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\begin{document}

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\section*{locally presentable categories - introduction}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

\hypertarget{category_theory_2}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory_2}

[[!include quasi-category theory contents]]

\begin{quote}%
This page means to give an introduction to the notion of \emph{[[locally presentable category]]}, and its related notions in [[higher category theory]] and survey some fundamental properties.

Expected background of the reader:

\begin{itemize}%
\item For the first section \emph{\hyperlink{BasicIdea}{Basic idea in category theory}} the reader is assumed to be familiar with basic notions of [[category theory]] such as \emph{[[presheaves]]} and \emph{[[colimits]]}.


\item For the section \emph{\hyperlink{BasicIdeaInModelCategoryTheory}{Basic idea in model category theory}} the reader is assumed to be familiar with basic notions in [[model category|model category theory]] such as \emph{[[cofibrantly generated model categories]]} and \emph{[[homotopy colimits]]}.


\item For the third section \emph{\hyperlink{BasicIdeaInInfinityCategoryTheory}{Basic idea in (∞,1)-category theory}} the reader is assumed to be familiar with basic concepts of [[(∞,1)-category|(?.1)-category theory]] such as \emph{[[(∞,1)-categories of (∞,1)-presheaves]]} and \emph{[[(∞,1)-colimits]]}.



\end{itemize}

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

\noindent\hyperlink{BasicIdea}{Basic idea in category theory}\dotfill \pageref*{BasicIdea} \linebreak
\noindent\hyperlink{generation_from_generators}{Generation from generators}\dotfill \pageref*{generation_from_generators} \linebreak
\noindent\hyperlink{small_data}{Small data}\dotfill \pageref*{small_data} \linebreak
\noindent\hyperlink{locally_presentable_category_generated_from_colimits_over_small_objects}{Locally presentable category: generated from colimits over small objects}\dotfill \pageref*{locally_presentable_category_generated_from_colimits_over_small_objects} \linebreak
\noindent\hyperlink{generation_exhibited_by_epimorphism_from_a_free_object}{Generation exhibited by epimorphism from a free object}\dotfill \pageref*{generation_exhibited_by_epimorphism_from_a_free_object} \linebreak
\noindent\hyperlink{left_exact_localizations}{Left exact localizations}\dotfill \pageref*{left_exact_localizations} \linebreak
\noindent\hyperlink{summary}{Summary and overview}\dotfill \pageref*{summary} \linebreak
\noindent\hyperlink{BasicIdeaInModelCategoryTheory}{Basic idea in model category theory}\dotfill \pageref*{BasicIdeaInModelCategoryTheory} \linebreak
\noindent\hyperlink{model_structure_on_simplicial_presheaves}{Model structure on simplicial presheaves}\dotfill \pageref*{model_structure_on_simplicial_presheaves} \linebreak
\noindent\hyperlink{left_bousfield_localization}{Left Bousfield localization}\dotfill \pageref*{left_bousfield_localization} \linebreak
\noindent\hyperlink{combinatorial_model_categories}{Combinatorial model categories}\dotfill \pageref*{combinatorial_model_categories} \linebreak
\noindent\hyperlink{duggers_theorem}{Dugger's theorem}\dotfill \pageref*{duggers_theorem} \linebreak
\noindent\hyperlink{BasicIdeaInInfinityCategoryTheory}{Basic idea in $(\infty,1)$-category theory}\dotfill \pageref*{BasicIdeaInInfinityCategoryTheory} \linebreak
\noindent\hyperlink{presheaves}{$(\infty,1)$-Presheaves}\dotfill \pageref*{presheaves} \linebreak
\noindent\hyperlink{localizations_of_categories}{Localizations of $(\infty,1)$-categories}\dotfill \pageref*{localizations_of_categories} \linebreak
\noindent\hyperlink{locally_presentable_categories}{Locally presentable $(\infty,1)$-categories}\dotfill \pageref*{locally_presentable_categories} \linebreak
\noindent\hyperlink{toposes}{$(\infty,1)$-Toposes}\dotfill \pageref*{toposes} \linebreak
\noindent\hyperlink{presentation_by_combinatorial_model_categories}{Presentation by combinatorial model categories}\dotfill \pageref*{presentation_by_combinatorial_model_categories} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{BasicIdea}{}\subsection*{{Basic idea in category theory}}\label{BasicIdea}

The general idea is that a \emph{[[locally presentable category]]} is a [[large category]] [[generators and relations|generated from]] small data: from [[small objects]] under [[small colimit]].

\hypertarget{generation_from_generators}{}\subsubsection*{{Generation from generators}}\label{generation_from_generators}

The notion of \emph{[[locally presentable category]]} is, at least roughly, an analogue for [[categories]] of the notion of a [[finitely generated module]].

\begin{example}
\label{FinitelyGeneratedAbelianGroup}\hypertarget{FinitelyGeneratedAbelianGroup}{}
An [[abelian group]] $A$ is called \emph{[[finitely generated module|finitely generated]]} if there is a [[finite set|finite]] [[subset]]

\begin{displaymath}
\iota \colon S \hookrightarrow U(A)
\end{displaymath}
of the underlying set $U(A)$ of $A$, such that every element of $A$ is a sum of such generating elements.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
We always have the maximal such presentation where $S = U(A)$ is the whole underlying set and $\iota \colon F(U(A)) \to A$ is the [[counit of an adjunction|counit]] of the [[free-forgetful adjunction]]. But the presentation is all the more interesting/useful the \emph{smaller} $S$ is.

\end{remark}
Now, the [[categorification]] of ``commutative sum'' is \emph{[[colimit]]}. Hence let now $\mathcal{C}$ be a category with all [[small colimits]].

\begin{defn}
\label{GenerationOfCategoryFromSubcategoryUnderColimits}\hypertarget{GenerationOfCategoryFromSubcategoryUnderColimits}{}
We say a subclass $S \hookrightarrow Obj(\mathcal{C})$ of [[objects]] or equivalently the [[full subcategory]] $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ on this subclass \emph{generates} $\mathcal{C}$ if every object in $\mathcal{C}$ is a [[colimit]] of objects in $\mathcal{C}^0$, hence the colimit over a [[diagram]] of the form

\begin{displaymath}
D \to \mathcal{C}^0 \hookrightarrow \mathcal{C}
  \,.
\end{displaymath}
\end{defn}
As before, such a presentation is all the more useful the ``smaller'' the generating data is. In order to grasp the various aspects of the notion of ``smallness'' in [[category theory]] we need to recall the notion of [[regular cardinal]].

\hypertarget{small_data}{}\subsubsection*{{Small data}}\label{small_data}

\begin{defn}
\label{}\hypertarget{}{}
The [[cardinality]] $\kappa = {\vert S\vert}$ of a [[set]] $S$ is \emph{[[regular cardinal|regular]]} if every [[coproduct]]/[[disjoint union]] of sets of cardinality smaller than $\kappa$ and indexed by a set of cardinality smaller than $\kappa$ is itself of cardinality smaller than $\kappa$.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
The smallest [[regular cardinal]] is [[?]]${}_0 = {\vert \mathbb{N}\vert}$: every finite union of finite sets is itself a finite set. (See the entry on [[regular cardinal]]s for a discussion as to whether one might consider some finite cardinals as being `regular'.)

\end{example}
We can now speak of objects that are ``$\kappa$-small sums'' using the notion of $\kappa$-[[filtered colimits]]:

\begin{defn}
\label{}\hypertarget{}{}
For $\kappa$ a [[regular cardinal]], a $\kappa$-[[filtered category]] is one where every [[diagram]] of size $\lt \kappa$ has a [[cocone]].

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
In an [[?]]${}_0$-[[filtered category]] every finite diagram has a cocone. This is equivalent to:

\begin{enumerate}%
\item for every pair of objects there is a third objct such that both have a morphism to it;


\item for every pair of [[parallel morphisms]] there is a morphism out of their codomain such that the two composites are equal.



\end{enumerate}
\end{example}
\begin{example}
\label{}\hypertarget{}{}
The [[tower diagram]] category $(\mathbb{N}, \leq)$

\begin{displaymath}
X_0 \to X_1 \to X_2 \to \cdots
\end{displaymath}
is filtered.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
For $\lambda \gt \kappa$ a bigger [[regular cardinal]], every $\lambda$-[[filtered category]] is in particular also $\kappa$-filtered.

\end{remark}
Using this we have the central definition now:

\begin{defn}
\label{}\hypertarget{}{}
A $\kappa$-[[filtered colimit]] is a [[colimit]] over a $\kappa$-filtered diagram.

\end{defn}
A crucial characterizing property of $\kappa$-filtered colimits is the following:

\begin{prop}
\label{FilteredColimitsAndFiniteLimits}\hypertarget{FilteredColimitsAndFiniteLimits}{}
A colimit in [[Set]] is $\kappa$-filtered precisely if it commutes with all $\kappa$-[[small limits]].

In particular a colimit in [[Set]] is filtered (meaning: [[?]]${}_0$-filtered) precisely if it commutes with all [[finite limits]].

\end{prop}
\begin{defn}
\label{}\hypertarget{}{}
An [[object]] $A \in \mathcal{C}$ is a $\kappa$-[[compact object]] if it commutes with $\kappa$-filtered colimits, hence if for $X \colon I \to \mathcal{C}$ any $\kappa$-[[filtered category|filtered]] [[diagram]], the canonical [[function]]

\begin{displaymath}
\underset{\to_i}{\lim} \mathcal{C}(A,X_i) \to \mathcal{C}(A, \underset{\to_i}{\lim} X_i)
\end{displaymath}
is a [[bijection]].

We say $X$ is a \textbf{[[small object]]} if it is $\kappa$-compact for \emph{some} [[regular cardinal]] $\kappa$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
If $\lambda \gt \kappa$, then being $\lambda$-compact is a weaker condition than being $\kappa$-compact.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The object $A$ commutes with the colimit over $I$ precisely if every morphism $A \to \underset{\to_i}{\lim} X_i$ lifts to a morphism $A \to X_j$ into one of the $X_j$. Schematically, depicting specifically a [[sequential colimit]], this means that we have:

\begin{displaymath}
\itexarray{
    \cdots&\to&X_{j-1} &\to& X_j &\to& X_{j+1} &\to& \cdots
    \\
    &&&{}^{\mathllap{\exists \hat f}}\nearrow&\downarrow & \swarrow
    \\
    &&A& \stackrel{f}{\to} & \underset{\to}{\lim} X_i
  }
  \,.
\end{displaymath}
Hence $A$ is ``small enough'' such that mapping it into the sum of all the $X_i$ it always entirely lands inside one of the $X_i$.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
There is a close relation between the notion of ``compact'' as in, on the one hand, \emph{[[compact topological space]]} and \emph{[[compact topos]]}, and on the other as in \emph{[[compact object]]} as above. This is mediated by proposition \ref{FilteredColimitsAndFiniteLimits}. But the relation is a bit more subtle and takes a bit more discussion than we maybe want to get into here.

\end{remark}
\hypertarget{locally_presentable_category_generated_from_colimits_over_small_objects}{}\subsubsection*{{Locally presentable category: generated from colimits over small objects}}\label{locally_presentable_category_generated_from_colimits_over_small_objects}

Using this we can now say:

\begin{defn}
\label{LocallyPresentableCategory}\hypertarget{LocallyPresentableCategory}{}
A [[locally small category]] $\mathcal{C}$ is a \emph{[[locally presentable category]]} if it has all small [[colimits]] and there is a [[small set]] $S \hookrightarrow Obj(\mathcal{C})$ of [[small objects]] such that this generates $\mathcal{C}$, by def. \ref{GenerationOfCategoryFromSubcategoryUnderColimits}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The adjective ``locally'' in ``[[locally presentable category]]'' is to indicate that the condition is all about the [[objects]], only. There is a different notion of ``presented category''.

\end{remark}
There are a bunch of equivalent reformulations of the notion of locally presentable category. One of the more important ones we again motivate first by analogy with presentable modules:

\hypertarget{generation_exhibited_by_epimorphism_from_a_free_object}{}\subsubsection*{{Generation exhibited by epimorphism from a free object}}\label{generation_exhibited_by_epimorphism_from_a_free_object}

\begin{example}
\label{}\hypertarget{}{}
If an [[abelian group]] $A$ is generated by a set $S \hookrightarrow U(A)$ as in example \ref{FinitelyGeneratedAbelianGroup}, this means equivalently that there is an [[epimorphism]]

\begin{displaymath}
L \colon F(S) \to A
\end{displaymath}
from the [[free abelian group]] $F(S)$ generated by $S$, hence the group obtained by forming [[formal linear combination|formal sums]] of elements in $S$. Here the epimorphism sends formal sums to actual sums in $A$:

\begin{displaymath}
L(\sum_k s_k) \coloneqq \sum_k \iota(s_k)
  \,.
\end{displaymath}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
The [[categorification]] of the notion \emph{[[free abelian group]]} is the notion of \emph{[[free cocompletion]]} of a category $\mathcal{C}^0$: the [[category of presheaves]] $PSh(\mathcal{C}^0)$.

\end{remark}
Accordingly:

\begin{example}
\label{}\hypertarget{}{}
If a [[full subcategory]] $\iota \colon \mathcal{C}^0 \hookrightarrow \mathcal{C}$ generates $\mathcal{C}$ under colimits as in defn. \ref{GenerationOfCategoryFromSubcategoryUnderColimits}, then there is a [[functor]]

\begin{displaymath}
L \colon PSh(\mathcal{C}^0) \to \mathcal{C}
\end{displaymath}
which sends formal colimits to actual colimits in $\mathcal{C}$

\begin{displaymath}
L(\underset{\to_k}{\lim} s_k) \coloneqq \underset{\to_k}{\lim} \iota(s_k)
  \,.
\end{displaymath}
Here $L$ by construction preserves all colimits.

\end{example}
Therefore conversely, given a colimit-preserving functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ we want to say that it \emph{locally presents} $\mathcal{C}$ if $L$ is ``suitably epi''.

It turns out that ``suitably epi'' is to be the following:

\begin{defn}
\label{Localization}\hypertarget{Localization}{}
A [[functor]] $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ from the [[category of presheaves]] over a [[small category]] $\mathcal{C}^0$ is an \textbf{[[accessible functor|accessible]] [[localization]]} if

\begin{itemize}%
\item $L$ has a [[section]], hence a [[functor]] $R \colon \mathcal{C} \to PSh(\mathcal{C}^0)$ with a [[natural isomorphism]] $L\circ R \simeq id_{\mathcal{C}}$;


\item such that

\begin{enumerate}%
\item $R$ is [[right adjoint]] to $L$;


\item $R\circ L$ preserves $\kappa$-[[filtered colimits]].



\end{enumerate}


\end{itemize}
\end{defn}
With this notion we have the following analog of the familiar statement that an abelian group is generated by $S$ precisely if there is an epimorphism $L \colon F(S) \to A$:

\begin{theorem}
\label{AdamekRosickyTheorem}\hypertarget{AdamekRosickyTheorem}{}
A category $\mathcal{C}$ is locally presentable according to def. \ref{LocallyPresentableCategory} precisely if it is an accessible localization, def. \ref{Localization},

\begin{displaymath}
L \colon PSh(\mathcal{C}^0) \to \mathcal{C}
\end{displaymath}
for some small category $\mathcal{C}^0$.

\end{theorem}
This is due to (\hyperlink{AdamekRosicky}{Ad\'a{}mek-Rosick\'y{}, prop 1.46}).

\hypertarget{left_exact_localizations}{}\subsubsection*{{Left exact localizations}}\label{left_exact_localizations}

\begin{remark}
\label{}\hypertarget{}{}
A [[locally presentable category]] $\mathcal{C}$ is called a \emph{[[topos]]}, precisely if the localization functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ from theorem \ref{AdamekRosickyTheorem} in addition is a [[left exact functor]], meaning that it preserves [[finite limits]].

\end{remark}
\hypertarget{summary}{}\subsection*{{Summary and overview}}\label{summary}

In summary the discussion \hyperlink{BasicIdea}{above} says that the notion of locally presentable categories sits in a sequence of notions as indicated in the row labeled ``category theory'' in the following table. The other rows are supposed to indicate that regarding a category as a [[(1,1)-category]] and simply varying in this story the parameters $(n,r)$ in ``[[(n,r)-category]]'' one obtains fairly straightforward analogs of the notion of locally presentable category in other fragments of [[higher category theory]]. These we discuss in more detail further below.

\textbf{Locally presentable categories:} [[large categories|Large categories]] whose [[objects]] arise from [[small object|small]] [[generators]] under [[small colimit|small]] [[relations]].

\newline | \textbf{[[(0,1)-category theory]]} | [[(0,1)-toposes]] | $\hookrightarrow$ | [[algebraic lattices]] | $\simeq$ \href{algebraic+lattice#RelationToLocallyFinitelyPresentableCategories}{Porst's theorem} | [[subobject lattices]] in [[accessible functor|accessible]] [[reflective subcategories]] of [[presheaf categories]] | | | | \textbf{[[category theory]]} | [[toposes]] | $\hookrightarrow$ | [[locally presentable categories]] | $\simeq$ \hyperlink{AdamekRosickyTheorem}{Ad\'a{}mek-Rosick\'y{}`s theorem} | [[accessible functor|accessible]] [[reflective subcategories]] of [[presheaf categories]] | $\hookrightarrow$ | [[accessible categories]] | | \textbf{[[model category|model category theory]]} | [[model toposes]] | $\hookrightarrow$ | [[combinatorial model categories]] | $\simeq$ \hyperlink{DuggerTheorem}{Dugger's theorem} | [[left Bousfield localization]] of global [[model structures on simplicial presheaves]] | | | | \textbf{[[(∞,1)-topos theory]]} | [[(∞,1)-toposes]] |$\hookrightarrow$ | [[locally presentable (∞,1)-categories]] | $\simeq$  \hyperlink{SimpsonTheorem}{Simpson's theorem} | [[accessible (∞,1)-functor|accessible]] [[reflective sub-(∞,1)-categories]] of [[(∞,1)-presheaf (∞,1)-categories]] | $\hookrightarrow$ |[[accessible (∞,1)-categories]] |

\hypertarget{BasicIdeaInModelCategoryTheory}{}\subsection*{{Basic idea in model category theory}}\label{BasicIdeaInModelCategoryTheory}

\hypertarget{model_structure_on_simplicial_presheaves}{}\subsubsection*{{Model structure on simplicial presheaves}}\label{model_structure_on_simplicial_presheaves}

The analog of a [[category of presheaves]] in [[model category]] theory is the [[model structure on simplicial presheaves]], which we now briefly indicate.

Write [[sSet]] for the category of [[simplicial sets]]. Here we always regard this as equipped with the standard [[model structure on simplicial sets]] $sSet_{Quillen}$.

\begin{defn}
\label{}\hypertarget{}{}
For $C$ a [[small category]] write $[C^{op}, sSet]\simeq [C^{op}, Set]^{\Delta^{op}}$ for the category of [[simplicial presheaves]]. The \textbf{global projective [[model structure on simplicial presheaves]]} $[C^{op}, sSet]_{proj}$ has as

\begin{itemize}%
\item [[weak equivalences]] the objectwise [[weak homotopy equivalences]] of simplicial sets


\item [[fibrations]] the objectwise [[Kan fibrations]].



\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Accordingly $[C^{op}, sSet]$ is a [[cofibrantly generated model category]] with generating (acyclic) cofibrations the [[tensoring]] of objects of $C$ with the generating (acyclic) cofibrations of $sSet_{Quillen}$.

\end{remark}
\hypertarget{left_bousfield_localization}{}\subsubsection*{{Left Bousfield localization}}\label{left_bousfield_localization}

\begin{defn}
\label{BousfieldLocalizationOfModelStructureOnSimplicialPresheaves}\hypertarget{BousfieldLocalizationOfModelStructureOnSimplicialPresheaves}{}
Given a [[model category]] $[C^{op}, Set]_{proj}$ and set $\mathcal{S} \subset Mor([C^{op}, Set])$ of morphisms, the [[Bousfield localization of model categories|left Bousfield localization]] is the model structure with the same cofibrations and weak equivalences the $\mathcal{S}$-[[local morphisms]].

\begin{displaymath}
[C^{op}, Set]_{proj,\mathcal{S}}
  \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}}
 [C^{op}, Set]_{proj}
  \,.
\end{displaymath}
\end{defn}
\hypertarget{combinatorial_model_categories}{}\subsubsection*{{Combinatorial model categories}}\label{combinatorial_model_categories}

The simple idea of the following definition is to say that the model category analog of \emph{[[locally presentable category]]} is simply a model structure on a locally presentable category.

\begin{defn}
\label{CombinatorialModelCategory}\hypertarget{CombinatorialModelCategory}{}
A model category is a \textbf{[[combinatorial model category]]} if

\begin{enumerate}%
\item the underlying category is a [[locally presentable category]];


\item the model structure is a [[cofibrantly generated model category]].



\end{enumerate}
\end{defn}
\hypertarget{duggers_theorem}{}\subsubsection*{{Dugger's theorem}}\label{duggers_theorem}

\begin{theorem}
\label{DuggerTheorem}\hypertarget{DuggerTheorem}{}
Every [[combinatorial model category]], def. \ref{CombinatorialModelCategory}, is [[Quillen equivalence|Quillen equivalent]] to a [[Bousfield localization of model categories|left Bousfield localization]] of a global [[model structure on simplicial presheaves]] as in def. \ref{BousfieldLocalizationOfModelStructureOnSimplicialPresheaves}.

\end{theorem}
See at \href{combinatorial%20model%20category#DuggerTheorem}{combinatorial model category - Dugger's theorem}.

\hypertarget{BasicIdeaInInfinityCategoryTheory}{}\subsection*{{Basic idea in $(\infty,1)$-category theory}}\label{BasicIdeaInInfinityCategoryTheory}

\hypertarget{presheaves}{}\subsubsection*{{$(\infty,1)$-Presheaves}}\label{presheaves}

\begin{defn}
\label{}\hypertarget{}{}
For $\mathcal{C}$ and $\mathcal{D}$ two [[(∞,1)-categories]] and $\mathcal{C}_{s}, \mathcla{D}_s \in sSet$ two models as [[quasi-categories]], an [[(∞,1)-functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is simply a homomorphism of simplicial set $\mathcal{C}_s \to \mathcal{D}_s$.

The [[(∞,1)-category of (∞,1)-functors]] $Func(\mathcal{C}, \mathcal{D})_s$ as a [[quasi-category]] is simply the [[hom object]] of simplicial set

\begin{displaymath}
Func(\mathcal{C}, \mathcal{D})_s = sSet(\mathcal{C}_s, \mathcal{D}_s)
  \in 
   QuasiCat \hookrightarrow sSet
  \,.
\end{displaymath}
\end{defn}
\begin{defn}
\label{InfinityPresheaves}\hypertarget{InfinityPresheaves}{}
For $\mathcal{D}$ an [[(∞,1)-category]], the \textbf{[[(∞,1)-category of (∞,1)-presheaves]]} on $\mathcal{D}$ is the functor category

\begin{displaymath}
PSh_\infty(\mathcal{D})
  =
  Func(\mathcal{D}^{op}, \infty Grpd)
\end{displaymath}
out of the [[opposite (∞,1)-category]] of $\mathcal{D}$ into the [[∞Grpd|(∞,1)-category of ∞-groupoids]].

\end{defn}
\hypertarget{localizations_of_categories}{}\subsubsection*{{Localizations of $(\infty,1)$-categories}}\label{localizations_of_categories}

The notions of [[adjoint functors]], [[full and faithful functors]] etc. have straightforward, essentially verbatim generalizations to $(\infty,1)$-categories:

\begin{defn}
\label{}\hypertarget{}{}
A pair of [[(∞,1)-functors]]

\begin{displaymath}
C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} 
  D
\end{displaymath}
is a pair of \textbf{[[adjoint (∞,1)-functors]]}, if there exists a \emph{unit transformation} $\epsilon : Id_C \to R \circ L$ -- a morphism in the [[(∞,1)-category of (∞,1)-functors]] $Func(C,D)$ -- such that for all $c \in C$ and $d \in D$ the induced morphism

\begin{displaymath}
Hom_C(L(c),d)
   \stackrel{R_{L(c), d}}{\to}  
  Hom_D(R(L(c)), R(d))
  \stackrel{Hom_D(\epsilon, R(d))}{\to}
  Hom_D(c,R(d))
\end{displaymath}
is an [[equivalence of ∞-groupoids]].

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
An [[(∞,1)-functor]] $F \colon \mathcal{C} \to \mathcal{D}$ is a \textbf{[[full and faithful (∞,1)-functor]]} if for all objects $X,Y \in \mathcal{C}$ the component

\begin{displaymath}
F_{X,Y} \colon \mathcal{C}(X,Y) \stackrel{\simeq}{\to} \mathcal{D}(F(X), F(Y))
\end{displaymath}
is an [[equivalence of ∞-groupoids]].

\end{defn}
\begin{defn}
\label{ReflectiveLocalization}\hypertarget{ReflectiveLocalization}{}
A [[reflective sub-(∞,1)-category]] $\mathcal{C} \hookrightarrow \mathcal{D}$ is a [[full and faithful (∞,1)-functor]] with a left [[adjoint (∞,1)-functor]].

\end{defn}
\hypertarget{locally_presentable_categories}{}\subsubsection*{{Locally presentable $(\infty,1)$-categories}}\label{locally_presentable_categories}

We have then the essentially verbatim analog of the situation for ordinary categories:

\begin{defn}
\label{LocallyPresentableInfinityCategory}\hypertarget{LocallyPresentableInfinityCategory}{}
An [[(∞,1)-category]] $\mathcal{C}$ is a \textbf{[[locally presentable (∞,1)-category]]} if there exists a [[small set]] of objects such that the [[full sub-(∞,1)-category]] $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ on it generates $\mathcal{C}$ under [[(∞,1)-colimits]].

\end{defn}
And the equivalent characterization is now as before

\begin{theorem}
\label{SimpsonTheorem}\hypertarget{SimpsonTheorem}{}
An [[(∞,1)-category]] is a [[locally presentable (∞,1)-category]], def. \ref{LocallyPresentableInfinityCategory}, precisely if it is [[equivalence of (∞,1)-categories|equivalent]] to [[localization of an (∞,1)-category|localization]], def. \ref{ReflectiveLocalization},

\begin{displaymath}
\mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}}
  PSh_\infty(\mathcal{C}^0)
\end{displaymath}
of an [[(∞,1)-category of (∞,1)-presheaves]], def. \ref{InfinityPresheaves}, such that $R \circ L$ preserves $\kappa$-[[filtered (∞,1)-colimits]] for some [[regular cardinal]] $\kappa$.

\end{theorem}
This appears as \hyperlink{Lurie}{Lurie, theorem 5.5.1.1}, attributed there to [[Carlos Simpson]].

\hypertarget{toposes}{}\subsubsection*{{$(\infty,1)$-Toposes}}\label{toposes}

As before, if a locally presentable $(\infty,1)$-category arises as the [[localization of an (∞,1)-category|localization]] $L \colon PSh_\infty(\mathcal{C}^0) \to \mathcal{C}$ of a [[left exact (∞,1)-functor]], then it is an [[(∞,1)-topos]].

\hypertarget{presentation_by_combinatorial_model_categories}{}\subsubsection*{{Presentation by combinatorial model categories}}\label{presentation_by_combinatorial_model_categories}

There is a close match between the theory of [[combinatorial model categories]] and [[locally presentable (∞,1)-categories]].

\begin{theorem}
\label{}\hypertarget{}{}
Every [[locally presentable (∞,1)-category]] arises, up to [[equivalence of (∞,1)-categories]], as the [[simplicial localization]] of a [[combinatorial model category]].

\end{theorem}
This is part of \hyperlink{Lurie}{Lurie, theorem 5.5.1.1}.

Accordingly, every [[simplicial Quillen adjunction]] between [[combinatorial model categories]] gives rise to a pair of [[adjoint (∞,1)-functors]] between the corresponding locally presentable $(\infty,1)$-categories.

Hence a [[left Bousfield localization]] of a [[model structure on simplicial presheaves]] presents a corresponding localization of an [[(∞,1)-category of (∞,1)-presheaves]] to a [[locally presentable (∞,1)-category]].

\begin{displaymath}
\itexarray{   
    \mathcal{C} &\stackrel{\overset{}{\leftarrow}}{\hookrightarrow}& PSh_\infty(C)
    \\
    \left[C^{op}, sSet\right]_{proj,\mathcal{S}}
    &\stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}}&
    [C^{op}, sSet]_{proj}
  }
\end{displaymath}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The standard textbook for [[locally presentable categories]] is

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

\end{itemize}
Decent accounts of [[combinatorial model categories]] include secton A.2.6 of

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

\end{itemize}
and

\begin{itemize}%
\item [[Clark Barwick]], \emph{On (Enriched) Left Bousfield Localization of Model Categories} (\href{http://arxiv.org/abs/0708.2067}{arXiv:0708.2067})

\end{itemize}
The standard text for [[locally presentable (∞,1)-categories]] is section 5 of \hyperlink{Lurie}{Lurie}.



\end{document}