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\section*{categorical homotopy groups in 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{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

This is a sub-entry of [[homotopy groups in an (∞,1)-topos]].

For the other notion of homotopy groups see [[geometric homotopy groups in an (∞,1)-topos]].

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{of_objects}{Of objects}\dotfill \pageref*{of_objects} \linebreak
\noindent\hyperlink{of_morphisms}{Of morphisms}\dotfill \pageref*{of_morphisms} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{in_}{In $\infty Grpd$}\dotfill \pageref*{in_} \linebreak
\noindent\hyperlink{of_homotopy_groups_of_morphisms}{Of homotopy groups of morphisms}\dotfill \pageref*{of_homotopy_groups_of_morphisms} \linebreak
\noindent\hyperlink{behaviour_under_geometric_morphisms}{Behaviour under geometric morphisms}\dotfill \pageref*{behaviour_under_geometric_morphisms} \linebreak
\noindent\hyperlink{connected_and_truncated_objects}{Connected and truncated objects}\dotfill \pageref*{connected_and_truncated_objects} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Recall that since an [[(∞,1)-topos]] $\mathbf{H}$ has all [[limits in a quasi-category|limits]], it is naturally \href{http://ncatlab.org/nlab/show/limit+in+a+quasi-category#Tensoring}{powered over ∞Grpd}:

\begin{displaymath}
(-)^{(-)} : \infty Grpd^{op} \times \mathbf{H} \to \mathbf{H}
  \,.
\end{displaymath}
Let $S^n = \partial \Delta[n+1]$ (or $S^n := Ex^\infty \partial \Delta[n+1]$) be the ([[Kan fibrant replacement]]) of the [[boundary of a simplex|boundary of the (n+1)-simplex]], i.e. the model in [[∞Grpd]] of the [[pointed object|pointed]] $n$-[[sphere]].

Then for $X \in \mathbf{H}$ an object, the power object $X^{S^n} \in \mathbf{H}$ plays the role of the space of of maps from the $n$-sphere into $X$, as in the definition of [[simplicial homotopy groups]], to which this reduces in the case that $\mathbf{H} =$ [[∞Grpd]].

By powering the canonical morphism $i_n : * \to S^n$ induces a morphism

\begin{displaymath}
X^{i_n} : X^{S^n} \to X
\end{displaymath}
which is restriction to the basepoint. This morphism may be regarded as an object of the [[over quasi-category|over]] [[(∞,1)-topos]] $\mathbf{H}_{/X}$.

\hypertarget{of_objects}{}\subsubsection*{{Of objects}}\label{of_objects}

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(categorical homotopy groups)}

For $n \in \mathbb{N}$ define

\begin{displaymath}
\pi_n(X) := \tau_{\leq 0} X^{i_n} \in \mathbf{H}_{/X}
\end{displaymath}
to be the [[n-truncated object of an (infinity,1)-topos|0-truncation]] of the object $X^{i_n}$.

\end{defn}
Passing to the 0-truncation here amounts to dividing out the [[homotopy|homotopies]] between maps from the $n$-sphere into $X$. The [[n-truncated object of an (infinity,1)-topos|0-truncated]] objects in $\mathbf{H}_{/X}$ have the interpretation of [[sheaf|sheaves]] on $X$. So in the world of [[∞-stack]]s a homotopy [[group object]] is a [[sheaf]] of groups.

To see that there is indeed a group structure on these \emph{homotopy sheaves} as usual, notice from the general properties of [[power]]ing we have that

\begin{displaymath}
X^{S^k \coprod_* S^l}
  \simeq
  X^{S_k} \times_X X^{S_l}
  \,.
\end{displaymath}
From the  we have that $\tau_{\leq n} : \mathbf{H} \to \mathbf{H}$ preserves such finite [[product]]s so that also

\begin{displaymath}
\tau_{\leq 0} X^{* \to S^k \coprod_* S^l}
  \simeq
  (\tau_{\leq 0} X^{* \to S^k} ) \times 
  (\tau_{\leq 0}) X^{* \to S^k}
  \,.
\end{displaymath}
Therefore the [[cogroup]] operations $S^n \to S^n \coprod_* S^n$ induce group operations

\begin{displaymath}
\pi_n(X) \times \pi_n(X) \to \pi_n(X)
\end{displaymath}
in the [[sheaf topos]] $\tau_{\leq 0} \mathbf{H}_{/X}$. By the usual argument about [[homotopy group]]s, these are trivial for $n = 0$ and [[abelian group|abelian]] for $n \geq 2$.

\hypertarget{of_morphisms}{}\subsubsection*{{Of morphisms}}\label{of_morphisms}

It is frequently useful to speak of homotopy groups of a [[morphism]] $f : X \to Y$ in an $(\infty,1)$-topos

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(homotopy groups of morphisms)}

For $f : X \to Y$ a [[morphism]] in an [[(∞,1)-topos]] $\mathbf{H}$, its \emph{homotopy groups} are the homotopy groups in the above sense of $f$ regarded as an object of the [[over quasi-category|over (∞,1)-category]] $\mathbf{H}_{/Y}$.

\end{defn}
So the homotopy sheaf $\pi_n(f)$ of a morphism $f$ is an object of the [[over quasi-category|over (∞,1)-category]] $Disc((\mathbf{H}_{/Y})_{/f}) \simeq Disc(\mathbf{H}_{/f})$. This in turn is equivalent to $\cdots \simeq \mathbf{H}_{/X}$ by the map that sends an object

\begin{displaymath}
\itexarray{
    && Q
    \\
    & \swarrow && \searrow
    \\
    X &&\stackrel{f}{\to}&& Y
  }
\end{displaymath}
in $\mathbf{H}_{/f}$ to

\begin{displaymath}
\itexarray{
    && Q
    \\
    & \swarrow 
    \\
    X
  }
  \,.
\end{displaymath}
The intuition is that the homotopy sheaf $\pi_n(f) \in Disc(\mathbf{H}_{/X})$ over a basepoint $x : * \in X$ is the homotopy group of the [[nLab:homotopy fiber]] of $f$ containing $x$ at $x$.

\begin{example}
\label{}\hypertarget{}{}
If $Y = *$ then there is an essentially unique morphism $f : X \to *$ whose [[homotopy fiber]] is $X$ itself. Accordingly $\pi_n(f) \simeq \pi_n(X)$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
If $X = *$ then the morphism $f : * \to Y$ is a point in $Y$ and the single [[homotopy fiber]] of $f$ is the [[loop space object]] $\Omega_f Y$.

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

\hypertarget{in_}{}\subsubsection*{{In $\infty Grpd$}}\label{in_}

For the case that $\mathbf{H} =$ [[∞Grpd]] $\simeq$ [[Top]], the $(\infty,1)$-topos theoretic definition of categorical homotopy groups in $\mathbf{H}$ reduces to the ordinary notion of [[homotopy group]]s in [[Top]]. For $\infty Grpd$ modeled by [[Kan complex]]es or the standard [[model structure on simplicial sets]], it reduces to the ordinary definition of [[simplicial homotopy group]]s.

\hypertarget{of_homotopy_groups_of_morphisms}{}\subsubsection*{{Of homotopy groups of morphisms}}\label{of_homotopy_groups_of_morphisms}

The definition of the homotopy groups of a morphism $f : X \to Y$ is equivalent to the following recursive definition

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(recursive homotopy groups of morphisms)}

For $n \geq 1$ we have

\begin{displaymath}
\pi_n(f) \simeq \pi_{n-1}(X \to X \times_Y X)
  \;\;\;
  \in 
  Disc(\mathbf{H}_{/X})
  \,.
\end{displaymath}
\end{defn}
This is [[Higher Topos Theory|HTT, remark 6.5.1.3]].

This is the generalization of the familiar fact that [[loop space object]]s have the same but shifted homotopy groups: In the special case that $X = *$ and $f$ is $f : * \to Y$ we have $X \times_Y X = \Omega_f Y$ and $X \to X \times_Y X$ is just $* \to \Omega_f Y$, so that

\begin{displaymath}
\pi_n(f) = \pi_n(Y)
\end{displaymath}
and

\begin{displaymath}
\pi_{n-1}(X \to X \times_Y X) \simeq \pi_{n-1} \Omega_f Y
  \,.
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
Given a sequence of morphisms $X \stackrel{f}{\to}Y \stackrel{g}{\to} Z$ in $\mathbf{H}$, there is a [[long exact sequence]]

\begin{displaymath}
\cdots
  \to
  f^* \pi_{n+1}(g) \stackrel{\delta_{n+1}}{\to}
  \pi_n(f)
  \stackrel{g \circ f}{\to}
  \to 
  f^* \pi_n(g)
  \stackrel{\delta_n}{\to}
  \pi_{n-1}(f)
  \to
  \cdots
\end{displaymath}
in the [[topos]] $Disc(\mathbf{H}_{/X})$.

\end{prop}
This is [[Higher Topos Theory|HTT, remark 6.5.1.5]].

\hypertarget{behaviour_under_geometric_morphisms}{}\subsubsection*{{Behaviour under geometric morphisms}}\label{behaviour_under_geometric_morphisms}

\begin{prop}
\label{}\hypertarget{}{}
Geometric morphisms of $(\infty,1)$-topos preserve homotopy groups.

If $k : \mathbf{H} \to \mathbf{K}$ is a [[geometric morphism]] of $(\infty,1)$-toposes then for $f : X \to Y$ any morphism in $\mathbf{H}$ there is a canonical [[isomorphism]]

\begin{displaymath}
k^* (\pi_n(f)) \simeq \pi_n(k^* f)
\end{displaymath}
in $Disc(\mathbf{H}_{/k^* Y})$.

\end{prop}
This is [[Higher Topos Theory|HTT, remark 6.5.1.4]].

\hypertarget{connected_and_truncated_objects}{}\subsubsection*{{Connected and truncated objects}}\label{connected_and_truncated_objects}

Let $X \in \mathbf{H}$.

\begin{itemize}%
\item The object $X$ is \textbf{$n$-[[truncated]]} if it is a [[n-truncated object in an (∞,1)-category|k-truncated object]] for some $k \gt n$ and if all its categorical homotopy groups above degree $n$ vanish.

Every object decomposes as a sequence of $n$-truncated objects: the [[Postnikov tower in an (∞,1)-category]].


\item The object $X$ is \textbf{$n$-[[connected]]} if the terminal morphism $X \to *$ is an [[effective epimorphism]] and if all categorical homotopy groups below degree $n$ are trivial.


\item The object $X$ is an \textbf{[[Eilenberg-MacLane object]]} of degree $n$ if it is both $n$-connected and $n$-truncated.



\end{itemize}
\hypertarget{models}{}\subsection*{{Models}}\label{models}

When the [[(∞,1)-topos]] $\mathbf{H}$ is [[presentable (∞,1)-category|presented]] by a [[model structure on simplicial presheaves]] $[C^{op}, sSet]_{loc}$, then since this is an [[sSet]]-[[enriched model category]] structure the [[power]]ing by $\infty Grpd$ is modeled, as described at,  by the ordinary powering

\begin{displaymath}
sSet^{op} \times [C^{op}, sSet] \to [C^{op}, sSet]
  \,,
\end{displaymath}
which is just objectwise the [[internal hom]] in [[sSet]]. Therefore the $(\infty,1)$-topos theoretical homotopy sheaves of an object in $([C^{op}, sSet]_{loc})^\circ$ are given by the following construction:

For $X \in [C^{op}, sSet]$ a presheaf, write

\begin{itemize}%
\item $\pi_0(X) \in [C^{op},Set]$ for the presheaf of connected components;


\item $\pi_n(X) = \coprod_{[x] \in \pi_0(X)} \pi_n(X,x)$ for the presheaf of [[simplicial homotopy group]]s with $n \geq 1$;


\item $\bar \pi_n(X) \to \bar \pi_0(X)$ for the [[sheafification]] of these presheaves.



\end{itemize}
Then these $\bar \pi_n(X) \to \bar \pi_0(X)$ are the sheaves of categorical homotopy groups of the object represented by $X$.

This definition of homotopy sheaves of [[simplicial presheaves]] is familiar from the Joyal-Jardine [[local model structure on simplicial presheaves]]. See for instance \href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf#page=4}{page 4} of \href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf}{Jard07}.

\begin{quote}%
this needs more discussion


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

The intrinsic $(\infty,1)$-theoretic description is the topic of section 6.5.1 of

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

\end{itemize}
The model in terms of the [[model structure on simplicial presheaves]] is duscussed for instance in

\begin{itemize}%
\item [[Rick Jardine]], \emph{Simplicial presheaves} (\href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf#page=4}{page 4})

\end{itemize}
[[!redirects categorical homotopy groups in an (∞,1)-topos]]

[[!redirects homotopy sheaf]] [[!redirects homotopy sheaves]]

[[!redirects categorical homotopy groups in an infinity-topos]] [[!redirects categorical homotopy groups]]



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