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\begin{document}

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\section*{accessible (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{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{StabilityUnderOperations}{Stability under various operations}\dotfill \pageref*{StabilityUnderOperations} \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}

The notion of \textbf{accessible $(\infty,1)$-category} is the generalization of the notion of [[accessible category]] from [[category theory]] to [[(∞,1)-category]] theory.

It is a means to handle $(\infty,1)$-categories that are not [[essentially small (∞,1)-category|essentially small]] in terms of small data.

An \emph{accessible} $(\infty,1)$-category is one which may be [[large category|large]], but can entirely be \emph{accessed} as an $(\infty,1)$-category of ``conglomerates of objects'' in a small $(\infty,1)$-category -- precisely: that it is a category of $\kappa$-small [[ind-object]]s in some small $(\infty,1)$-category $C$.

A $\kappa$-accessible $(\infty,1)$-category which in addition has all [[(∞,1)-colimits]] is called a \emph{[[locally presentable (∞,1)-category|locally ∞-presentable]]} or a \emph{$\kappa$-[[compactly generated (∞,1)-category]]}.

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

Let $\kappa$ be a [[regular cardinal]]. spring

\begin{defn}
\label{}\hypertarget{}{}
A [[(∞,1)-category]] $\mathcal{C}$ is \textbf{$\kappa$-accessible} if it satisfies the following equivalent conditions:

\begin{enumerate}%
\item There is a [[small (∞,1)-category]] $\mathcal{C}^0$ and an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
\mathcal{C} \simeq Ind_\kappa(C^0)
\end{displaymath}
of $\mathcal{C}$ with the [[(∞,1)-category of ind-objects]], relative $\kappa$, in $\mathcal{C}^0$.


\item The $(\infty,1)$-category $\mathcal{C}$

\begin{enumerate}%
\item is [[locally small (∞,1)-category|locally small]]


\item has all $\kappa$-[[filtered colimits]]


\item the full [[sub-(∞,1)-category]] $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ of $\kappa$-[[compact objects]] is an [[essentially small (∞,1)-category]];


\item $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ generates $\mathcal{C}$ under $\kappa$-[[filtered (∞,1)-colimits]].



\end{enumerate}

\item The $(\infty,1)$-category $\mathcal{C}$

\begin{enumerate}%
\item is [[locally small (∞,1)-category|locally small]]


\item has all $\kappa$-[[filtered colimits]]


\item there is \emph{some} [[essentially small (∞,1)-category|essentially small]][[sub-(∞,1)-category]] $\mathcal{C}' \hookrightarrow \mathcal{C}$ of $\kappa$-[[compact objects]] which generates $\mathcal{C}$ under $\kappa$-[[filtered (∞,1)-colimits]].



\end{enumerate}


\end{enumerate}
The notion of accessibility is mostly interesting for \emph{large} (∞,1)-categories. For

\begin{itemize}%
\item If $\mathcal{C}$ is small, then there exists a $\kappa$ such that $\mathcal{C}$ is $\kappa$-accessible if and only if $\mathcal{C}$ is an [[idempotent-complete (∞,1)-category]].

\end{itemize}
Generally, $\mathcal{C}$ is called an \textbf{accessible $(\infty,1)$-category} if it is $\kappa$-accessible for some regular cardinal $\kappa$.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
These conditions are indeed equivalent.

\end{prop}
For the first few this is [[Higher Topos Theory|HTT, prop. 5.4.2.2]]. The last one is in [[HTT|HTT, section 5.4.3]].

\begin{defn}
\label{}\hypertarget{}{}
An [[(∞,1)-functor]] between accessible $(\infty,1)$-categories that preserves $\kappa$-filtered colimits is called an \textbf{[[accessible (∞,1)-functor]]} .

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Write $(\infty,1)AccCat \subset (\infty,1)Cat$ for the 2-[[sub-(∞,1)-category]] of [[(∞,1)Cat]] on

\begin{itemize}%
\item those objects that are accessible $(\infty,1)$-categories;


\item those morphisms for which there is a $\kappa$ such that the [[(∞,1)-functor]] is $\kappa$-continuous and preserves $\kappa$-[[compact object]]s.



\end{itemize}
\end{defn}
So morphisms are the [[accessible (∞,1)-functor]]s that also preserves [[compact object]]s. (?)

This is [[Higher Topos Theory|HTT, def. 5.4.2.16]].

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

\hypertarget{StabilityUnderOperations}{}\subsubsection*{{Stability under various operations}}\label{StabilityUnderOperations}

\begin{theorem}
\label{}\hypertarget{}{}
If $C$ is an accessible $(\infty,1)$-category then so are

\begin{itemize}%
\item for $K$ a small simplicial set the [[(∞,1)-category of (∞,1)-functors]] $Func(K,C)$;


\item for $p : K \to C$ a small [[diagram]], the [[over quasi-category]] $C_{/p}$ and under-quasi-category $C_{p/}$.



\end{itemize}
\end{theorem}
This is [[Higher Topos Theory|HTT]] section 5.4.4, 5.4.5 and 5.4.6.

\begin{theorem}
\label{}\hypertarget{}{}
The [[(∞,1)-pullback]] of accessible $(\infty,1)$-categories in [[(∞,1)Cat]] is again accessible.

\end{theorem}
This is [[Higher Topos Theory|HTT, section 5.4.6]].

Generally:

\begin{theorem}
\label{}\hypertarget{}{}
The $(\infty,1)$-category $(\infty,1)AccCat$ has all small [[(∞,1)-limit]]s and the inclusion

\begin{displaymath}
(\infty,1)AccCAT \hookrightarrow (\infty,1)CAT
\end{displaymath}
preserves these.

\end{theorem}
This is [[Higher Topos Theory|HTT, proposition 5.4.7.3]].

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

\begin{itemize}%
\item [[compactly generated (∞,1)-category]]

\end{itemize}
[[!include locally presentable categories - table]]

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

The theory of accessible 1-categories is described in

\begin{itemize}%
\item [[Ji?í Adámek]], [[Ji?í Rosický]], \emph{[[Locally presentable and accessible categories]]}, Cambridge University Press, (1994)

\end{itemize}
The theory of accessible $(\infty,1)$-categories is the topic of section 5.4 of

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

\end{itemize}
[[!redirects accessible (infinity,1)-categories]] [[!redirects accessible (∞,1)-category]] [[!redirects accessible (∞,1)-categories]]



\end{document}