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\begin{document}

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\section*{locally n-connected (n+1,1)-topos}

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

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

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{LocallyContractibleExamples}{Examples}\dotfill \pageref*{LocallyContractibleExamples} \linebreak
\noindent\hyperlink{over_locally_connected_sites}{Over locally $\infty$-connected sites}\dotfill \pageref*{over_locally_connected_sites} \linebreak
\noindent\hyperlink{over_locally_connected_topological_spaces}{Over locally $n$-connected topological spaces}\dotfill \pageref*{over_locally_connected_topological_spaces} \linebreak
\noindent\hyperlink{locally_connected_overtoposes}{Locally $\infty$-connected over-$(\infty,1)$-toposes}\dotfill \pageref*{locally_connected_overtoposes} \linebreak
\noindent\hyperlink{LocInfConnProperties}{Properties}\dotfill \pageref*{LocInfConnProperties} \linebreak
\noindent\hyperlink{relation_to_slicing}{Relation to slicing}\dotfill \pageref*{relation_to_slicing} \linebreak
\noindent\hyperlink{RelationToLocallyConnectedToposes}{Relation to locally connected toposes}\dotfill \pageref*{RelationToLocallyConnectedToposes} \linebreak
\noindent\hyperlink{further_structures}{Further structures}\dotfill \pageref*{further_structures} \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}

A [[topos]] may be thought of as a generalized [[topological space]]. Accordingly, the notions of

\begin{itemize}%
\item [[locally connected space]]


\item [[locally simply connected space]]


\item locally $2$-connected space


\item locally $n$-connected space


\item [[locally contractible space]]



\end{itemize}
have analogs for [[topos]]es, [[(n,1)-toposes]] and [[(∞,1)-topos]]es

\begin{itemize}%
\item [[locally connected topos]]


\item [[locally simply connected (2,1)-topos]]


\item \textbf{locally $n$-connected $(n+1,1)$-topos}


\item \textbf{locally $\infty$-connected $(\infty,1)$-topos}



\end{itemize}
The numbering mismatch is traditional from [[topology]]; see [[n-connected space]]. It reads a bit better if we say \emph{locally $n$-simply connected} for \emph{locally $n$-connected}, since \emph{locally $1$-(simply) connected} is \emph{locally simply connected}, but being locally $n$-simply connected is still a property of an $(n+1,1)$-topos.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\begin{defn}
\label{}\hypertarget{}{}
A [[(∞,1)-sheaf (∞,1)-topos]] $\mathbf{H}$ is called \textbf{locally $\infty$-connected} if the (essentially unique) [[global section]] [[(∞,1)-geometric morphism]]

\begin{displaymath}
(\Delta\dashv\Gamma):
  \mathbf{H} \xrightarrow{\Gamma}\infty\Grpd
\end{displaymath}
extends to an \textbf{[[essential geometric morphism]] $(\infty,1)$-geometric morphism}, i.e. there is a further [[left adjoint]] $\Pi$

\begin{displaymath}
(\Pi \dashv \Delta \dashv \Gamma) : 
  \mathbf{H}
  \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}}
  \infty Grpd
  \,.
\end{displaymath}
If in addition $\Pi$ preserves the [[terminal object]] we say that $\mathbf{H}$ is an \textbf{[[∞-connected (∞,1)-topos]]}.

If $\Pi$ preserves even all [[finite limit|finite]] [[(∞,1)-product]]s we say that $\mathbf{H}$ is a [[strongly ∞-connected (∞,1)-topos]].

If $\Pi$ preserves even all [[finite limit|finite]] [[(∞,1)-limit]]s we say that $\mathbf{H}$ is a [[totally ∞-connected (∞,1)-topos]].

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
In (\hyperlink{Lurie}{Lurie, section A.1}) this is called an $(\infty,1)$-topos of \textbf{locally constant [[shape of an (infinity,1)-topos|shape]]}.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
For $\mathbf{H}$ a locally $\infty$-connected $(\infty,1)$-topos and $X \in \mathbf{H}$ an [[object]], we call $\Pi X \in$ [[∞Grpd]] the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]] of $X$. The ([[categorical homotopy groups in an (∞,1)-topos|categorical]]) [[homotopy group]]s of $\Pi(X)$ we call the [[geometric homotopy groups in an (∞,1)-topos|geometric homotopy groups]] of $X$

\begin{displaymath}
\pi_\bullet^{geom}(X) := \pi_\bullet(\Pi (X))
  \,.
\end{displaymath}
\end{defn}
Similarly we have:

\begin{defn}
\label{}\hypertarget{}{}
For $n \in \mathbb{N}$ an $(n+1,1)$-[[(n,1)-topos|topos]] $\mathbf{H}$ is called \textbf{locally $n$-connected} if the (essentially unique) [[global section]] geometric morphism is has an extra left adjoint.

\end{defn}
For $n = 0$ this reproduces the case of a [[locally connected topos]].

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

\hypertarget{over_locally_connected_sites}{}\subsubsection*{{Over locally $\infty$-connected sites}}\label{over_locally_connected_sites}

The follow proposition gives a large supply of examples.

\begin{prop}
\label{}\hypertarget{}{}
Let $C$ be a [[locally ∞-connected (∞,1)-site]]/[[∞-connected (∞,1)-site]]. Then the [[(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(C)$ is a locally $\infty$-connected $(\infty,1)$-topos.

\end{prop}
See [[locally ∞-connected (∞,1)-site]]/[[∞-connected (∞,1)-site]] for the proof.

\begin{remark}
\label{}\hypertarget{}{}
In (\hyperlink{SimpsonTeleman}{SimpsonTeleman, prop. 2.18}) is stated essentially what the above proposition asserts at the level of [[homotopy category|homotopy categories]]: if $C$ has contractible objects, then there exists a [[left adjoint]] $Ho(\Pi):Ho(Sh_{(\infty,1)}(C)) \to Ho(\infty Grpd)$.

\end{remark}
This includes the following examples.

\begin{example}
\label{}\hypertarget{}{}
The sites [[CartSp]]${}_{top}$ $CartSp_{smooth}$ $CartSp_{synthdiff}$ are locally $\infty$-connected. The corresponding $(\infty,1)$-toposes are the [[cohesive (∞,1)-topos]]es [[ETop∞Grpd]], [[Smooth∞Grpd]] and [[SynthDiff∞Grpd]].

\end{example}
\hypertarget{over_locally_connected_topological_spaces}{}\subsubsection*{{Over locally $n$-connected topological spaces}}\label{over_locally_connected_topological_spaces}

\begin{example}
\label{}\hypertarget{}{}
For $X$ a [[locally contractible space]], $Sh_{(\infty,1)}(X)$ is a locally $\infty$-connected $(\infty,1)$-topos.

\end{example}
\begin{proof}
The full subcategory $cOp(X) \hookrightarrow Op(X)$ of the [[category of open subsets]] on the contractible subsets is another site of definition for $Sh_{(\infty,1)}(X)$. And it is a [[locally ∞-connected (∞,1)-site]].

\end{proof}
By the same kind of argument:

\begin{exampke}
\label{}\hypertarget{}{}
For $n \in \mathbb{N}$ and for $X$ a locally $n$-[[n-connected space|connected]] [[topological space]], $Sh_{(n+1,1)}(X)$ is a locally $n$-connected $(n+1)$-topos.

\end{exampke}
\begin{prop}
\label{}\hypertarget{}{}
For $X$ a [[locally contractible topological space]] we have that the [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos]] computes the correct [[homotopy type]] of $X$:

the image of $X$ as the [[terminal object in an (∞,1)-category|terminal object]] in $Sh_{(\inffty,1)}(C)$ under the [[fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos]]-functor

\begin{displaymath}
\Pi : Sh_{(\infty,1)}(X) \to \infty Grpd
\end{displaymath}
is equivalent to the ordinary [[fundamental ∞-groupoid]] given by the [[singular simplicial complex]]

\begin{displaymath}
\Pi(X) \simeq Sing X
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By using the [[presentable (∞,1)-category|presentations]] of $Sh_{(\infty,1)}(X)$ by the [[model structure on simplicial presheaves]] as discussed at [[locally ∞-connected (∞,1)-site]] one finds that this boils down to the old Artin-Mazur theorem. More on this at [[geometric homotopy groups in an (∞,1)-topos]].

\end{proof}
\hypertarget{locally_connected_overtoposes}{}\subsubsection*{{Locally $\infty$-connected over-$(\infty,1)$-toposes}}\label{locally_connected_overtoposes}

\begin{prop}
\label{}\hypertarget{}{}
For $\mathbf{H}$ a locally $\infty$-connected $(\infty,1)$-topos, also all its objects $X \in \mathbf{H}$ are locally $\infty$-connected, in that their [[petit topos|petit]] [[over-(∞,1)-toposes]] $\mathbf{H}/X$ are locally $\infty$-connected.

The two notions of fundamental $\infty$-groupoids of $X$ induced this way do agree, in that there is a natural equivalence

\begin{displaymath}
\Pi_X(X \in \mathbf{H}/X) \simeq \Pi(X \in \mathbf{H})
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the general facts recalled at [[etale geometric morphism]] we have a composite [[essential geometric morphism]]

\begin{displaymath}
(\Pi_X \dashv \Delta_X \dashv \Gamma_X) : 
  \mathbf{H}_{/X}
   \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{\X_*}{\to}}}
  \mathbf{H}
   \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}}  
   \infty Grpd
\end{displaymath}
and $X_!$ is given by sending $(Y \to X) \in \mathbf{H}/X$ to $Y \in \mathbf{H}$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
If in the above $X$ is contractible in that $\Pi X \simeq *$ then $\mathbf{H}/X$ is even an [[∞-connected (∞,1)-topos]].

\end{remark}
\begin{proof}
By the discussion there we need to check that $\Pi_X$ preserves the terminal object:

\begin{displaymath}
\Pi_X (X \to X) \simeq \Pi X_! (X \to X) \simeq \Pi X \simeq *
  \,.
\end{displaymath}
\end{proof}
\hypertarget{LocInfConnProperties}{}\subsection*{{Properties}}\label{LocInfConnProperties}

\hypertarget{relation_to_slicing}{}\subsubsection*{{Relation to slicing}}\label{relation_to_slicing}

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{X}$ be an $(\infty,1)$-topos and $\{U_i\}_i$ a collection of [[objects]] such that

\begin{itemize}%
\item the canonical morphism $\coprod_i U_i \to *$ out of their [[coproduct]] to the [[terminal object]] is an [[effective epimorphism in an (∞,1)-category|effective epimorphism]];


\item all the [[slice-(∞,1)-toposes]] $\mathcal{X}_{/U_i}$ are locally $\infty$-connected.



\end{itemize}
Then also $\mathcal{X}$ itself is locally $\infty$-connected.

\end{prop}
This appears as (\hyperlink{Lurie}{Lurie, corollary A.1.7}).

\hypertarget{RelationToLocallyConnectedToposes}{}\subsubsection*{{Relation to locally connected toposes}}\label{RelationToLocallyConnectedToposes}

\begin{prop}
\label{}\hypertarget{}{}
For $(\Pi \dashv \Delta \dashv \Gamma) :  \mathbf{H} \to \infty Grpd$ a locally $\infty$-connected $(\infty,1)$-topos, its underlying [[(n,1)-topos|(1,1)-topos]] $\tau_{\leq 0} \mathbf{H}$ is a [[locally connected topos]]. Moreover, if $\mathbf{H}$ is strongly connected (the extra left adjoint preserves finite products), then so is $\tau_{\leq 0} \mathbf{H}$.

\end{prop}
\begin{proof}
The [[global sections]] [[geometric morphism]] $\Gamma \simeq \mathbf{H}(*,-)$ is given by homming out of the terminal object and hence preserves [[n-truncated object in an (infinity,1)-category|0-truncated]] objects by definition. Also, by the $(\Pi \dahsv \Delta)$-[[adjunction]] so does $\Delta$: for every $S \in Set \simeq \tau_{\leq }\infty Grpd \hookrightarrow \infty Grpd$ and every $X \in \mathbf{H}$ we have

\begin{displaymath}
\mathbf{H}(X, \Delta(S))
  \simeq
  \infty Grpd(\Pi(X), S)
  \simeq
  Set(\tau_{\leq 0} \Pi(X), S)
  \in 
  Set \hookrightarrow \infty Grpd
  \,.
\end{displaymath}
Therefore by essential uniqueness of [[adjoints]] the restriction $\Delta|_{\leq 0} \colon Set \hookrightarrow \infty Grpd \stackrel{\Delta}{\to} \mathbf{H}$ has a [[left adjoint]] given by

\begin{displaymath}
\Pi_0 \coloneqq \tau_{\leq 0} \circ \Pi
  \,.
\end{displaymath}
Finally, by the discussion \href{n-truncated+object+of+an+%28infinity%2C1%29-category#PropertiesGeneral}{here}, $\tau_{\leq 0}$ preserves finite limits. Hence $\Pi_0$ does so if $\Pi$ does.

\end{proof}
\hypertarget{further_structures}{}\subsection*{{Further structures}}\label{further_structures}

The fact that the terminal geometric morphism is essential gives rise to various induced structures of interest. For instance it induces a notion of

\begin{itemize}%
\item [[Whitehead tower in an (∞,1)-topos]].

\end{itemize}
For a more exhaustive list of extra structures see [[cohesive (∞,1)-topos]].

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

\begin{itemize}%
\item [[locally connected topos]] / \textbf{locally ∞-connected (∞,1)-topos}

\begin{itemize}%
\item [[connected topos]] / [[∞-connected (∞,1)-topos]]


\item [[strongly connected topos]] / [[strongly ∞-connected (∞,1)-topos]]


\item [[totally connected topos]] / [[totally ∞-connected (∞,1)-topos]]



\end{itemize}

\item [[local topos]] / [[local (∞,1)-topos]].


\item [[cohesive topos]] / [[cohesive (∞,1)-topos]]



\end{itemize}
and

\begin{itemize}%
\item [[locally connected site]], [[locally ∞-connected site]]


\item [[connected site]]


\item [[local site]]


\item [[cohesive site]], [[(∞,1)-cohesive site]]



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

Some discussion of the [[homotopy category of an (∞,1)-category|homotopy category]] of locally $\infty$-connected $(\infty,1)$-toposes is around proposition 2.18 of

\begin{itemize}%
\item [[Carlos Simpson]], [[Constantin Teleman]], \emph{de Rham's theorem for $\infty$-stacks} (\href{http://math.berkeley.edu/~teleman/math/simpson.pdf}{pdf})

\end{itemize}
Under the term \emph{locally constant shape} the notion appears in section A.1 of

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

\end{itemize}
See also

\begin{itemize}%
\item [[Marc Hoyois]], \emph{A note on \'E{}tale homotopy}, 2013 (\href{http://math.northwestern.edu/~hoyois/papers/etalehomotopy.pdf}{pdf})

\end{itemize}
For related references see

\begin{itemize}%
\item [[geometric homotopy groups in an (∞,1)-topos]]


\item [[cohesive (∞,1)-topos]].



\end{itemize}
[[!redirects locally n-connected (n,1)-topos]] [[!redirects locally n-connected (n,1)-toposes]] [[!redirects locally n-connected (n,1)-topoi]] [[!redirects locally n-connected (n+1,1)-topos]] [[!redirects locally n-connected (n+1,1)-toposes]] [[!redirects locally n-connected (n+1,1)-topoi]] [[!redirects locally n-simply connected (n+1,1)-topos]] [[!redirects locally n-simply connected (n+1,1)-toposes]] [[!redirects locally n-simply connected (n+1,1)-topoi]]

[[!redirects locally infinity-connected (n,1)-topos]] [[!redirects locally infinity-connected (n,1)-toposes]] [[!redirects locally infinity-connected (n,1)-topoi]] [[!redirects locally ∞-connected (n,1)-topos]] [[!redirects locally ∞-connected (n,1)-toposes]] [[!redirects locally ∞-connected (n,1)-topoi]] [[!redirects locally contractible (n,1)-topos]] [[!redirects locally contractible (n,1)-toposes]] [[!redirects locally contractible (n,1)-topoi]]

[[!redirects locally n-connected (∞,1)-topos]] [[!redirects locally n-connected (∞,1)-toposes]] [[!redirects locally n-connected (∞,1)-topoi]] [[!redirects locally n-connected (infinity,1)-topos]] [[!redirects locally n-connected (infinity,1)-toposes]] [[!redirects locally n-connected (infinity,1)-topoi]] [[!redirects locally n-connected (infinity,1)-tops]]

[[!redirects locally ∞-connected (∞,1)-topos]] [[!redirects locally ∞-connected (∞,1)-toposes]] [[!redirects locally ∞-connected (∞,1)-topoi]] [[!redirects locally ∞-connected (infinity,1)-topos]] [[!redirects locally ∞-connected (infinity,1)-toposes]] [[!redirects locally ∞-connected (infinity,1)-topoi]] [[!redirects locally infinity-connected (∞,1)-topos]] [[!redirects locally infinity-connected (∞,1)-toposes]] [[!redirects locally infinity-connected (∞,1)-topoi]] [[!redirects locally infinity-connected (infinity,1)-topos]] [[!redirects locally infinity-connected (infinity,1)-toposes]] [[!redirects locally infinity-connected (infinity,1)-topoi]] [[!redirects locally contractible (∞,1)-topos]] [[!redirects locally contractible (∞,1)-toposes]] [[!redirects locally contractible (∞,1)-topoi]] [[!redirects locally contractible (infinity,1)-topos]] [[!redirects locally contractible (infinity,1)-toposes]] [[!redirects locally contractible (infinity,1)-topoi]]

[[!redirects n-connected (n,1)-topos]] [[!redirects n-connected (n,1)-toposes]] [[!redirects n-connected (n,1)-topoi]] [[!redirects n-connected (n+1,1)-topos]] [[!redirects n-connected (n+1,1)-toposes]] [[!redirects n-connected (n+1,1)-topoi]] [[!redirects n-simply connected (n+1,1)-topos]] [[!redirects n-simply connected (n+1,1)-toposes]] [[!redirects n-simply connected (n+1,1)-topoi]]

[[!redirects ∞-connected (n,1)-topos]] [[!redirects ∞-connected (n,1)-toposes]] [[!redirects ∞-connected (n,1)-topoi]] [[!redirects infinity-connected (n,1)-topos]] [[!redirects infinity-connected (n,1)-toposes]] [[!redirects infinity-connected (n,1)-topoi]] [[!redirects contractible (n,1)-topos]] [[!redirects contractible (n,1)-toposes]] [[!redirects contractible (n,1)-topoi]]

[[!redirects n-connected (∞,1)-topos]] [[!redirects n-connected (∞,1)-toposes]] [[!redirects n-connected (∞,1)-topoi]] [[!redirects n-connected (infinity,1)-topos]] [[!redirects n-connected (infinity,1)-toposes]] [[!redirects n-connected (infinity,1)-topoi]]

[[!redirects ∞-connected (∞,1)-topos]] [[!redirects ∞-connected (∞,1)-toposes]] [[!redirects ∞-connected (∞,1)-topoi]] [[!redirects ∞-connected (infinity,1)-topos]] [[!redirects ∞-connected (infinity,1)-toposes]]

[[!redirects ∞-connected (infinity,1)-topoi]] [[!redirects infinity-connected (∞,1)-topos]] [[!redirects infinity-connected (∞,1)-toposes]] [[!redirects infinity-connected (∞,1)-topoi]] [[!redirects infinity-connected (infinity,1)-topos]] [[!redirects infinity-connected (infinity,1)-toposes]] [[!redirects infinity-connected (infinity,1)-topoi]] [[!redirects contractible (∞,1)-topos]] [[!redirects contractible (∞,1)-toposes]] [[!redirects contractible (∞,1)-topoi]] [[!redirects contractible (infinity,1)-topos]] [[!redirects contractible (infinity,1)-toposes]] [[!redirects contractible (infinity,1)-topoi]]

[[!redirects locally connected (∞,1)-geometric morphism]] [[!redirects locally connected (∞,1)-geometric morphisms]] [[!redirects locally connected (infinity,1)-geometric morphism]] [[!redirects locally connected (infinity,1)-geometric morphisms]]

[[!redirects connected (∞,1)-geometric morphism]] [[!redirects connected (∞,1)-geometric morphisms]] [[!redirects connected (infinity,1)-geometric morphism]] [[!redirects connected (infinity,1)-geometric morphisms]]

[[!redirects essential (∞,1)-geometric morphism]] [[!redirects essential (∞,1)-geometric morphisms]] [[!redirects essential (infinity,1)-geometric morphism]] [[!redirects essential (infinity,1)-geometric morphisms]]

[[!redirects locally constant shape]] [[!redirects ((∞,1)-topos of locally constant shape)]] [[!redirects ((infinity,1)-topos of locally constant shape)]] [[!redirects ((∞,1)-toposes of locally constant shape)]] [[!redirects ((infinity,1)-toposes of locally constant shape)]]

[[!redirects locally ∞-connected geometric morphism]] [[!redirects locally ∞-connected geometric morphisms]]



\end{document}