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\section*{locally presentable (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{EquivalentCharacterizationsDetails}{Equivalent characterizations}\dotfill \pageref*{EquivalentCharacterizationsDetails} \linebreak
\noindent\hyperlink{StabilityUnderVariousConstructions}{Stability under various constructions}\dotfill \pageref*{StabilityUnderVariousConstructions} \linebreak
\noindent\hyperlink{LimitsAndColimits}{Limits and colimits}\dotfill \pageref*{LimitsAndColimits} \linebreak
\noindent\hyperlink{PresentedByCombinatorialSimplicialModelCategories}{As $(\infty,1)$-categories presented by combinatorial simplicial model categories}\dotfill \pageref*{PresentedByCombinatorialSimplicialModelCategories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

An [[(∞,1)-category]] is called \emph{locally presentable} if it has all small [[(∞,1)-colimits]] and its [[objects]] are [[generators and relations|presented]] under [[(∞,1)-colimits]] by a [[small set]] of [[small objects]]. This is the direct analog in [[(∞,1)-category]] theory of the notion of \emph{[[locally presentable category]]} in [[category theory]].

There is a wealth of equivalent ways to make precise what this means, which are listed \hyperlink{Definition}{below}. Two particularly useful ones are:

\begin{enumerate}%
\item A locally presentable $(\infty,1)$-category is an [[accessible (∞,1)-category]] that admits all small [[(∞,1)-colimits]].


\item The locally presentable $(\infty,1)$-categories $\mathcal{C}$ are precisely the [[accessible (∞,1)-functor|accessibly embedded]] [[localization of an (∞,1)-category|localizations]]/[[reflective sub-(∞,1)-category|reflections]] $\mathcal{C} \stackrel{\overset{}{\leftarrow}}{\hookrightarrow} PSh_\infty(K)$ of an [[(∞,1)-category of (∞,1)-presheaves]]. In particular, if the reflector of this reflection is a [[left exact (∞,1)-functor]], then $\mathcal{C}$ is an [[(∞,1)-topos]].



\end{enumerate}
See also at \emph{[[locally presentable categories - introduction]]}.

\textbf{Warning on terminology.} In \hyperlink{Lurie}{Lurie} the term \emph{presentable $(\infty,1)$-category} is used for what we call a \emph{locally presentable $(\infty,1)$-category} here, in order to be in line with the established terminology of \emph{[[locally presentable category]]} in ordinary [[category theory]].

\textbf{Terminological variant.} The term ``$\kappa$-[[compactly generated (∞,1)-category]]'' is sometimes used to mean ``locally $\kappa$-presentable (∞,1)-category. See there for a discussion of usage differences.

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

\begin{defn}
\label{}\hypertarget{}{}
An [[(∞,1)-category]] $\mathcal{C}$ is called \textbf{locally presentable} if

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


\item it has all small [[(∞,1)-colimits]].



\end{enumerate}
\end{defn}
\begin{prop}
\label{EquivalentCharacterizations}\hypertarget{EquivalentCharacterizations}{}
That $\mathcal{C}$ is locally presentable is equivalent to each of the following equivalent characterizations.

\begin{enumerate}%
\item $\mathcal{C}$ is [[locally small (infinity,1)-category|locally small]], with all small [[(∞,1)-colimits]] such that there is a [[small set]] $S \hookrightarrow Obj(\mathcal{C})$ of [[small objects]] which generates all of $\mathcal{C}$ under [[(∞,1)-colimits]].


\item $\mathcal{C}$ is the [[localization of an (∞,1)-category|localization]] of an [[(∞,1)-category of (∞,1)-presheaves]] $PSh_\infty(K)$ along an [[accessible (∞,1)-functor]]:

there exists a [[small (∞,1)-category]] $K$ and a pair of [[adjoint (∞,1)-functors]]

\begin{displaymath}
\mathcal{C} \stackrel{\overset{}{\leftarrow}}{\hookrightarrow} PSh_\infty(K)
\end{displaymath}
such that the [[right adjoint]] $\mathcal{C} \hookrightarrow PSh_\infty(K)$ is [[full and faithful (∞,1)-functor|full and faithful]] and [[accessible (∞,1)-functor|accessible]].

(if here in addition $f$ is [[exact functor|left exact]] then $\mathcal{C}$ is an [[(∞,1)-category of (∞,1)-sheaves]] on $K$).


\item There exists a [[combinatorial simplicial model category]] $A$ and and [[equivalence of (infinity,1)-categories]] $\mathcal{C} \simeq L_W A$ with the [[simplicial localization]] of $A$.

More explicitly: with $\mathcal{C}$ incarnated as a [[quasi-category]] there is [[equivalence of quasi-categories]] $\mathcal{C} \simeq N(A^\circ)$ of $\mathcal{C}$ with the [[homotopy coherent nerve]] of the full [[sSet]]-[[enriched category|enriched]] [[subcategory]] of $A$ on [[fibrant object|fibrant]] and [[cofibrant objects]].


\item $\mathcal{C}$ is [[accessible (infinity,1)-category|accessible]] and for every [[cardinal number|regular cardinal]] $\kappa$ the [[full sub-(∞,1)-category]] $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ on the $\kappa$ [[compact object in an (∞,1)-category|compact objects]] admits $\kappa$-small [[(∞,1)-colimits]].


\item There exists a [[cardinal number|regular cardinal]] $\kappa$ such that $\mathcal{C}$ is $\kappa$-[[accessible (infinity,1)-category|accessible]] and $C^\kappa$ admits $\kappa$-small [[limit in quasi-categories|colimits]];


\item There exists a [[cardinal number|regular cardinal]] $\kappa$, a [[small (∞,1)-category]] $D$ with $\kappa$-small [[limit in quasi-categories|colimits]] and an equivalence $Ind_\kappa D \stackrel{\simeq}{\to} \mathcal{C}$ with the category of $\kappa$-[[ind-object]]s of $D$.



\end{enumerate}
\end{prop}
This is \hyperlink{Lurie}{Lurie, theorem 5.5.1.1}, following (\hyperlink{Simpson}{Simpson}).

\begin{remark}
\label{}\hypertarget{}{}
That [[localization of an (infinity,1)-category|localizations]] $\mathcal{C} \stackrel{\leftarrow}{\hookrightarrow} PSh_{(\infty,1)}(K)$ correspond to combinatorial simplicial model categories is essentially \textbf{[[Dugger's theorem]]} (\hyperlink{Dugger}{Dugger}): every [[combinatorial model category]] arises, up to Quillen equivalence, as the left [[Bousfield localization of model categories|left Bousfield localization]] of the global projective [[model structure on simplicial presheaves]].

\end{remark}
Locally presentable $(\infty,1)$-categories have a number of nice properties, and therefore it is of interest to consider as morphisms between them only those [[(∞,1)-functor]]s that preserve these properties. It turns out that it is useful to consider \emph{[[limit in a quasi-category|colimit]] preserving} functors. By the [[adjoint (∞,1)-functor theorem]] these are precisely the functors that have a right [[adjoint (∞,1)-functor]].

\begin{defn}
\label{}\hypertarget{}{}
Write [[Pr(∞,1)Cat]] $\subset$ [[(∞,1)Cat]] for the (non-full) [[sub-quasi-category|sub-(∞,1)-category]] of [[(∞,1)Cat]] (the collection of not-necessarily small $(\infty,1)$-categories) on

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


\item those morphisms that are colimit-preserving [[(∞,1)-functor]]s.



\end{itemize}
\end{defn}
This is \hyperlink{Lurie}{Lurie, def. 5.5.3.1}.

This $(\infty,1)$-category $Pr(\infty,1)Cat$ in turn as special properties. More on that is at \emph{[[symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]]}.

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

\hypertarget{EquivalentCharacterizationsDetails}{}\subsubsection*{{Equivalent characterizations}}\label{EquivalentCharacterizationsDetails}

We indicate stepts in the proof of prop. \ref{EquivalentCharacterizations}.

\begin{lemma}
\label{}\hypertarget{}{}
Let $f \colon \mathcal{C} \to \mathcal{D}$ be an [[(∞,1)-functor]] which exhibits $\mathcal{D}$ as an [[idempotent completion]] $\mathcal{C}$. Let $\kappa$ be a [[regular cardinal]]. Then the induced functor on [[(∞,1)-categories of ind-objects]]

\begin{displaymath}
Ind_\kappa(f) \colon Ind_\kappa(\mathcal{C}) \to Ind_\kappa(\mathcal{D})
\end{displaymath}
is an [[equivalence of (∞,1)-categories]].

\end{lemma}
This is (\hyperlink{Lurie}{Lurie, lemma 5.5.1.3}).

\begin{lemma}
\label{}\hypertarget{}{}
Let $L \colon \mathcal{C} \to \mathcal{D}$ be an [[(∞,1)-functor]] between [[(∞,1)-categories]] which have $\kappa$-[[filtered (∞,1)-colimits]], and let $R$ be a [[right adjoint|right]] [[adjoint (∞,1)-functor]] of $L$. If $R$ preserves $\kappa$-[[filtered (∞,1)-colimits]] then $L$ preserves $\kappa$-[[compact objects]].

\end{lemma}
This is \hyperlink{Lurie}{Lurie, lemma 5.5.1.4}.

(\ldots{})

\hypertarget{StabilityUnderVariousConstructions}{}\subsubsection*{{Stability under various constructions}}\label{StabilityUnderVariousConstructions}

\begin{prop}
\label{}\hypertarget{}{}
For $C$ a locally presentable $(\infty,1)$-category and $p : K \to C$ a [[diagram]] in $C$, also the [[over quasi-category|over (∞,1)-category]] $C_{/pp}$ as well as the under-$(\infty,1)$-category $C_{p/}$ are locally presentable.

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 5.5.3.10, prop. 5.5.3.11]].

\begin{example}
\label{}\hypertarget{}{}
Since [[Pr(∞,1)Cat]] admits all small limits, we obtain new locally presentable $(\infty,1)$-categories by forming limits over given ones. In particular the [[product]] of locally presentable $(\infty,1)$-categories is again locally presentable.

\end{example}
\hypertarget{LimitsAndColimits}{}\subsubsection*{{Limits and colimits}}\label{LimitsAndColimits}

In the first definition of locally presentable $(\infty,1)$-category above only the existence of colimits is postulated. An important fact is that it follows automatically that also all small limits exist:

A [[representable functor]] $C^{op} \to \infty Grpd$ preserves [[limit in a quasi-category|limits]] (see [[(∞,1)-Yoneda embedding]]). If $C$ is locally presentable, then also the converse holds:

\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{C}$ is a locally presentable $(\infty,1)$-category then an [[(∞,1)-functor]] $C^{op} \to \infty Grpd$ is a [[representable functor]] precisely if it preserves [[limit in a quasi-category|limits]].

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 5.5.2.2]].

\begin{proof}
We need to prove that a limit-preserving functor $F : C^{op} \to \infty Grpd$ is [[representable functor|representable]]. By the above characterizations we know that $C$ is an accessible localization of a presheaf category.

So consider first the case that $C = PSh(D)$ \emph{is} a presheaf category. Write

\begin{displaymath}
f : D^{op} \stackrel{j^{op}}{\to} PSh(D)^{op} \stackrel{F}{\to} \infty Grpd
\end{displaymath}
for the precomposition of $F$ with the [[(∞,1)-Yoneda embedding]]. Then let

\begin{displaymath}
F' := Hom_{C}(-,f) : PSh(D)^{op} \to \infty Grpd
\end{displaymath}
the functor represented by $f$.

We claim that $F \simeq F'$, which proves that $F$ is represented by $F \circ j^{op}$: since both $F$ and $F'$ preserve limits (hence colimits as functors on $PSh(D)$) it follows from the fact that the Yoneda embedding exhibits the universal co-completion of $D$ that it is sufficient to show that $F \circ j^{op} \simeq F' \circ j^{op}$. But this is the case precisely by the statement of the full [[(∞,1)-Yoneda lemma]].

Now consider more generally the case that $C$ is a [[reflective sub-(∞,1)-category]] of $PSh(D)$. Let $L : PSh(D) \to C$ be the [[left adjoint]] reflector. Since it respects all colimits, the composite

\begin{displaymath}
F \circ L^{op} : PSh(D)^{op} \stackrel{L^{op}}{\to} C^{op} \stackrel{F}{\to} \infty Grpd
\end{displaymath}
respects all limits. By the above it is therefore represented by some object $X \in PSh(D)$.

By the general properties of [[reflective sub-(∞,1)-categories]], we have that $C$ is the full [[sub-(∞,1)-category]] of $PSh(D)$ on those objects that are [[local object]]s with respect to the morphisms that $L$ sends to equivalences. But $X$, since it presents $F \circ L^{op}$, is manifestly local in this sense and therefore also represents $F \circ L^{op}|_{C}$. But on $C$ the functor $L$ is equivalent to the identity, so that this is equivlent to $F$.

\end{proof}
This statement has the following important consequence:

\begin{cor}
\label{}\hypertarget{}{}
A locally presentable $(\infty,1)$-category $C$ has all small [[limit in a quasi-category|limits]].

\end{cor}
This is [[Higher Topos Theory|HTT, prop. 5.5.2.4]].

\begin{proof}
We may compute the limit after applying the [[(∞,1)-Yoneda embedding]] $j : C \to PSh_{(\infty,1)}(c)$. Since this is a [[full and faithful (∞,1)-functor]] it is sufficient to check that the limit computed in $PSh(C)$ lands in the essential image of $j$. But by the above lemma, this amounts to checking that the limit over limit-preserving functors is itself a limit-preserving functor. This follows using that limits of functors are computed objectwise and that generally limits commute with each other (see [[limit in a quasi-category]]):

to check for $I \to PSh(C)$ a diagram of limit-preserving functors that $\lim_i F_i$ is a functor that commutes with all limits, let $a : J \to C$ be a diagram and compute (verbatim as in ordinary category theory)

\begin{displaymath}
\begin{aligned}
    \lim_j (\lim_i F_i)(a_j)
    & \simeq 
    \lim_j (\lim_i F_i(a_j))
    \\
    & \simeq
    \lim_i (\lim_j F_i(a_j))
    \\
    & \simeq
    \lim_i F_i(\lim a_j)
    \\
    & \simeq
    (\lim_i F_i)(\lim a_j)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{PresentedByCombinatorialSimplicialModelCategories}{}\subsubsection*{{As $(\infty,1)$-categories presented by combinatorial simplicial model categories}}\label{PresentedByCombinatorialSimplicialModelCategories}

By prop. \ref{EquivalentCharacterizations} locally presentable $(\infty,1)$-categories are equivalently those [[(∞,1)-categories]] which are \emph{presented} by a [[combinatorial simplicial model category]] $C$ in that they are the full [[simplicially enriched category|simplicial subcategory]] $C^\circ \hookrightarrow C$ on fibrant-cofibrant objects of $C$ (or, equivalently, the [[quasi-category]] associated to this [[simplicially enriched category]]).

\begin{remark}
\label{QuillenEquivZigZag}\hypertarget{QuillenEquivZigZag}{}
Under this presentation, [[equivalence of (∞,1)-categories]] between locally presentable $(\infty,1)$-categories corresponds to [[zigzags]] of [[Quillen equivalences]] between presenting [[combinatorial simplicial model category|combinatorial simplicial model categories]]:

$C^\circ$ and $D^\circ$ are [[equivalence of (∞,1)-categories|equivalent as (∞,1)-categories]] precisely if they are connected by a [[zig-zag]] of [[simplicial Quillen adjunction|simplicial]] [[Quillen equivalences]]

\begin{displaymath}
C 
  \stackrel{\leftarrow}{\to}
  \stackrel{\to}{\leftarrow}
  \stackrel{\leftarrow}{\to}
  \cdots  
  D.
\end{displaymath}
\end{remark}
This is \hyperlink{Lurie}{Lurie, remark A.3.7.7}.

\begin{remark}
\label{}\hypertarget{}{}
Partly due to the fact that [[simplicial model category|simplicial model categories]] have been studied for a longer time -- partly because they are simply more tractable than [[(∞,1)-categories]] -- many $(\infty,1)$-categories are indeed handled in terms of such a presentation by a [[simplicial model category]].

The canonical example is the presentation of the [[(∞,1)-category of (∞,1)-sheaves]] on an ordinary (1-categorical) [[site]] $S$ by the simplicial [[model structure on simplicial presheaves|model category of simplicial presheaves]] on $S$.

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

The basic example is:

\begin{example}
\label{}\hypertarget{}{}
[[∞Grpd]] is locally presentable.

\end{example}
(\hyperlink{Lurie}{Lurie, example 5.5.1.8})

\begin{proof}
According to the discussion at \href{limit+in+a+quasi-category#Tensoring}{(∞,1)-colimit -- Tensoring with an ∞-groupoid} every [[∞-groupoid]] is the colimit over itself of the functor contant on the point, the terminal $\infty$-groupoid. This is clearly compact, and hence generates [[∞Grpd]].

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
An [[(∞,1)-topos]] is precisely a locally presentable $(\infty,1)$-category where the [[localization of an (∞,1)-category|localization]] functor also preserves finite limits.

\end{example}
\begin{prop}
\label{}\hypertarget{}{}
For $C$ and $D$ locally presentable $(\infty,1)$-categories, write $Func^L(C,D) \subset Func(C,D)$ for the full sub-$(\infty,1)$-category on left-adjoint $(\infty,1)$-functors. This is itself locally presentable

\end{prop}
This is [[Higher Topos Theory|HTT, prop 5.5.3.8]]

Notice that this makes the [[symmetric monoidal (∞,1)-category of presentable (∞,1)-categories]] \emph{[[closed monoidal category|closed]]} .

\begin{prop}
\label{}\hypertarget{}{}
For $C$ an $(\infty,1)$-category with finite [[product]]s, the $(\infty,1)$-category $Alg_{(\infty,1)}(C)$ of algebras over $C$ regarded as an [[(∞,1)-algebraic theory]] is locally presentable.

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

\begin{itemize}%
\item [[Ho(CombModCat)]]

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

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

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

The theory of locally presentable $(\infty,1)$-categories was first implicitly conceived in terms of [[model category]] presentations in

\begin{itemize}%
\item [[Carlos Simpson]], \emph{A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi (\href{http://arxiv.org/abs/math/9903167}{arXiv:math/9903167})}

\end{itemize}
The full intrinsic $(\infty,1)$-categorical theory appears in section 5

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

\end{itemize}
with section A.3.7 establishing the relation [[combinatorial model categories]] and \href{http://ncatlab.org/nlab/show/combinatorial+model+category#DuggerTheorem}{Dugger's theorem} in [[Higher Topos Theory|HTT, prop A.3.7.6]]

The statement of \href{combinatorial+model+category#DuggerTheorem}{Dugger's theorem} of which the characterization of locally presentable $(\infty,1)$-categories as localizations of $(\infty,1)$-presheaf categories is a variant is due to

\begin{itemize}%
\item [[Dan Dugger]], \emph{[[Combinatorial model categories have presentations]]}

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

[[!redirects locally presentable (infinity,1)-category]] [[!redirects locally presentable (infinity,1)-categories]] [[!redirects locally presentable (∞,1)-category]] [[!redirects locally presentable (∞,1)-categories]]

[[!redirects presentable (infinity,1)-category]]

[[!redirects locally presentable infinity-category]]



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