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\begin{document}

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\section*{n-connected object of an (infinity,1)-topos}

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

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - 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{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{PropertiesGeneral}{General}\dotfill \pageref*{PropertiesGeneral} \linebreak
\noindent\hyperlink{RecursiveCharacterization}{Recursive characterization}\dotfill \pageref*{RecursiveCharacterization} \linebreak
\noindent\hyperlink{factorization_system}{Factorization system}\dotfill \pageref*{factorization_system} \linebreak
\noindent\hyperlink{blakersmassey_theorem}{Blakers-Massey theorem}\dotfill \pageref*{blakersmassey_theorem} \linebreak
\noindent\hyperlink{Clock}{The truncated / connected clock}\dotfill \pageref*{Clock} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{InTop}{In $Top$}\dotfill \pageref*{InTop} \linebreak
\noindent\hyperlink{ExamplesInGrpd}{In $Grpd$}\dotfill \pageref*{ExamplesInGrpd} \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{$n$-connected object} is an object all whose [[homotopy group]]s \emph{equal to or below} degree $n$ are trivial.

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

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

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

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

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

\begin{enumerate}%
\item the [[terminal object|terminal]] morphism $X \to *$ is an [[effective epimorphism in an (infinity,1)-category|effective epimorphism]];


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

\begin{displaymath}
\pi_k(X) = * \;\;\; for \; k \leq n
  \,.
\end{displaymath}


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

\begin{enumerate}%
\item it is an [[effective epimorphism in an (∞,1)-category]]


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



\end{enumerate}
\end{defn}
This appears as [[Higher Topos Theory|HTT, def. 6.5.1.10]], but under the name ``$(n+1)$-connective''. Another possible term is ``$n$-simply connected''; see [[n-connected space]] for discussion.

One adopts the following convenient terminology.

\begin{itemize}%
\item Every object is $(-2)$-connected.


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


\item A 0-connected object is simply called a \textbf{connected object}.



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

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

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

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

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 6.5.1.12]].

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

\end{lemma}
This is [[Higher Topos Theory|HTT, prop. 6.5.1.16, item 2]].

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

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
The class of $n$-connected morphisms is stable under [[pullback]] and [[pushout]].

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

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 6.5.1.16, item 6]].

\hypertarget{RecursiveCharacterization}{}\subsubsection*{{Recursive characterization}}\label{RecursiveCharacterization}

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

\begin{displaymath}
\Delta_f : X \to X \times_Y X
\end{displaymath}
is $(n-1)$-connected.

\end{prop}
This appears as [[Higher Topos Theory|HTT, prop. 6.5.1.18]].

\hypertarget{factorization_system}{}\subsubsection*{{Factorization system}}\label{factorization_system}

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

\end{prop}
See also [[n-connected/n-truncated factorization system]].

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

\hypertarget{blakersmassey_theorem}{}\subsubsection*{{Blakers-Massey theorem}}\label{blakersmassey_theorem}

\begin{itemize}%
\item [[Blakers-Massey theorem]]

\end{itemize}
\hypertarget{Clock}{}\subsubsection*{{The truncated / connected clock}}\label{Clock}

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

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

\begin{itemize}%
\item any morphism = $(-2)$-connected


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


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


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


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


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


\item $\infty$-truncated = any morphism



\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{InTop}{}\subsubsection*{{In $Top$}}\label{InTop}

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

\begin{itemize}%
\item a $(-1)$-connected object is an [[inhabited set|inhabited space]].


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


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


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



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

\hypertarget{ExamplesInGrpd}{}\subsubsection*{{In $Grpd$}}\label{ExamplesInGrpd}

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

\end{prop}
\begin{proof}
As discussed there, an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] in [[∞Grpd]] between 1-groupoids is precisely an [[essentially surjective functor]].

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

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

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

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

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

\begin{displaymath}
\itexarray{
     f(x_1) &\stackrel{f(\hat \alpha)}{\to}& f(x_2)
     \\
     \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}}
     \\
     f(x_2) &\stackrel{id}{\to}& f(x_2)
  }
\end{displaymath}
to an object in the diagonal.

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

\begin{displaymath}
\itexarray{
    f(x_1) &\stackrel{f(h_1)}{\to}& f(x_2)
    \\
    \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}}
    \\
    f(x_2) &\stackrel{f(h_2)}{\to}& f(x_2)
  }
  \,.
\end{displaymath}
Therefore $h_2^{-1}\circ h_1$ is a preimage of $\alpha$ under $f$, and hence $f$ is full.

\end{proof}
See also [[(eso+full, faithful) factorization system]].

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

\begin{itemize}%
\item [[n-truncated object in an (infinity,1)-category]]


\item [[Eilenberg subcomplex]]


\item [[connective spectrum]], [[connective cover]]


\item [[Freudenthal suspension theorem]]


\item [[Blakers-Massey theorem]]



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

Section 6.5.1 of

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

\end{itemize}
A discussion in terms of [[model category]] [[presentable (∞,1)-category|presentations]] is in section 8 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}
[[!redirects (-2)-connected]] [[!redirects (-1)-connected]] [[!redirects 0-connected]] [[!redirects 0-connective]] [[!redirects 1-connected]] [[!redirects 1-connective]] [[!redirects 2-connected]] [[!redirects 2-connective]] [[!redirects 3-connected]] [[!redirects 3-connective]] [[!redirects 4-connected]] [[!redirects 5-connected]] [[!redirects ∞-connected]] [[!redirects ∞-connective]] [[!redirects n-connected]] [[!redirects n-connective]]

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

[[!redirects n-connected object in an (∞,1)-topos]] [[!redirects n-connected object in an (infinity,1)-topos]] [[!redirects n-connected object of an (∞,1)-topos]] [[!redirects n-connected objects of an (∞,1)-topos]] [[!redirects n-connected objects of (∞,1)-toposes]] [[!redirects n-connected objects of (∞,1)-topoi]] [[!redirects n-connected object of an (infinity,1)-topos]] [[!redirects n-connected objects of an (infinity,1)-topos]] [[!redirects n-connected objects of (infinity,1)-toposes]] [[!redirects n-connected objects of (infinity,1)-topoi]] [[!redirects n-simply connected object of an (∞,1)-topos]] [[!redirects n-simply connected objects of an (∞,1)-topos]] [[!redirects n-simply connected objects of (∞,1)-toposes]] [[!redirects n-simply connected objects of (∞,1)-topoi]] [[!redirects n-simply connected object of an (infinity,1)-topos]] [[!redirects n-simply connected objects of an (infinity,1)-topos]] [[!redirects n-simply connected objects of (infinity,1)-toposes]] [[!redirects n-simply connected objects of (infinity,1)-topoi]]

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

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

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

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



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