \textbf{$n$-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]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_terms_of_truncations}{}\subsubsection*{{In terms of truncations}}\label{in_terms_of_truncations}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{($n$-truncated $\infty$-groupoid)}

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

Precisely: in the model of [[∞-groupoid]]s given by [[Kan complex]]es $A$ is \textbf{$n$-truncated} if the [[simplicial homotopy group]]s $\pi_k(A,x)$ are trivial for all $x$ and all $k \gt n$.

\end{defn}
It makes sense for the following to adopt the convention that $A$ is called

\begin{itemize}%
\item \emph{$(-1)$-truncated} if it is empty or contractible -- this is a [[(-1)-groupoid]].


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



\end{itemize}
(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 shall be useful to slightly reformulate the above as follows:

\begin{lemma}
\label{}\hypertarget{}{}
An [[∞-groupoid]] $A$ is $n$-truncated precisely for all other [[∞-groupoid]]s $X$ the hom-$\infty$-groupoid $\infty Grpd(X,A)$ is $n$-truncated.

\end{lemma}
In categorical terms this just says that [[(n,k)-transformation|(∞,k)-transformation]] between $X$ and $A$ whose components a [[k-morphism]]s in $A$ cannot be nontrivial for $k \gt n$ if there are no nontrivial [[k-morphism]]s 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-[[∞-groupoid]]s:

\begin{defn}
\label{nTruncatedObject}\hypertarget{nTruncatedObject}{}
\textbf{($n$-truncated object in an $(\infty,1)$-category)}

An [[object]] $A \in C$ of an [[(∞,1)-category]] $C$ is \textbf{$n$-truncated}, for $n \in \mathbb{N}$, if for all $X \in C$ the hom-[[∞-groupoid]] $C(X,A)$ is $n$-truncated.

\end{defn}
This is [[Higher Topos Theory|HTT, def. 5.5.6.1]].

Some terminology:

\begin{itemize}%
\item A 0-truncated object is also called \textbf{discrete} . Notice that this is \emph{categorically discrete} as in [[discrete category]], not discrete in the sense of [[topological space]]s. An object in an [[(∞,1)-topos]] is discrete in this sense if, regarded as an [[∞-groupoid]] with extra structure it has only trivial morphisms.


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


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



\end{itemize}
\begin{defn}
\label{nTruncatedMorphism}\hypertarget{nTruncatedMorphism}{}
\textbf{($n$-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{nTruncatedObject}.

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

\begin{displaymath}
C(W,f) : C(W,X) \to C(W,Y)
\end{displaymath}
is $n$-truncated in [[∞Grpd]].

\end{defn}
By the \href{http://ncatlab.org/nlab/show/fiber+sequence#OfFuncCats}{characterization of homotopy fiber of functor categories} 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]] $C_{/Y}$.

\hypertarget{in_terms_of_categorical_homotopy_groups}{}\subsubsection*{{In terms of categorical homotopy groups}}\label{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]]:

\begin{prop}
\label{RecognizngnTuncationOnSimplicialHomotopyGroups}\hypertarget{RecognizngnTuncationOnSimplicialHomotopyGroups}{}
Suppose that an object $X$ in an [[∞-stack]] [[(∞,1)-topos]] is $k$-truncated for \emph{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.

\end{prop}
\begin{proof}
This is [[Higher Topos Theory|HTT, prop 6.5.1.7]].

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Notice that this expected statement does require the assumption that $X$ is $k$-truncated for some $k$. Without any \emph{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]].

\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RecursiveDefinition}{}\subsubsection*{{Recursive definition}}\label{RecursiveDefinition}

\begin{prop}
\label{DiagonalCharacterization}\hypertarget{DiagonalCharacterization}{}
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.

\end{prop}
This is [[Higher Topos Theory|HTT, lemma 5.5.6.15]].

\begin{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]]s $C(d,-)$ 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]]s. We may find a fibration $\bar \phi : \bar X \to \bar Y$ between [[Kan complex]]es in [[sSet]] that models $f : X \to Y$ in the standard [[model structure on simplicial sets]], and by the standard rules for [[homotopy pullback]]s 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]]s of $\bar X_y \to \bar X_y \times \bar X_y$ are $(k-1)$-truncated.

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

\begin{displaymath}
\itexarray{
     Q &\to& *
     \\
     \downarrow && \downarrow^{\mathrlap{(a,b)}}
     \\
     X &\to& X \times X
  }
\end{displaymath}
as usual by replacing the right vertical morphism by 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]]s 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 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.

\end{proof}
\hypertarget{PropertiesGeneral}{}\subsubsection*{{General}}\label{PropertiesGeneral}

\begin{prop}
\label{}\hypertarget{}{}
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|limits]] in $C$.

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 5.5.6.5]].

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathbf{H}$ be an [[(∞,1)-topos]]. For all $(-2) \leq n \leq \infty$ 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 $\mathbf{H}$.

\end{prop}
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 (\hyperlink{Rezk}{Rezk, prop. 8.5}). See also [[n-connected/n-truncated factorization system]].

\hypertarget{ModelCategoryPresentations}{}\subsubsection*{{Model category presentations}}\label{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.

\hypertarget{truncation}{}\subsection*{{Truncation}}\label{truncation}

\hypertarget{definition_5}{}\subsubsection*{{Definition}}\label{definition_5}

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-morphism]]s 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 $\tau_{\leq n} \infty Grpd = n Grpd$.

\begin{prop}
\label{}\hypertarget{}{}
If $C$ is an [[(∞,1)-category]] that is [[presentable (∞,1)-category|presentable]] then the canonical inclusion [[(∞,1)-functor]]

\begin{displaymath}
\tau_{\leq n} C \hookrightarrow C
\end{displaymath}
has an [[accessible (infinity,1)-functor|accessible]] [[left adjoint]]

\begin{displaymath}
\tau_{\leq n} : C \to C_{\leq n}
  \,.
\end{displaymath}
\end{prop}
This is [[Higher Topos Theory|HTT 5.5.6.18]].

Indeed, as the notation suggests, $C_{\leq n}$ 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 \textbf{$n$-truncation} of $A$.

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

\begin{displaymath}
\tau_{\leq n} C
  \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow}
  C
  \,.
\end{displaymath}
\hypertarget{Properties}{}\subsubsection*{{Properties}}\label{Properties}

\hypertarget{GeneralPropsTruncation}{}\paragraph*{{General}}\label{GeneralPropsTruncation}

\begin{corollary}
\label{}\hypertarget{}{}
A left [[exact functor]] $F : C \to D$ between $(\infty,1)$-categories with finite limits sends $k$-truncated objects/morphisms to $k$-truncated objects/morphisms.

\end{corollary}
This is [[Higher Topos Theory|HTT, prop. 5.5.6.1.6]].

\begin{proof}
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}
\begin{prop}
\label{}\hypertarget{}{}
A presentable $(\infty,1)$-functor $F : C \to D$ between [[locally presentable (∞,1)-categories]] $C$ and $D$ commutes with truncation:

\begin{displaymath}
F \circ \tau^C_{\leq k} \simeq \tau^D_{\leq k} \circ F
  \,.
\end{displaymath}
\end{prop}
This is [[Higher Topos Theory|HTT, prop. 5.5.6.28]].

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

\begin{displaymath}
\itexarray{
    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
  }
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    C &\stackrel{G}{\leftarrow}& D
    \\
    \uparrow 
    & & 
    \uparrow
    \\
    \tau_{\leq k } C &\stackrel{G}{\leftarrow}&
    \tau_{\leq k} D
  }
  \,.
\end{displaymath}
evidently commutes, since it just expresses this restriction.

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

\end{prop}
This appears as [[Higher Topos Theory|HTT, lemma 6.5.1.2]].

\begin{proof}
First notice that the statement is true for $C =$ [[∞Grpd]]. For instance we can use the example \hyperlink{InInfGrpd}{In ∞Grpd and Top}, model [[∞-groupoid]]s by [[Kan complex]]es and notice that then $\tau_{\leq n}$ is given by the truncation functor $tr_{n+1} : sSet \to [\Delta^{op}_{\leq n+1}, sSet]$. This is also a [[right adjoint]] and as such preserves in particular product 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,

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

\begin{displaymath}
\tau_{\leq k} \prod_j i(X_j)
  \stackrel{\simeq}{\to}
  \prod_j \tau_{\leq} i(X_j)
  \,.
\end{displaymath}
Now applying $L$ to this equivalence and using now that $L$ preserves the \emph{finite} product, this gives an equivalence

\begin{displaymath}
L \tau_{\leq k} \prod_j i(X_j)
  \stackrel{\simeq}{\to}
  L \prod_j \tau_{\leq} i(X_j)
  \simeq
  \prod_j L \tau_{\leq} i(X_j)
\end{displaymath}
in $C$. The claim follows now with the above result that $L \circ \tau_{\leq n} \simeq \tau_{\leq n} \circ L$.

\end{proof}
\hypertarget{postnikov_tower}{}\paragraph*{{Postnikov tower}}\label{postnikov_tower}

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

\begin{displaymath}
\tau_{\leq n}A  \to A
\end{displaymath}
as the [[adjunct]] of the [[identity]] on $A$, and then by iteration also canonical morphisms

\begin{displaymath}
\tau_{\leq (n+1)} A \to \tau_{\leq n} A
  \,.
\end{displaymath}
For any $A \in C$ the sequence

\begin{displaymath}
\cdots
   \to \tau_{\leq 2}A \to \tau_{\leq 1} A \to \tau_{\leq 0} A
\end{displaymath}
is the [[Postnikov tower in an (∞,1)-category]] of $A$. See there for more details.

\hypertarget{TruncationHomotopyTypeTheorySyntax}{}\paragraph*{{Homotopy type theory syntax}}\label{TruncationHomotopyTypeTheorySyntax}

Discussion of $n$-truncation of [[types]] in [[homotopy type theory]] via [[higher inductive types]] is in (\hyperlink{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 \emph{[[n-truncation modality]]}.

\hypertarget{RelationToHomotopyGroups}{}\subsubsection*{{Relation to homotopy groups}}\label{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]]es this coincides with the notion of [[simplicial homotopy groups]]:

\begin{lemma}
\label{}\hypertarget{}{}
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 $k \gt n$.

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

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

Then

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


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



\end{enumerate}
\end{prop}
This implies

\begin{corollary}
\label{}\hypertarget{}{}
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 $k \gt n$.

\end{corollary}
\textbf{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]]es.

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{truncated_morphisms}{}\subsubsection*{{Truncated morphisms}}\label{truncated_morphisms}

\hypertarget{general_3}{}\paragraph*{{General}}\label{general_3}

A morphism $f : X \to 0$ is

\begin{itemize}%
\item (-2)-truncated precisely if it is an [[equivalence in a quasi-category|equivalence]];


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



\end{itemize}
\hypertarget{TruncatedMorphismsBetweenGroupoids}{}\paragraph*{{Between groupoids}}\label{TruncatedMorphismsBetweenGroupoids}

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

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

\begin{itemize}%
\item for $n = -2$ -- an [[equivalence of categories]];


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


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



\end{itemize}
\end{prop}
\begin{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

\begin{displaymath}
\cdots \to \pi_1(F) \to \pi_1(X) \hookrightarrow \pi_1(Y,f(x)) \to \cdots
  \,,
\end{displaymath}
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.

\end{proof}
\hypertarget{TruncatedMorphismsBetweenStacks}{}\paragraph*{{Between stacks}}\label{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]] $Sh_{(\infty,1)}(C)$.

\begin{prop}
\label{}\hypertarget{}{}
Let $f : X \to Y$ be a morphism of stacks which 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 $Sh_{(\infty,1)}(C)$.

\end{prop}
\begin{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}.

\end{proof}
\hypertarget{InInfGrpd}{}\subsubsection*{{In $\infty Grpd$ and in $Top$}}\label{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]]

\begin{displaymath}
n Grpd \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow}
  \infty Grpd
  \,.
\end{displaymath}
\begin{lemma}
\label{}\hypertarget{}{}
If we model the [[(∞,1)-category]] [[∞Grpd]] as the [[Kan complex]]-[[enriched category]]/fibrant [[simplicial category]] $KanCplx \subset$ [[sSet]] of [[Kan complex]]es, then the truncation adjunction

\begin{displaymath}
(\tau_{\leq n } \dashv i)
  :
  n Grpd \stackrel{\overset{\tau_{\leq n}}{\leftarrow}}{\hookrightarrow}
  \infty Grpd
  \,.
\end{displaymath}
is modeled by the [[simplicial skeleton|simplicial coskeleton]] [[sSet]]-enriched adjunction

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

\end{lemma}
\begin{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 \href{http://ncatlab.org/nlab/show/simplicial+skeleton#Truncation}{properties} of $cosk_{n+1}$ we have that the unit

\begin{displaymath}
X \to cosk_{n+1} tr_{n+1} X
\end{displaymath}
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  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]]s on [[∞Grpd]]. So $tr_{n+1} : KanCplx \to KanCplx$ indeed presents the [[left adjoint]] $\tau_{\leq} : \infty Grpd \to n Grpd$ to the inclusion $n Grpd \hookrightarrow \infty Grpd$.

\end{proof}
\hypertarget{DiagonalExamples}{}\subsubsection*{{Diagonals}}\label{DiagonalExamples}

\begin{example}
\label{}\hypertarget{}{}
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]]).

\end{example}
\begin{example}
\label{}\hypertarget{}{}
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

\begin{displaymath}
\Delta : X \to X \times X
\end{displaymath}
is an $(n-1)$-truncated morphism, and precisely if it is $(n-1)$-truncated is $X$ $n$-truncated.

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

\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[n-truncation modality]]


\item \textbf{$n$-truncated morphism} / [[n-connected]] morphism


\item [[n-groupoid]], [[homotopy n-type]]


\item [[model structure for homotopy n-types]]



\end{itemize}
[[!include homotopy n-types - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The discussion of truncated objects in an $(\infty,1)$-category is in section 5.5.6 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]}

\end{itemize}
The discussion of categorical [[homotopy groups in an (∞,1)-topos]] is in section 6.5.1.

A discussion in terms of [[model category]] [[presentable (∞,1)-category|presentations]] is in section 7 of

\begin{itemize}%
\item [[Charles Rezk]], \emph{Toposes and homotopy toposes} (\href{http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf}{pdf})

\end{itemize}
A classical article that amplifies the expression of Postnikov towers in terms of [[coskeleton]]s is

\begin{itemize}%
\item [[William Dwyer]], [[Dan Kan]], \emph{An obstruction theory for diagrams of simplicial sets} (\href{http://www.nd.edu/~wgd/Dvi/ObstructionTheoryForDiagrams.pdf}{pdf})

\end{itemize}
Discussion in the context of [[modal type theory|modal]] [[homotopy type theory]] is in

\begin{itemize}%
\item [[Univalent Foundations Project]], section 7.6 of \emph{[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]}


\item [[Guillaume Brunerie]], \emph{Truncations and truncated higher inductive types} (\href{http://homotopytypetheory.org/2012/09/16/truncations-and-truncated-higher-inductive-types/}{web})



\end{itemize}
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