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\section*{locally presentable category}

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

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

[[!include category theory - contents]]

\hypertarget{compact_objects}{}\paragraph*{{Compact objects}}\label{compact_objects}

[[!include compact object - 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{EquivalentCharacterizations}{Equivalent characterizations}\dotfill \pageref*{EquivalentCharacterizations} \linebreak
\noindent\hyperlink{as_limitpreserving_functor_categories}{As limit-preserving functor categories}\dotfill \pageref*{as_limitpreserving_functor_categories} \linebreak
\noindent\hyperlink{AsLocalizationsOfPresheafCategories}{As localizations of presheaf categories}\dotfill \pageref*{AsLocalizationsOfPresheafCategories} \linebreak
\noindent\hyperlink{GabrielUlmerDuality}{Finite presentability and Gabriel--Ulmer duality}\dotfill \pageref*{GabrielUlmerDuality} \linebreak
\noindent\hyperlink{stability_of_presentability_under_various_operations}{Stability of presentability under various operations}\dotfill \pageref*{stability_of_presentability_under_various_operations} \linebreak
\noindent\hyperlink{wellpoweredness_and_wellcopoweredness}{Well-poweredness and well-copoweredness}\dotfill \pageref*{wellpoweredness_and_wellcopoweredness} \linebreak
\noindent\hyperlink{examples_and_applications}{Examples and applications}\dotfill \pageref*{examples_and_applications} \linebreak
\noindent\hyperlink{locally_finitely_presentable_categories}{Locally finitely presentable categories}\dotfill \pageref*{locally_finitely_presentable_categories} \linebreak
\noindent\hyperlink{locally_presentable_categories}{Locally presentable categories}\dotfill \pageref*{locally_presentable_categories} \linebreak
\noindent\hyperlink{combinatorial_model_categories}{Combinatorial model categories}\dotfill \pageref*{combinatorial_model_categories} \linebreak
\noindent\hyperlink{orthogonal_subcategory_problem}{Orthogonal subcategory problem}\dotfill \pageref*{orthogonal_subcategory_problem} \linebreak
\noindent\hyperlink{functor_categories}{Functor categories}\dotfill \pageref*{functor_categories} \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}

A \emph{locally presentable category} is a [[category]] which contains a [[small set]] $S$ of [[small objects]] such that every [[object]] is a nice [[colimit]] over objects in this set.

This says equivalently that a presentable category $\mathcal{C}$ is a [[reflective localization]] $\mathcal{C} \hookrightarrow PSh(S)$ of a [[category of presheaves]] over $S$. Since here $PSh(S)$ is the [[free colimit completion]] of $S$ and the localization imposes \emph{relations}, this is a presentation of $\mathcal{C}$ by \emph{[[generators and relations]]}, hence the name \emph{(locally) presentable category}.

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

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

There are many equivalent characterizations of \emph{locally presentable categories}. The following is one of the most intuitive, equivalent characterizations are discussed \hyperlink{EquivalentCharacterizations}{below}.

\begin{defn}
\label{PresentableCategory}\hypertarget{PresentableCategory}{}
\textbf{(locally presentable category)}

A [[category]] $\mathcal{C}$ is called \textbf{locally presentable} if

\begin{enumerate}%
\item it is an [[accessible category]];


\item it has all small [[colimits]].



\end{enumerate}
This means

\begin{enumerate}%
\item $\mathcal{C}$ is a [[locally small category]];


\item $\mathcal{C}$ has all small [[colimits]];


\item there exists a [[small set]] $S \hookrightarrow Obj(\mathcal{C})$ of $\lambda$-small [[objects]] that generates $\mathcal{C}$ under $\lambda$-filtered colimits for some regular cardinal $\lambda$.

(meaning that every object of $\mathcal{C}$ may be written as a colimit over a [[diagram]] with objects in $S$);


\item every object in $\mathcal{C}$ is a [[small object]] (assuming 3, this is equivalent to the assertion that every object in $S$ is small).



\end{enumerate}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The \emph{locally} in \emph{locally presentable category} refers to the fact that it is the \emph{objects} that are presentable, not the category as such.

For instance, consider the notion of ``locally finitely presentable category'', def. \ref{LocallyFinitelyPresentable} below, in which the generating set $S$ consists of [[finitely presentable objects]], i.e. $\omega$-small ones. If one dropped the word ``locally'' then one would get the notion ``[[finitely presentable category]]'' which means something completely different, namely a [[finitely presentable object|finitely presentable]] ($\omega$-small) object of [[Cat]].

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Since a [[small object]] is one which is $\kappa$-[[compact object|compact]] for some $\kappa$, and any $\kappa$-compact object is also $\lambda$-compact for any $\lambda\gt\kappa$, it follows that there exists some $\kappa$ such that every object of the colimit-generating set $S$ is $\kappa$-compact.

\end{remark}
This provides a ``stratification'' of the class of locally presentable categories, as follows.

\begin{defn}
\label{}\hypertarget{}{}
\textbf{(locally $\kappa$-presentable category)}

For $\kappa$ a [[regular cardinal]], a \textbf{locally $\kappa$-presentable category} is a locally presentable category, def. \ref{PresentableCategory}, such that the colimit-generating set $S$ may be taken to consist of $\kappa$-compact objects.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
Thus, a locally presentable category is one which is locally $\kappa$-presentable for \emph{some} [[regular cardinal]] $\kappa$ (hence also for every $\lambda\gt\kappa$). In fact, in this case the fourth condition is redundant; once we know that there is a colimit-generating set consisting of $\kappa$-compact objects, it follows automatically that every object is $\lambda$-compact for some $\lambda$ (though there is no uniform upper bound on the required size of $\lambda$). Moreover, colimit-generation is also stronger than necessary; it suffices to have a [[strong generator]] consisting of small objects.

\end{remark}
\begin{defn}
\label{LocallyFinitelyPresentable}\hypertarget{LocallyFinitelyPresentable}{}
A locally ${\aleph}_0$-presentable category is called a \textbf{[[locally finitely presentable category]]}.

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

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

There are various equivalent characterizations of locally presentable categories.

\hypertarget{as_limitpreserving_functor_categories}{}\paragraph*{{As limit-preserving functor categories}}\label{as_limitpreserving_functor_categories}

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(as limit sketches)}

Locally presentable categories are precisely the categories of [[sketch|models of limit-sketches]].

\end{prop}
This is (\hyperlink{AdamekRosicky}{Adamek-Rosicky, corollary 1.52}).

Restricted to locally finitely presentable categories this becomes:

\begin{prop}
\label{}\hypertarget{}{}
Locally finitely presentable categories, def. \ref{LocallyFinitelyPresentable}, are equivalently the categories of [[finite limit]] preserving functors $C \to Set$, for small finitely complete categories $C$.

\end{prop}
For the more detailed statement see below at \emph{\hyperlink{GabrielUlmerDuality}{Gabriel-Ulmer duality}}. Equivalently this says that:

\begin{remark}
\label{}\hypertarget{}{}
Locally finitely presentable categories are equivalently [[models]] of finitary [[essentially algebraic theories]].

\end{remark}
\hypertarget{AsLocalizationsOfPresheafCategories}{}\paragraph*{{As localizations of presheaf categories}}\label{AsLocalizationsOfPresheafCategories}

\begin{prop}
\label{AsLocalizationOfPresheafCategories}\hypertarget{AsLocalizationOfPresheafCategories}{}
\textbf{(as accessible reflective subcategories of presheaves)}

Locally presentable categories are precisely the [[accessible functor|accessibly embedded]] full [[reflective subcategories]]

\begin{displaymath}
(L \dashv i) 
  :
  C \stackrel{\overset{L}{\leftarrow}}{\underset{i}{\hookrightarrow}}
  PSh(K)
\end{displaymath}
of [[categories of presheaves]] on some category $K$.

\end{prop}
This appears as (\hyperlink{AdamekRosicky}{Ad\'a{}mek-Rosick\'y{}, prop 1.46}).

\begin{remark}
\label{}\hypertarget{}{}
Here \emph{accessibly embedded} means that $C \hookrightarrow Psh(K)$ is an [[accessible functor]], which in turn means that $C$ is closed in $Psh(K)$ under $\kappa$-[[filtered colimits]] for some [[regular cardinal]] $\kappa$.

\end{remark}
See also at \emph{[[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]}.

\hypertarget{GabrielUlmerDuality}{}\subsubsection*{{Finite presentability and Gabriel--Ulmer duality}}\label{GabrielUlmerDuality}

\begin{defn}
\label{}\hypertarget{}{}
Write $Lex$ for the [[2-category]] of [[small categories]] with [[finite limits]], with finitely continuous (i.e., finite limit preserving) [[functors]] between them, and [[natural transformations]] between those.

Write $LFP$ for the [[2-category]] of locally finitely presentable categories, def. \ref{LocallyFinitelyPresentable}, [[right adjoint]] functors which preserve [[filtered colimits]], and natural transformations between them.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{([[Gabriel-Ulmer duality]])}

There is an [[equivalence of 2-categories]]

\begin{displaymath}
Lex^{op} \stackrel{\simeq}{\to} LFP
\end{displaymath}
\begin{displaymath}
C \mapsto Lex(C,Set)
\end{displaymath}
which sends a finitely complete category $C$ to the category of [[models]] of $C$, i.e., the category of [[left exact functors]] $C \to$ [[Set]].

\end{prop}
\hypertarget{stability_of_presentability_under_various_operations}{}\subsubsection*{{Stability of presentability under various operations}}\label{stability_of_presentability_under_various_operations}

\begin{lemma}
\label{}\hypertarget{}{}
A [[slice category]] of a locally presentable category is again locally presentable.

\end{lemma}
This appears for instance as (\href{CentazzoRosickyVitale}{Centazzo-Rosick\'y{}-Vitale, remark 3}).

\begin{theorem}
\label{}\hypertarget{}{}
If $A$ is locally presentable and $C$ is a [[small category]], then the [[functor category]] $A^C$ is locally presentable.

\end{theorem}
\hypertarget{wellpoweredness_and_wellcopoweredness}{}\subsubsection*{{Well-poweredness and well-copoweredness}}\label{wellpoweredness_and_wellcopoweredness}

\begin{itemize}%
\item Every [[locally presentable category]] is well-powered, since it is a full reflective subcategory of a presheaf topos, so its subobject lattices are subsets of those of the latter.


\item Every locally presentable category is also well-\emph{copowered}. This is shown in \hyperlink{AdamekRosicky}{Adamek-Rosicky, Proposition 1.57 and Theorem 2.49}.



\end{itemize}
\hypertarget{examples_and_applications}{}\subsection*{{Examples and applications}}\label{examples_and_applications}

\hypertarget{locally_finitely_presentable_categories}{}\subsubsection*{{Locally finitely presentable categories}}\label{locally_finitely_presentable_categories}

We list examples of locally finitely presentable categories, def. \ref{LocallyFinitelyPresentable}.

\begin{example}
\label{}\hypertarget{}{}
The category [[Set]] of [[sets]] is locally finitely presentable.

For notice that every [[set]] is the [[directed colimit]] over the [[poset]] of all its [[finite set|finite]] [[subsets]].

Moreover, a set $S \in Set$ is a $\kappa$-[[compact object]] precisely if it has cardinality $|S| \lt \kappa$. So all finite sets are [[?]]$_0$-compact.

Hence a a set of generators that exhibits $Set$ as a locally finitely complete category is given by the set containing one finite set of [[cardinality]] $n \in \mathbb{N}$ for all $n$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
More generally, for $C$ any [[small category]] the [[category of presheaves]] $Set^C$ is locally finitely presentable if $C$ is small.

This follows with [[Gabriel-Ulmer duality]]: the [[lex completion|finite limit completion]] of $C$, $Lex(C)$, is also small, and $Set^C$ is [[equivalence of categories|equivalent]] to the category of finitely continuous functors $Lex(C) \to Set$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
More generally still, if $A$ is locally finitely presentable and $C$ is [[small category|small]], then $A^C$ is locally finitely presentable.

To see this, embed $A$ as a [[accessible functor|finitely-accessible]] [[reflective subcategory]] of a [[presheaf topos]] $Set^B$, and then note that by [[2-functor|2-functoriality]] of $(-)^C$ we get $A^C$ as a finitely-accessible reflective subcategory of $Set^{B \times C}$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The category of [[algebra over a Lawvere theory|algebras of]] a [[Lawvere theory]], for example [[Grp]], is locally finitely presentable. A $T$-algebra $A$ is finitely presented if and only if the [[hom-functor]] $Alg_T(A, -)$ preserves [[filtered colimits]], and any $T$-[[algebra over an algebraic theory|algebra]] can be expressed as a filtered colimit of finitely presented algebras.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The category of [[coalgebras]] over a [[field]] $k$ is locally finitely presentable; similarly the category of commutative coalgebras over $k$ is locally finitely presentable.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
A [[poset]], regarded as a category, is locally finitely presentable if it is a complete [[lattice]] which is [[algebraic lattice|algebraic]] (each element is a directed [[join]] of finite elements).

\end{example}
\begin{example}
\label{}\hypertarget{}{}
\begin{itemize}%
\item The category [[FinSet]] of \emph{finite} sets is not locally finitely presentable, as it does not have all countable colimits.


\item The category of fields and field homomorphisms is not locally presentable, as it does not have all binary coproducts (for instance, there are none between fields of differing characteristics).


\item [[Top]] is not locally finitely presentable.


\item The [[opposite category]] of a locally presentable category (in particular, a locally finitely presentable category) is \emph{never} locally presentable, unless it is a poset. This is \hyperlink{GabrielUlmer}{Gabriel-Ulmer, Satz 7.13}.



\end{itemize}
\end{example}
\hypertarget{locally_presentable_categories}{}\subsubsection*{{Locally presentable categories}}\label{locally_presentable_categories}

\begin{example}
\label{}\hypertarget{}{}
A [[poset]], considered as a category, is locally presentable precisely if it is a complete [[lattice]].

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The following three examples, being [[presheaf categories]], are locally finitely presentable, thus \emph{a fortiori} locally presentable. They are important for the general study of [[(∞,1)-categories]].

\begin{itemize}%
\item the category [[sSet]] of [[simplicial sets]];


\item the category [[dSet]] of [[dendroidal sets]].


\item for $C$ a [[small category]] the [[functor category]] $Funct(C,sSet)$ of [[simplicial presheaves]].



\end{itemize}
\end{example}
More generally,

\begin{prop}
\label{}\hypertarget{}{}
Every [[sheaf topos]] is locally presentable.

\end{prop}
This appears for instance as (\hyperlink{Borceux}{Borceux, prop. 3.4.16, page 220}). It follows directly with prop. \ref{AsLocalizationOfPresheafCategories} and using that every [[sheaf topos]] is an accessibly embedded [[subtopos]] of a [[presheaf topos]] (see at \emph{[[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]})

The main ingredient of a direct proof is:

\begin{prop}
\label{}\hypertarget{}{}
For $C$ a [[site]] and $\kappa$ a [[regular cardinal]] strictly larger than the [[cardinality]] of $Mor(C)$, every $\kappa$-[[filtered colimit]] in the [[sheaf topos]] $Sh(C)$ is computed objectwise.

\end{prop}
This implies that all [[representable functor|representables]] in a [[sheaf topos]] are $\kappa$-[[compact objects]].

\begin{theorem}
\label{AlgebrasOverAnAccessibleMonad}\hypertarget{AlgebrasOverAnAccessibleMonad}{}
If $T$ is an [[accessible monad]] (a [[monad]] whose underlying [[functor]] is an [[accessible functor]]) on a locally presentable category $A$, then the category $A^T$ of [[algebras over a monad|algebras over the monad]] is locally presentable. In particular, if $A$ is locally presentable and $i: B \to A$ is a [[reflective subcategory]], then $B$ is locally presentable if $i$ is accessible.

\end{theorem}
This appears in (\hyperlink{AdamekRosicky}{Adamek-Rosicky, 2.78}).

This is actually somewhat subtle and gets into some transfinite combinatorics, from what I can gather.

\hypertarget{combinatorial_model_categories}{}\subsubsection*{{Combinatorial model categories}}\label{combinatorial_model_categories}

A [[combinatorial model category]] is a [[model category]] that is in particular a locally presentable category.

\hypertarget{orthogonal_subcategory_problem}{}\subsubsection*{{Orthogonal subcategory problem}}\label{orthogonal_subcategory_problem}

Given a class of morphisms $\Sigma$ in a locally presentable category, the answer to the [[orthogonal subcategory problem]] for $\Sigma^\perp$ is affirmative if $\Sigma$ is small, and is affirmative for any class $\Sigma$ assuming the large cardinal axiom known as [[Vopenka's principle]].

\hypertarget{functor_categories}{}\subsubsection*{{Functor categories}}\label{functor_categories}

See at \emph{\href{http://ncatlab.org/nlab/show/functor+category#LocalPresentability}{Functor category -- Local presentability}}.

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

\begin{itemize}%
\item [[PrCat]], [[Pr(∞,1)Cat]], [[Ho(CombModCat)]]


\item Another notion of ``presentable category'' is that of an \emph{[[equationally presentable category]]}.


\item Locally presentable categories are a special case of \emph{[[locally bounded category|locally bounded categories]]}.


\item [[class-locally presentable category]]



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

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

The definition is due to

\begin{itemize}%
\item [[Pierre Gabriel]], [[Friedrich Ulmer]], \emph{Lokal pr\"a{}sentierbare Kategorien}, Springer LNM 221, 1971

\end{itemize}
The standard textbook is

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

\end{itemize}
More details are in

\begin{itemize}%
\item C. Centazzo, [[Jiří Rosický]], [[Enrico Vitale]], \emph{A characterization of locally $\mathbb{D}$-presentable categories} (\href{http://perso.uclouvain.be/enrico.vitale/ClaudiaJiri.pdf}{pdf})

\end{itemize}
Some further discussion is in

\begin{itemize}%
\item [[Francis Borceux]], \emph{[[Handbook of Categorical Algebra]]: III Categories of Sheaves} (proposition 3.4.16), page 220.

\end{itemize}
See also section A.1.1 of

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

\end{itemize}
where locally presentable categories are called just [[presentable category|presentable categories]].

[[!redirects locally presentable]] [[!redirects locally presentable category]] [[!redirects locally presentable categories]] [[!redirects presentable category]] [[!redirects presentable categories]] [[!redirects locally representable category]]



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