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\begin{document}

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\section*{shape 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{shape_of_a_locally_connected_topos}{Shape of a locally $\infty$-connected topos}\dotfill \pageref*{shape_of_a_locally_connected_topos} \linebreak
\noindent\hyperlink{shape_of_a_topological_space}{Shape of a topological space}\dotfill \pageref*{shape_of_a_topological_space} \linebreak
\noindent\hyperlink{Retract}{Shape of an essential retract}\dotfill \pageref*{Retract} \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}

If an [[(∞,1)-topos]] $\mathbf{H}$ is that of [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaves]] on (the [[site]] of [[category of open subsets|open subsets]] of) a [[paracompact space|paracompact]] [[topological space]] -- $\mathbf{H} = Sh_{(\infty,1)}(X)$ -- then its \textbf{shape} is the \emph{strong shape} of $X$ in the sense of [[shape theory]]: a [[pro-object]] $Shape(X)$ in the category of [[CW-complex]]es.

It turns out that $Shape(X)$ may be extracted in a canonical fashion from just the [[(∞,1)-topos]] $Sh_{(\infty,1)}(X)$, and in a way that makes sense for any [[(∞,1)-topos]]. This then gives a definition of shape of general $(\infty,1)$-toposes.

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

\begin{udef}
The composite [[(∞,1)-functor]]

\begin{displaymath}
\Pi : (\infty,1)Topos 
     \stackrel{Y}{\to}  Func((\infty,1)Topos, \infty Grpd)^{op}
      \stackrel{Lex(PSh(-), \infty Grpd)}{\to}
      AccLex(\infty Grpd, \infty Grpd)^{op}
     \simeq
       Pro \infty Grpd
\end{displaymath}
is the \textbf{shape functor} . Its value

\begin{displaymath}
\Pi(\mathbf{H}) = 
  (\infty,1)Topos(\mathbf{H}, PSh(-))
\end{displaymath}
on an $(\infty,1)$-topos $\mathbf{H}$ is the \textbf{shape} of $\mathbf{H}$.

\end{udef}
Here

\begin{itemize}%
\item [[(∞,1)Topos]] is the [[(∞,1)-category]] of [[(∞,1)-topos]]es;


\item [[∞Grpd]] is the $(\infty,1)$-category of [[∞-groupoid]]s;


\item $Y$ is the [[(∞,1)-Yoneda embedding]];


\item $Func(-,-)$ is the [[(∞,1)-category of (∞,1)-functors]];


\item $AccLex(-,-) \subset (\infty,1)Func(-,-)$ is the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those which are left [[exact functor]]s (preserve finite [[(∞,1)-limit]]s) and also [[accessible (∞,1)-functor|accessible]].


\item $PSh(-) : \infty Grpd \to (\infty,1)Topos$ is the functor that produces the [[(∞,1)-category of (∞,1)-presheaves]] $Func(X^{op}, \infty Grpd)$ on $X$ (equivalently on the equivalent [[opposite (∞,1)-category|opposite ∞-groupoid]] $X^{op}$);


\item $Pro \infty Grpd$ is the [[pro-object in an (∞,1)-category|(∞,1)-category of pro-objects]] in $\infty Grpd$.



\end{itemize}
That this does indeed land in accessible left exact functors is shown below.

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

Notice that for every [[(∞,1)-topos]] $\mathbf{H}$ there is a unique [[geometric morphism]]

\begin{displaymath}
(LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
\end{displaymath}
where [[∞Grpd]] is the $(\infty,1)$-topos of [[∞-groupoids]], $\Gamma$ is the [[global section]]s [[(∞,1)-functor]] and $LConst$ is the [[constant ∞-stack]] functor.

\begin{uprop}
The \textbf{shape} of $\mathbf{H}$ is the composite functor

\begin{displaymath}
\Pi(\mathbf{H}) := \Gamma \circ LConst
  \;\;:\;\;
  \infty Grpd \stackrel{LConst}{\to}
  \mathbf{H}
  \stackrel{\;\;\Gamma\;\;}{\to}
  \infty Grpd
\end{displaymath}
regarded as an object

\begin{displaymath}
\Pi(\mathbf{H})
  \in
  Pro(\infty Grpd)
  =
  Lex(\infty Grpd, \infty Grpd)^{op}
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
For $X \in$ [[∞Grpd]] we have by the [[(∞,1)-Grothendieck construction]]-theorem and using that up to equivalence every morphism of $\infty$-groupoids is a [[Cartesian fibration]] (see there) that

\begin{displaymath}
Func(X,\infty Grpd) \simeq \infty Grpd/X
\end{displaymath}
is the [[over-(∞,1)-category]]. Moreover, by the  we have that the terminal geometric morphism $Hom(*,-): [X, \infty Grpd] \to \infty Grpd$ is the canonical projection $\infty Grpd/ X \to \infty Grpd$. This means that it is an [[etale geometric morphism]]. So for any [[geometric morphism]] $f : \mathbf{H} \to [X, \infty Grpd]$ we have a system of [[adjoint (∞,1)-functor]]s

\begin{displaymath}
(LConst \dashv \Gamma)
   :
  \mathbf{H}
    \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}
   \infty Grpd/X
    \stackrel{\overset{\pi^*}{\leftarrow}}{\underset{\pi_*}{\to}}    
   \infty Grpd
  \,.
\end{displaymath}
whose composite is the [[global section]] geometric morphism as indicated, because that is terminal.

Notice that in $\infty Grpd/X$ there is a canonical morphism

\begin{displaymath}
(* \to  \pi^* X) := (X \stackrel{(Id,Id)}{\to} X \times X)
  \,.
\end{displaymath}
The image of this under $f^*$ is (using that this preserves the terminal object) a morphism

\begin{displaymath}
* \to f^* \pi^* X = LConst X
\end{displaymath}
in $\mathbf{H}$.

Conversely, given a morphism of the form $* \to LConst X$ in $\mathbf{H}$ we obtain the [[base change geometric morphism]]

\begin{displaymath}
\mathbf{H} \simeq \mathbf{H}/* \to \mathbf{H}/LConst X
   \stackrel{\Gamma}{\to}
  \infty Grpd/X
  \,.
\end{displaymath}
One checks that these constructions establish an equivalence

\begin{displaymath}
(\infty,1)Topos(\mathbf{H}, \infty Grpd/X)
  \simeq
  \mathbf{H}(*, LConst X)
  \,.
\end{displaymath}
Using this, we see that

\begin{displaymath}
\begin{aligned}
    \Pi (\mathbf{H}) : X \mapsto & (\infty,1)Topos(\mathbf{H}, X)
     \\
      & \simeq \mathbf{H}(*,LConst X)
     \\
      & \simeq \mathbf{H}(LConst *, LConst X)
     \\
      & \simeq \infty Grpd(*, \Gamma LConst X)
    \\
      & \simeq \Gamma LConst X
  \end{aligned}   
  \,.
\end{displaymath}
\end{proof}
\begin{uremark}
In particular this does show that $\Pi(\mathbf{H}) : \infty Grpd \to \infty Grpd$ does preserve finite $(\infty,1)$-limits, since $\Gamma$ preserves all limits and $LConst$ is a left [[exact functor]]. It also shows that it is accessible, since $\Gamma$ and $LConst$ are both accessible.

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

\hypertarget{shape_of_a_locally_connected_topos}{}\subsubsection*{{Shape of a locally $\infty$-connected topos}}\label{shape_of_a_locally_connected_topos}

Suppose that $\mathbf{H}$ is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]], meaning that $\LConst$ has a left adjoint $\Pi$ which constructs the [[geometric homotopy groups in an (∞,1)-topos|homotopy ∞-groupoids]] of objects of $\mathbf{H}$. Then $\Shape(\mathbf{H})$ is [[representable functor|represented]] by $\Pi(*)\in \infty Grpd$, for we have the following sequence of [[natural equivalence|natural]] [[equivalences of ∞-groupoids]]:

\begin{displaymath}
\begin{aligned}
Shape(\mathbf{H})(A) &\simeq \Gamma(LConst(A))\\
&\simeq Hom_{\infty Grpd}(*, \Gamma(LConst(A)))\\
&\simeq Hom_{\mathbf{H}}(LConst(*), LConst(A)) \\
&\simeq Hom_{\mathbf{H}}(*, LConst(A)) \\
&\simeq Hom_{\infty Grpd}(\Pi(*),A).
\end{aligned}
\end{displaymath}
Thus, if we regard $\Pi(*)$ as ``the [[fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos|fundamental ∞-groupoid]] of $\mathbf{H}$'' --- which is reasonable since when $\mathbf{H}=Sh(X)$ consists of sheaves on a locally contractible [[topological space]] $X$, $\Pi_{\mathbf{H}}(*)$ is equivalent to the usual [[fundamental ∞-groupoid]] of $X$ --- then we can regard the shape of an $(\infty,1)$-topos as a generalized version of the ``homotopy $\infty$-groupoid'' which nevertheless makes sense even for non-locally-contractible toposes, by taking values in the larger category of ``pro-$\infty$-groupoids.''

It follows also that $\mathbf{H}$ is not only locally ∞-connected but also [[∞-connected (∞,1)-topos|∞-connected]], then it has the shape of a point.

\hypertarget{shape_of_a_topological_space}{}\subsubsection*{{Shape of a topological space}}\label{shape_of_a_topological_space}

For a discussion of how the $(\infty,1)$-topos theoretic shape of $Sh_{(\infty,1)}(X)$ relates to the ordinary shape-theoretic \emph{strong shape} of the topological space $X$ see [[shape theory]].

\hypertarget{Retract}{}\subsubsection*{{Shape of an essential retract}}\label{Retract}

The following is trivial to observe, but may be useful to note.

\begin{ulemma}
Let $(f_! \dashv f^* \dashv f_*) : \mathbf{H} \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} \mathbf{B}$ be an [[essential geometric morphism]] of $(\infty,1)$-toposes that exhibits $\mathbf{B}$ as an essential [[retract]] of $\mathbf{H}$ in that

\begin{displaymath}
(Id \dashv Id)
  \;\;
  \simeq
  \;\;
  \mathbf{B}
    \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} 
  \mathbf{H}
    \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} 
  \mathbf{B}
  \,.
\end{displaymath}
Then the shape of $\mathbf{B}$ is equivalent to that of $\mathbf{H}$.

\end{ulemma}
\begin{proof}
Since $\infty Grpd$ is the [[terminal object]] in the category of Grothendieck $(\infty,1)$-toposes and [[geometric morphisms]], we have

\begin{displaymath}
\begin{aligned}
    (\infty Grpd \stackrel{LConst_{\mathbf{B}}}{\to} \mathbf{B} 
     \stackrel{\Gamma_\mathbf{B}}{\to} \infty Grpd)
    &\simeq
    (\infty Grpd 
      \stackrel{LConst_{\mathbf{B}}}{\to} 
      \mathbf{B} 
      \stackrel{f^*}{\to} \mathbf{H}   
     \stackrel{f_*}{\to} \mathbf{B}
     \stackrel{\Gamma_\mathbf{B}}{\to}
     \infty Grpd)
    \\
    &\simeq
    (\infty Grpd 
     \stackrel{LConst_\mathbf{H}}{\to} 
     \mathbf{H} 
      \stackrel{\Gamma_\mathbf{H}}{\to}
      \infty Grpd)    
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{uexample}
Every

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


\item and hence also every [[cohesive (∞,1)-topos]]



\end{itemize}
over $\infty Grpd$ has the shape of the point.

\end{uexample}
\begin{proof}
By definition $\mathbf{H}$ is $\infty$-connected if the [[constant ∞-stack]] [[inverse image]] $f^* = L Const$ is

\begin{enumerate}%
\item not only a left but also a [[right adjoint]];


\item is a [[full and faithful (∞,1)-functor]].



\end{enumerate}
By standard properties of [[adjoint (∞,1)-functor]]s we have that a [[right adjoint]] $f^*$ is a [[full and faithful (∞,1)-functor]] precisely if the counit $f_! f^* \to Id$ is an [[equivalence in a quasi-category|equivalence]].

Equivalently, we can observe that a locally ∞-connected (∞,1)-topos is ∞-connected precisely when $\Pi$ preserves the terminal object, and apply the above observation that the shape of a locally ∞-connected (∞,1)-topos is represented by $\Pi(*)$.

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

\begin{itemize}%
\item \textbf{shape of an $(\infty,1)$-topos}


\item [[coshape of an (∞,1)-topos]]



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

The definition of shape of $(\infty,1)$-toposes as $\Gamma \circ LConst$ is due to

\begin{itemize}%
\item [[Bertrand Toen]], [[Gabriele Vezzosi]], \emph{Segal topoi and stacks over Segal categories} , in Proceedings of the Program \emph{Stacks, Intersection theory and Non-abelian Hodge Theory} , MSRI, Berkeley, January-May 2002 (\href{http://arxiv.org/abs/math/0212330}{arXiv:math/0212330}).

\end{itemize}
This and the relation to [[shape theory]], more precisely the [[strong shape]], of topological spaces is further discussed in section 7.1.6 of

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

\end{itemize}
See also

\begin{itemize}%
\item [[Marc Hoyois]], \emph{Higher Galois theory} (\href{http://math.mit.edu/~hoyois/papers/highergalois.pdf}{pdf}).

\end{itemize}
[[!redirects shape of an (∞,1)-topos]]



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