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\begin{document}

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\section*{homotopy dimension}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{}\hypertarget{}{}
An [[(∞,1)-topos]] $\mathcal{X}$ has \textbf{homotopy dimension} $\leq n \in \mathbb{N}$ if every [[n-connected|(n-1)-connected]] [[object]] $A$ has a [[global element]], a [[morphism]] $* \to A$ from the [[terminal object in an (∞,1)-category|terminal object]] into it.

\end{defn}
This appears as [[Higher Topos Theory|HTT, def. 7.2.1.1]].

\begin{defn}
\label{}\hypertarget{}{}
An [[(∞,1)-topos]] $\mathcal{X}$ is \textbf{locally of homotopy dimension} $\leq n \in \mathbb{N}$ if there exists a collection $\{U_i \in \mathcal{X}\}$ of [[object]]s such that

\begin{itemize}%
\item the $\{U_i\}$ generate $\mathcal{X}$ under [[(∞,1)-colimit]]s;


\item each [[over-(∞,1)-topos]] $\mathcal{X}/U_i$ has homotopy dimension $\leq n$.



\end{itemize}
\end{defn}
This appears as [[Higher Topos Theory|HTT, def. 7.2.1.8]].

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

\begin{prop}
\label{}\hypertarget{}{}
If an [[(∞,1)-topos]] $\mathcal{X}$ is locally of homotopy dimension $\leq n$ for some $n \in \mathbb{N}$ then it is a [[hypercomplete (∞,1)-topos]].

\end{prop}
This appears as [[Higher Topos Theory|HTT, cor. 7.2.1.12]].

\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{X}$ has homotopy dimension $\leq n$ then it also has [[cohomological dimension]] $\leq n$.

The converse holds if $\mathcal{X}$ has finite homotopy dimension and $n \geq 2$.

\end{prop}
This appears as [[Higher Topos Theory|HTT, cor. 7.2.2.30]].

\begin{prop}
\label{RecognitionByGlobalSections}\hypertarget{RecognitionByGlobalSections}{}
An [[(∞,1)-topos]] $\mathcal{X}$ has homotopy dimension $\leq n$ precisely if the [[global section]] [[(∞,1)-geometric morphism]]

\begin{displaymath}
(\Delta \dashv \Gamma) : \mathcal{X} 
    \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} 
  \infty Grpd
\end{displaymath}
has the property that $\Gamma$ sends $(k\geq n)$-[[n-connective|connective]] morphisms to $(k-n)$-[[n-connective|connective]] morphisms.

\end{prop}
This is [[Higher Topos Theory|HTT, lemma 7.2.1.7]]

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

\begin{prop}
\label{}\hypertarget{}{}
Up to [[equivalence in an (∞,1)-category|equivalence]], the unique [[(∞,1)-topos]] of homotopy dimension $\leq -1$ is the the [[terminal category]] $* \simeq Sh_{(\infty,1)}(\emptyset)$.

\end{prop}
This is [[Higher Topos Theory|HTT, example. 7.2.1.2]].

\begin{proof}
An object $X \in \mathcal{X}$ is $(-1)$-[[connected]] if the morphism $X \to *$to the [[terminal object in an (∞,1)-category]] is. This is the case if it is an [[effective epimorphism]].

Since the [[global section]] [[(∞,1)-functor]] is corepresented by the terminal object, $X$ is 0-connective precisely if $\Gamma(X) \to \Gamma(*) = *$ is an epimorphism on connected components. By the discussion at [[effective epimorphism]], this is the case precisely if $\Gamma(X) \to *$ is an effective epimorphism in [[∞Grpd]].

So $\mathcal{X}$ has homotopy dimension $\leq 0$ if $\Gamma$ preserves effective epimorphisms. This is the case if it preserves [[finite limit|finit]] [[(∞,1)-limit]]s (the [[(∞,1)-pullback]]s defining a [[Cech nerve]]) and all [[(∞,1)-colimit]]s (over the resulting Cech nerve). being a [[right adjoint|right]] [[adjoint (∞,1)-functor]] $\Gamma$ always preserves [[(∞,1)-limit]]s. If $\mathcal{X}$ is [[local (∞,1)-topos|local]] then $\Gamma$ is by definition also a [[left adjoint]] and hence also preserves [[(∞,1)-colimit]]s.

\end{proof}
\begin{prop}
\label{LocalToposesHaveHomotopDimension0}\hypertarget{LocalToposesHaveHomotopDimension0}{}
Every [[local (∞,1)-topos]] has homotopy dimension $\leq 0$.

\end{prop}
\begin{proof}
Let

\begin{displaymath}
(\Delta \dashv \Gamma \dashv \nabla)
  : 
  \mathbf{H} \to \infty Grpd
\end{displaymath}
be [[generalized the|the]] [[global section geometric morphism|terminal geometric morphism]] of the local $(\infty,1)$-topos, with $\nabla$ being the extra [[right adjoint]] to the [[global section]] [[(∞,1)-geometric morphism]] functor that characterizes locality.

By prop \ref{RecognitionByGlobalSections} it is sufficient to show that $\Gamma$ send [[(-1)-connected]] morphisms to (-1)-connected morphisms, hence [[effective epimorphism in an (∞,1)-category|effective epimorphisms]] to effective epimorphisms.

By the existence of $\nabla$ we have that $\Gamma$ preserves not only [[(∞,1)-limit]]s but also [[(∞,1)-colimit]]s. Since effective epimorphisms are defined as certain colimits over diagrams of certain limits, $\Gamma$ preserves effective epimorphisms.

\end{proof}
So in particular for $C$ any [[(∞,1)-category]] with a [[terminal object in an (∞,1)-category|terminal object]], the [[(∞,1)-category of (∞,1)-presheaves]] $PSh_{(\infty,1)}(C)$ is an [[(∞,1)-topos]] of homotopy dimension $\leq 0$. Notably [[Top]] $\simeq$ [[∞Grpd]] $\simeq PSh_{(\infty,1)}(*)$ has homotopy dimension $\leq 0$.

This is [[Higher Topos Theory|HTT, example. 7.2.1.3]].

\begin{prop}
\label{}\hypertarget{}{}
Every [[(∞,1)-category of (∞,1)-presheaves]] is an [[(∞,1)-topos]] of local homotopy dimension $\leq 0$.

\end{prop}
This appears as [[Higher Topos Theory|HTT, example. 7.2.1.9]].

\begin{theorem}
\label{}\hypertarget{}{}
If a [[paracompact topological space]] $X$ has [[covering dimension]] $\leq n$, then the [[(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(X) := Sh_{(\infty,11)}(Op(X))$ is an [[(∞,1)-topos]] of homotopy dimension $\leq n$.

\end{theorem}
This is [[Higher Topos Theory|HTT, theorem 7.2.3.6]].

\begin{prop}
\label{}\hypertarget{}{}
For $X \in$ [[∞Grpd]] $\simeq$ [[Top]] an object, the [[over-(∞,1)-topos]] $\infty Grpd/X$ has homotopy dimension $\leq n$ precisely if $X \in Top$ a [[retract]] in the [[homotopy category of an (∞,1)-category|homotopy category]] $Ho(Top)$ of a [[CW-complex]] of [[dimension]] $\leq n$.

\end{prop}
This is [[Higher Topos Theory|HTT, example 7.2.1.4]].

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

\begin{itemize}%
\item [[dimension]]

\begin{itemize}%
\item \textbf{homotopy dimension}


\item [[cohomological dimension]]


\item [[covering dimension]]


\item [[Heyting dimension]]



\end{itemize}


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

The [[(∞,1)-topos theory|(∞,1)-topos theoretic]] notion is discuss in section 7.2.1 of

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

\end{itemize}


\end{document}