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\section*{(infinity,1)Cat}

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

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

[[!include quasi-category theory contents]]

\hypertarget{categories_of_categories__contentscategories_of_categories}{}\paragraph*{{[[categories of categories - contents|categories of categories]]}}\label{categories_of_categories__contentscategories_of_categories}

[[!include categories of categories - contents]]

\textbf{$(\infty,1)Cat$} is the [[(∞,2)-category]] of all small [[(∞,1)-categories]].

Its full [[subcategory]] on [[∞-groupoid]]s is [[∞Grpd]].

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

\noindent\hyperlink{the_category}{The $(\infty,2)$-category}\dotfill \pageref*{the_category} \linebreak
\noindent\hyperlink{as_an_category}{As an $SSet$-category}\dotfill \pageref*{as_an_category} \linebreak
\noindent\hyperlink{as_an_enriched_model_category}{As an enriched model category}\dotfill \pageref*{as_an_enriched_model_category} \linebreak
\noindent\hyperlink{the_category_2}{The $(\infty,1)$-category}\dotfill \pageref*{the_category_2} \linebreak
\noindent\hyperlink{as_an_category_2}{As an $SSet$-category}\dotfill \pageref*{as_an_category_2} \linebreak
\noindent\hyperlink{as_an_enriched_model_category_2}{As an enriched model category}\dotfill \pageref*{as_an_enriched_model_category_2} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{LimitsAndColimits}{Limits and colimits in $(\infty,1)$Cat}\dotfill \pageref*{LimitsAndColimits} \linebreak
\noindent\hyperlink{Automorphisms}{Automorphisms}\dotfill \pageref*{Automorphisms} \linebreak
\noindent\hyperlink{presentations}{Presentations}\dotfill \pageref*{presentations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{the_category}{}\subsection*{{The $(\infty,2)$-category}}\label{the_category}

\hypertarget{as_an_category}{}\subsubsection*{{As an $SSet$-category}}\label{as_an_category}

One incarnation of [[(∞,2)-categories]] is given by [[quasi-category]]-enriched categories (see there for details). As such $(\infty,1)Cat$ is the full [[SSet]]-[[enriched category|enriched]] [[subcategory]] of [[SSet]] on those [[simplicial set]]s that are [[quasi-categories]]. By the fact described at [[(∞,1)-category of (∞,1)-functors]] this is indeed a [[quasi-category]]-enriched category.

\hypertarget{as_an_enriched_model_category}{}\subsubsection*{{As an enriched model category}}\label{as_an_enriched_model_category}

The [[model category]] presenting this [[(∞,2)-category]] is the Joyal [[model structure for quasi-categories]] $sSet_{Joyal}$. Its full [[sSet]]-[[subcategory]] is the [[quasi-category]] enriched category of quasi-categories from above.

\hypertarget{the_category_2}{}\subsection*{{The $(\infty,1)$-category}}\label{the_category_2}

Sometimes it is useful to consider inside the full $(\infty,2)$-catgeory of $(\infty,1)$-categories just the maximal $(\infty,1)$-category and discarding all non-invertible [[k-morphism|2-morphisms]]. This is the [[(∞,1)-category of (∞,1)-categories]].

\hypertarget{as_an_category_2}{}\subsubsection*{{As an $SSet$-category}}\label{as_an_category_2}

As an [[SSet]]-[[enriched category]] the [[(∞,1)-category of (∞,1)-categories]] is obtained from the quasi-category-enriched version by picking in each [[hom-object]] simplicial set of $(\infty,1)Cat$ the maximal [[Kan complex]].

\hypertarget{as_an_enriched_model_category_2}{}\subsubsection*{{As an enriched model category}}\label{as_an_enriched_model_category_2}

One [[model category]] structure presenting this is the [[model structure on marked simplicial over-sets|model structure on marked simplicial sets]]. As a plain [[model category]] this is [[Quillen equivalence|Quillen equivalent]] to $sSet_{Joyal}$, but as an [[enriched model category]] it is $sSet_{Quillen}$ enriched, so that its full [[SSet]]-subcategory on fibrant-cofibrant objects presents the $(\infty,1)$-category of $(\infty,1)$-categories.

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

\hypertarget{LimitsAndColimits}{}\paragraph*{{Limits and colimits in $(\infty,1)$Cat}}\label{LimitsAndColimits}

[[limit in a quasi-category|Limits and colimits]] over a [[(∞,1)-functor]] with values in $(\infty,1)Cat$ may be reformulation in terms of the [[universal fibration of (infinity,1)-categories]] $Z \to (\infty,1)Cat^{op}$

Then let $X$ be any [[(∞,1)-category]] and

\begin{displaymath}
F : X \to (\infty,1)Cat
\end{displaymath}
an [[(∞,1)-functor]]. Recall that the [[Cartesian fibration|coCartesian fibration]] $E_F \to X$ classified by $F$ is the pullback of the [[universal fibration of (∞,1)-categories]] $Z$ along F:

\begin{displaymath}
\itexarray{
    E_F &\to& Z
    \\
    \downarrow && \downarrow
    \\
    X &\stackrel{F}{\to}& (\infty,1)Cat
  }
\end{displaymath}
\begin{uprop}
Let the assumptions be as above. Then:

\begin{itemize}%
\item The colimit of $F$ is equivalent to $E_F$:

\begin{displaymath}
E_F \simeq colim F
\end{displaymath}

\item The limit of $F$ is equivalent to the [[(infinity,1)-category of cartesian section]] of $E_F \to X$

\begin{displaymath}
\Gamma_X(E_F) \simeq lim F
  \,.
\end{displaymath}


\end{itemize}
\end{uprop}
\begin{proof}
This is [[Higher Topos Theory|HTT, section 3.3]].

\end{proof}
\hypertarget{Automorphisms}{}\paragraph*{{Automorphisms}}\label{Automorphisms}

\begin{utheorem}
The full subcategory of the [[(∞,1)-category of (∞,1)-categories]] $Func((\infty,1)Cat, (\infty,1)Cat)$ on those [[(∞,1)-functor]]s that are equivalences is equivalent to $\{Id, op\}$: it contains only the identity functor and the one that sends an $(\infty,1)$-category to its [[opposite (infinity,1)-category]].

\end{utheorem}
\begin{proof}
This is due to

\begin{itemize}%
\item [[Bertrand Toen]], \emph{Vers une axiomatisation de la th\'e{}orie des cat\'e{}gories sup\'e{}rieures} , K-theory 34 (2005), no. 3, 233-263.

\end{itemize}
It appears as [[Higher Topos Theory|HTT, theorem 5.2.9.1]] (\href{http://arxiv.org/abs/math.CT/0608040}{arxiv v4+} only)

First of all the statement is true for the ordinary category of [[poset]]s. This is \href{http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=311}{prop. 5.2.9.14}.

From this the statement is deduced for $(\infty,1)$ -categories by observing that posets are characterized by the fact that two parallel functors into them that are objectwise equivalent are already equivalent, \href{http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=310}{prop. 5.2.9.11}, which means that posets $C$ are characterized by the fact that

\begin{displaymath}
\pi_0 (\infty,1)Cat(D,C)
  \to 
  Hom_{Set}(
   \pi_0 (\infty,1)Cat(*,D)
   ,
   \pi_0 (\infty,1)Cat(*,C)
  )
\end{displaymath}
is an injection for all $D \in (\infty,1)Cat$.

This is preserved under automorphisms of $(\infty,1)Cat$, hence any such automorphism preserves posets, hence restricts to an automorphism of the category of posets, hence must be either the identity or $(-)^{op}$ there, by the above statement for posets.

Now finally the main point of the proof is to see that the linear posets $\Delta \subset (\infty,1)Cat$ are [[dense functor|dense]] in $(\infty,1)Cat$, i.e. that the identity transformation of the inclusion functor $\Delta \hookrightarrow (\infty,1)Cat$ exhibits $Id_{(\infty,1)Cat}$ as the left [[Kan extension]]

\begin{displaymath}
\itexarray{
    \Delta &\hookrightarrow& (\infty,1)Cat
    \\
    \downarrow & \nearrow_{Lan = \mathrlap{Id}}
    \\
    (\infty,1)Cat
  }
  \,.
\end{displaymath}
\end{proof}
\hypertarget{presentations}{}\subsection*{{Presentations}}\label{presentations}

[[!include table - models for (infinity,1)-operads]]

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

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


\item [[Set]]


\item [[Grpd]], [[∞Grpd]]


\item [[Cat]], [[Operad]]


\item [[2Cat]]


\item \textbf{$(\infty,1)$Cat}, [[(∞,1)Operad]]


\item [[(∞,n)Cat]]


\item [[(infinity, 1)Prof]]



\end{itemize}
category: category

[[!redirects (∞,1)Cat]]



\end{document}