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\begin{document}

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\section*{simplicial object in an (infinity,1)-category}

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

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory contents]]

\hypertarget{internal_categories}{}\paragraph*{{Internal $(\infty,1)$-Categories}}\label{internal_categories}

[[!include internal infinity-categories 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{Powering}{Powering over simplicial sets}\dotfill \pageref*{Powering} \linebreak
\noindent\hyperlink{cohesion}{Cohesion}\dotfill \pageref*{cohesion} \linebreak
\noindent\hyperlink{internal_language}{Internal language}\dotfill \pageref*{internal_language} \linebreak
\noindent\hyperlink{geometric_realization_and_filtering}{Geometric realization and filtering}\dotfill \pageref*{geometric_realization_and_filtering} \linebreak
\noindent\hyperlink{doldkan_correspondence}{$\infty$-Dold-Kan correspondence}\dotfill \pageref*{doldkan_correspondence} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{internal_category_objects}{Internal category objects}\dotfill \pageref*{internal_category_objects} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{reference}{Reference}\dotfill \pageref*{reference} \linebreak


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

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{C}$ an [[(∞,1)-category]], a [[simplicial object]] in $\mathcal{C}$ is an [[(∞,1)-functor]]

\begin{displaymath}
X \colon \Delta^{op} \to \mathcal{C}
\end{displaymath}
from the [[opposite category]] of the [[simplex category]] into $\mathcal{C}$.

The $(\infty,1)$-category of simplicial objects in $\mathcal{C}$ and morphisms between them is the [[(∞,1)-category of (∞,1)-functors]]

\begin{displaymath}
\mathcal{C}^{\Delta^{op}}
  = 
  Func_\infty(\Delta^{op}, \mathcal{C})
  \,.
\end{displaymath}
\end{defn}
For instance (\hyperlink{Lurie}{Lurie, def. 6.1.2.2}).

\begin{remark}
\label{}\hypertarget{}{}
For $\mathcal{C}$ a [[1-category]] a simplicial object in $\mathcal{C}$ is a [[simplicial object]] in the traditional sense of [[category theory]].

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
A \emph{cosimplicial object} in $\mathcal{C}$ is a simplicial object in the [[opposite category]] $\mathcal{C}^{op}$.

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

\hypertarget{Powering}{}\subsubsection*{{Powering over simplicial sets}}\label{Powering}

Assume that $\mathcal{C}$ has all [[(∞,1)-limits]]. The following is a model for the [[powering]] of simplicial objects in $\mathcal{C}$ by simplicial sets.

\begin{defn}
\label{PoweringByLimitOverCategoryOfSimplices}\hypertarget{PoweringByLimitOverCategoryOfSimplices}{}
Let $\mathcal{C} \in QCat \hookrightarrow sSet$ be an [[(∞,1)-category]] incarnated as a [[quasi-category]], and let $X \colon \Delta^{op} \to \mathcal{C}$ be a simplicial object. Then for $K \in sSet$ any [[simplicial set]], write

\begin{displaymath}
X[K] \colon \Delta_{/K}^{op} \to \Delta^{op} \stackrel{X}{\to} \mathcal{C}
\end{displaymath}
for the the composite [[(∞,1)-functor]] of $X_\bullet$ with the projection from (the [[opposite category]] of) the [[category of simplices]] of $K$, and write

\begin{displaymath}
X(K) 
    \colon 
  \underset{\leftarrow}{\lim}
  \left(
    \Delta_{/K}^{op} \to \Delta^{op} \stackrel{X}{\to} \mathcal{C}
  \right)
\end{displaymath}
for the [[(∞,1)-limit]] over it (if it exists).

\end{defn}
This is discussed in (\hyperlink{Lurie}{Lurie HTT 4.2.3, notation 6.1.2.5}). See also around (\hyperlink{LurieGood}{Lurie 2, notation 1.1.7}).

\begin{remark}
\label{}\hypertarget{}{}
The inclusion $\Delta_{/K}^{nd} \hookrightarrow \Delta_{K}$ of the [[full subcategory]] on non-degenerate simplicies is a [[homotopy cofinal functor]] (as discussed there). Therefore the $(\infty,1)$-limit in def. \ref{PoweringByLimitOverCategoryOfSimplices} may equivalently be taken over this category of non-degenerate simplices.

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
For $K = \Delta^1 \coprod_{\Delta^0} \Delta^1$ the simplicial set consisting of two consecutive edges, we have for $X_\bullet \in \mathcal{C}^{\Delta^\bullet}$ that

\begin{displaymath}
X(K) \simeq X_1 \underset{X_0}{\times} X_1
\end{displaymath}
is the [[homotopy fiber product]] in

\begin{displaymath}
\itexarray{
    && X(K)
    \\
    & \swarrow && \searrow
    \\
    X_1 &&&& X_1
    \\
    & {}_{\mathllap{\partial_1}}\searrow && \swarrow_{\mathrlap{\partial_0}}
    \\
    && X_0
  }
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
For $K = \Delta^n$ itself an $n$-[[simplex]], for some $n \in \mathbb{N}$ the powering reduces to evaluation on that simplex:

\begin{displaymath}
X(\Delta^n) \simeq X_n
  \,.
\end{displaymath}
This is because the [[category of non-degenerate simplices]] of an $n$-simplex has a [[terminal object]] (namely that $n$-simplex itself), and so its [[opposite category]] has an [[initial object]] and the $(\infty,1)$-limit over a [[diagram]] with initial object is given by evaluation at that initial object.

\end{example}
\begin{remark}
\label{EquivalenceOfConeCategoriesAndLimits}\hypertarget{EquivalenceOfConeCategoriesAndLimits}{}
For $X_\bullet \in \mathcal{C}^{\Delta^{op}}$ and $K \to K'$ the following are equivalent

\begin{enumerate}%
\item the induced morphism of cone $(\infty,1)$-categoris $\mathcal{C}_{X[K]} \to \mathcal{C}_{X[K']}$ is an [[equivalence of (∞,1)-categories]];


\item the induced morphism of [[(∞,1)-limits]] $X(K) \to X(K')$ is an [[equivalence in an (∞,1)-category|equivalence]].



\end{enumerate}
\end{remark}
(The first perspective is used in (\hyperlink{Lurie}{Lurie}), the second in (\hyperlink{Lurie2}{Lurie2}).)

\begin{proof}
In one direction: the limit is the [[terminal object in an (∞,1)-category|terminal object]] in the cone category, and so is preserved by equivalences of cone categories. (This direction appears as (\hyperlink{Lurie}{Lurie, prop. 4.1.1.8})). Conversely, the limits is the object representing cones, and hence an equivalence of limits induces an equivalence of cone categories.

\end{proof}
\begin{prop}
\label{SlicingOverPoweringOfSimplicialObjects}\hypertarget{SlicingOverPoweringOfSimplicialObjects}{}
Let $X \colon \Delta^{op} \to \mathcal{C}$ be a simplicial object which is a [[groupoid object in an (∞,1)-category]].

If $K \to K'$ is a morphism in [[sSet]] which is a [[weak homotopy equivalence]] and a [[bijection]] on [[vertices]], then the induced morphism on [[slice-(∞,1)-categories]]

\begin{displaymath}
\mathcal{C}_{/X[K]}
  \to 
  \mathcal{C}_{/X[K']}
\end{displaymath}
is an [[equivalence of (∞,1)-categories]] (a [[weak equivalence]] in the [[model structure for quasi-categories]]).

Equivalently, by remark \ref{SlicingOverPoweringOfSimplicialObjects}, we have an equivalence

\begin{displaymath}
X(K) \to X(K')
  \,.
\end{displaymath}
\end{prop}
This is (\hyperlink{Lurie}{Lurie, prop. 6.1.2.6}).

\hypertarget{cohesion}{}\subsubsection*{{Cohesion}}\label{cohesion}

If $\mathcal{C} = \mathbf{H}$ is an [[(∞,1)-topos]] then $\mathcal{C}^{\Delta^{op}}$ is a [[cohesive (∞,1)-topos]] over $\mathbf{H}$. For more see at \emph{\href{cohesive+%28infinity%2C1%29-topos#SimplicialObjctsInACohesiveTopos}{cohesive (∞,1)-topos - Examples - Simplicial objects}}.

\hypertarget{internal_language}{}\subsubsection*{{Internal language}}\label{internal_language}

If $\mathcal{C}$ is a [[locally cartesian closed (∞,1)-category]] whose [[internal language]] is [[homotopy type theory]], then the internal language of $\mathcal{C}^{\Delta^{op}}$ is that homotopy type theory equipped with the axioms for a [[linear interval]] object. (\ldots{})

\hypertarget{geometric_realization_and_filtering}{}\subsubsection*{{Geometric realization and filtering}}\label{geometric_realization_and_filtering}

The \emph{[[geometric realization]]} ${\vert X_\bullet \vert}$ of a simplicial object $X_\bullet$ is, if it exists, the [[(∞,1)-colimit]] over the corresponding [[(∞,1)-functor]] $X_\bullet \;\colon\; \Delta^{op} \to \mathcal{C}$.

\begin{displaymath}
{\vert X_\bullet \vert}
  \coloneqq
  \underset{\longrightarrow}{\lim}_n X_n
  \,.
\end{displaymath}
Hence the geometric realization of a cosimplicial object $\Delta^{op} \to \mathcal{C}^{op}$ -- called its [[totalization]] -- is the [[(∞,1)-limit]] over $\Delta \to \mathcal{C}$.

The [[geometric realization]] of the [[simplicial skeleta]] of $X_\bullet$

\begin{displaymath}
{\vert sk_0 X_\bullet \vert} \to {\vert sk_1 X_\bullet \vert} \to {\vert sk_2 X_\bullet \vert} \to \cdots
  \,.
\end{displaymath}
constitutes a [[filtered object in an (infinity,1)-category|filtering]] on the [[geometric realization]] of $X_\bullet$ itself

\begin{displaymath}
{\vert X_\bullet \vert}
  \simeq
  \underset{\longrightarrow}{\lim}_n {\vert sk_n X_\bullet \vert}
  \,.
\end{displaymath}
If $\mathcal{C}$ is a [[stable (∞,1)-category]], then the the corresponding [[spectral sequence of a filtered stable homotopy type]] is the [[spectral sequence of a simplicial stable homotopy type]].

\hypertarget{doldkan_correspondence}{}\subsubsection*{{$\infty$-Dold-Kan correspondence}}\label{doldkan_correspondence}

The following statement is the \emph{[[infinity-Dold-Kan correspondence]]}.

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a [[stable (∞,1)-category]]. Then the [[(∞,1)-categories]] of non-negatively graded [[sequential diagram|sequences]] in $C$ is [[equivalence of (∞,1)-categories|equivalent]] to the [[(∞,1)-category]] of [[simplicial objects in an (∞,1)-category]] in $\mathcal{C}$

\begin{displaymath}
Fun(N(\mathbb{Z}_{\geq 0}), C)
  \simeq
  Fun(N(\Delta)^{op}, C)  
  \,.
\end{displaymath}
Under this equivalence, a [[simplicial object]] $X_\bullet$ is sent to the [[sequential diagram|sequence]] of [[geometric realizations]] ([[(∞,1)-colimits]]) of its [[simplicial skeleta]]

\begin{displaymath}
{\vert sk_0 X_\bullet \vert} \to {\vert sk_1 X_\bullet \vert} \to {\vert sk_2 X_\bullet \vert} \to \cdots
  \,.
\end{displaymath}
This constitutes a [[filtered object in an (infinity,1)-category|filtering]] on the [[geometric realization]] of $X_\bullet$ itself

\begin{displaymath}
{\vert X_\bullet \vert}
  \simeq
  \underset{\longrightarrow}{\lim}_n {\vert sk_n X_\bullet \vert}
  \,.
\end{displaymath}
\end{prop}
([[Higher Algebra|Higher Algebra, theorem 1.2.4.1]])

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

\hypertarget{internal_category_objects}{}\subsubsection*{{Internal category objects}}\label{internal_category_objects}

\begin{itemize}%
\item A [[pre-category object in an (∞,1)-category]] $\mathcal{C}$ is a simplicial object which satisfies the [[Segal conditions]];


\item a [[category object in an (∞,1)-category]] is a pre-category object which also satisfies the [[univalence axiom]];


\item a [[groupoid object in an (∞,1)-category]] is a category object all of whose [[morphisms]] are [[equivalences]] under composition.



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

\begin{itemize}%
\item [[simplicial object]]

\begin{itemize}%
\item [[simplicial set]]

\end{itemize}

\item [[semi-simplicial object]]

\begin{itemize}%
\item [[semisimplicial set]]

\end{itemize}

\item [[simplicial homotopy theory]]



\end{itemize}
\hypertarget{reference}{}\subsection*{{Reference}}\label{reference}

Simplicial objects in general [[(∞,1)-categories]] are discussed in

\begin{itemize}%
\item [[Jacob Lurie]], section 6.1.2 of \emph{[[Higher Topos Theory]]}

\end{itemize}
Related discussion is also in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[(∞,2)-Categories and the Goodwillie Calculus]]} (\href{http://arxiv.org/abs/0905.0462}{arXiv:0905.0462})

\end{itemize}
Simplicial obects in [[stable (∞,1)-categories]] are discussed in

\begin{itemize}%
\item [[Jacob Lurie]], section 1.2.4 of \emph{[[Higher Algebra]]}

\end{itemize}
[[!redirects simplicial objects in an (∞,1)-category]] [[!redirects simplicial objects in an (infinity,1)-category]]

[[!redirects simplicial object in an (∞,1)-category]]

[[!redirects cosimplicial object in an (∞,1)-category]] [[!redirects cosimplicial objects in an (∞,1)-category]]

[[!redirects cosimplicial object in an (infinity,1)-category]] [[!redirects cosimplicial objects in an (infinity,1)-category]]



\end{document}