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\section*{compact object}

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

\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{FinitelyPresentableObject}{Definition}\dotfill \pageref*{FinitelyPresentableObject} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{SubtletiesAndDifferentMeanings}{Subtleties and different meanings}\dotfill \pageref*{SubtletiesAndDifferentMeanings} \linebreak
\noindent\hyperlink{CompactnessInAdditiveCategories}{Compactness in additive categories}\dotfill \pageref*{CompactnessInAdditiveCategories} \linebreak
\noindent\hyperlink{CompactObjectsInTop}{Compact objects in $Top$}\dotfill \pageref*{CompactObjectsInTop} \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 [[object]] of a [[category]] is called \emph{compact} if it is ``finite'' or ``small'' in some precise sense. There are however different formalizations of this idea. Here discussed is the notion, usually going by this term, where an object $X$ is called \emph{compact} if mapping out of it commutes with [[filtered colimits]].

This means that if any other object $A$ is given as the [[colimit]] of a ``suitably increasing'' family of objects $\{A_i\}$, then every morphism

\begin{displaymath}
X \to A = \lim_{\to_i} A_i
\end{displaymath}
out of the compact object $X$ into that colimit factors through one of the inclusions $A_i \to \underset{\to_i}\lim A_i$.

The notion of \emph{[[small object]]} is essentially the same, with a bit more flexibility on when the family $\{A_i\}$ is taken to be ``suitably increasing''. An important application of the above factorization property is accordingly named the \emph{[[small object argument]]}. On the other hand, there is also the notion of \emph{[[finite object]]} (in a [[topos]]) which, while closely related, is different. See also \emph{\hyperlink{SubtletiesAndDifferentMeanings}{Subtleties and different meanings}} below.

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

\begin{defn}
\label{}\hypertarget{}{}
Let $C$ be a [[locally small category]] that admits [[filtered colimits]]. Then an [[object]] $X \in C$ is \textbf{compact}, or \textbf{finitely presented} or \textbf{finitely presentable}, or \textbf{of finite presentation}, if the [[corepresentable functor]]

\begin{displaymath}
Hom_C(X,-) \colon C \to Set
\end{displaymath}
[[preserved limit|preserves]] these [[filtered colimits]]. This means that for every [[filtered category]] $D$ and every functor $F : D \to C$, the canonical morphism

\begin{displaymath}
\underset{\to_d}{\lim} C(X,F(d)) \xrightarrow{\simeq}
  C(X, \underset{\to_d}{\lim} F(d))
\end{displaymath}
is an [[isomorphism]].

More generally, if $\kappa$ is a [[regular cardinal]], then an object $X$ such that $C(X,-)$ commutes with $\kappa$-[[filtered colimits]] is called \textbf{$\kappa$-compact}, or \textbf{$\kappa$-presented}, or \textbf{$\kappa$-presentable}. An object which is $\kappa$-compact for some regular $\kappa$ is called a [[small object]].

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

\begin{prop}
\label{}\hypertarget{}{}
A $\kappa$-small [[colimit]] of $\kappa$-compact objects is again a $\kappa$-compact object.

\end{prop}
\begin{proof}
Let $D$ be a $\kappa$-[[small category]] and $X : D \to C$ a [[diagram]] of $\kappa$-compact objects. Let $I$ be a $\kappa$-[[filtered category]] and $A : I \to C$ a $\kappa$-filtered diagram in $C$. Then

\begin{displaymath}
Hom(\lim_{\to_d} X_d, \lim_{\to_i} A_i)
  \simeq
  \lim_{\leftarrow_d} Hom(X_d, \lim_{\to_i} A_i)
\end{displaymath}
by general properties of the [[hom functor]]. Now using that every $X_d$ is $\kappa$-compact and $I$ is $\kappa$-filtered this is

\begin{displaymath}
\cdots 
  \simeq
  \lim_{\leftarrow_d} \lim_{\to_i} Hom(X_d, A_i)
  \,.
\end{displaymath}
Since this (co)limit is taken in [[Set]] ,the $\kappa$-small limit over $D$ commutes with the $\kappa$-filtered colimit

\begin{displaymath}
\cdots 
  \simeq
  \lim_{\to_i} \lim_{\leftarrow_d}  Hom(X_d, A_i)
  \,.
\end{displaymath}
We can take the limit again to a colimit in the first argument

\begin{displaymath}
\cdots 
  \simeq
  \lim_{\to_i}  Hom(\lim_{\to_d}  X_d, A_i)
  \,,
\end{displaymath}
which proves the claim.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
In a $\lambda$-[[accessible category]], if $\lambda$ is [[sharply smaller cardinal|sharply smaller]] than $\kappa$, then every $\kappa$-compact object is a [[retract]] of a $\kappa$-small $\lambda$-filtered colimit of $\lambda$-compact objects.

\end{prop}
\begin{proof}
See \hyperlink{AdamekRosicky94}{Adamek-Rosicky, Remark 2.15}. It is noted there that with the more technical proof of \hyperlink{MakkaiPare89}{Makkai-Pare, Proposition 2.3.11} the words ``a retract of'' can be omitted.

\end{proof}
If we weaken the hypothesis to $\lambda\le \kappa$, then we retain all of the result except for $\lambda$-filteredness of the colimit.

\begin{prop}
\label{}\hypertarget{}{}
In a locally $\lambda$-[[locally presentable category|presentable category]], if $\lambda\le \kappa$, then every $\kappa$-compact object is a [[retract]] of a $\kappa$-small colimit of $\lambda$-compact objects.

\end{prop}
\begin{proof}
See \hyperlink{AdamekRosicky94}{Adamek-Rosicky, Remark 1.30}. It is claimed there that the words ``a retract of'' can be omitted by reference to an argument in Makkai-Pare, but it seems unclear how this argument is intended to be used. An alternative proof of this improvement is proposed \href{https://mathoverflow.net/q/325278}{at this mathoverflow question}.

\end{proof}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{itemize}%
\item In $C =$ [[Set]] an object is compact precisely if it is a [[finite set]]. For this to hold constructively, [[filtered categories]] (appearing in the definition of \emph{[[filtered colimit]]}) have to be understood as categories admitting cocones of every \emph{Bishop-finite} diagram. (An object of Set is a [[finite set|Kuratowski-finite]] precisely if it is a [[finitely generated object]], or equivalently if it is [[compact space|compact]] when regarded as a [[discrete object|discrete]] topological space.)


\item For $C$ a [[topos]], $X$ is compact if:

\begin{itemize}%
\item $C$ is a [[Grothendieck topos | sheaf topos]] on a site whose topology is generated by finite covering families and $X$ is a representable sheaf;


\item in particular if $C$ is a [[coherent topos]] and $X$ is a [[coherent object]].



\end{itemize}


\end{itemize}
However, there exist compact objects which are not coherent, c.f. the [[Elephant]], D3.3.12.

\begin{itemize}%
\item In $C =$ [[Grp]] an object is compact precisely if it is [[finitely presented group|finitely presented]] as a group.


\item More generally, if $C$ is any [[variety of algebras]], then an object is compact precisely if it is [[finitely presented algebra|finitely presented]] as an algebra. A proof may be found in \hyperlink{AdamekRosicky94}{Ad\'a{}mek-Rosick\'y{} 94, Corollary 3.13}.


\item Let $X$ be a [[topological space]] and let $C = Op(X)$ be the [[category of open subsets]] of $X$. Then an [[open subset]] $U \in C$ is a compact object in $C$ precisely if it is a [[compact space|compact topological space]]. (It is \emph{not} true that $X$ is a compact object of $Top$ iff it is a compact topological space; see below.)


\item A [[finite-dimensional vector space]] is compact in [[Vect]], see \href{finite-dimensional+vector+space#CompactClosure}{here}.



\end{itemize}
\hypertarget{SubtletiesAndDifferentMeanings}{}\subsection*{{Subtleties and different meanings}}\label{SubtletiesAndDifferentMeanings}

One has to be careful about the following variations of the theme of compactness.

(Some of these subtleties are resolved by noticing that there is a hierarchy of notions of compact objects that are secretly different but partly go by the same name. Some discussion of this is currently at \emph{[[compact topos]]}, but more detailed discussion should eventually be somewhere\ldots{})

In the [[Elephant]], what Johnstone calls \emph{compact objects} are those objects such that the [[top]] element of the [[poset of subobjects]] $\operatorname{Sub}(C)$ is a [[compact element]]; he reserves the term \emph{finitely-presented} for the notion of compact on this page.

\hypertarget{CompactnessInAdditiveCategories}{}\subsubsection*{{Compactness in additive categories}}\label{CompactnessInAdditiveCategories}

When $C$ is an [[additive category]] (often a [[triangulated category]]), an object $x$ in $C$ is called \textbf{compact} if for every set $S$ of objects of $C$ such that the coproduct $\coprod_{s\in S} s$ exists, the canonical map

\begin{displaymath}
\coprod_{s\in S} C(x,s)\to C(x,\coprod_{s\in S}s)
\end{displaymath}
is an [[isomorphism]] of [[commutative monoid|commutative monoids]].

Here is an application of this concept to characterize which abelian categories are categories of modules of some ring:

\begin{utheorem}
Let $C$ be an abelian category. If $C$ has all [[small category|small]] [[coproducts]] and has a compact [[projective object| projective]] [[generator]], then $C \simeq R Mod$ for some ring $R$. In fact, in this situation we can take $R = C(x,x)^{op}$ where $x$ is any compact projective generator. Conversely, if $C \simeq R Mod$, then $C$ has all small coproducts and $x = R$ is a compact projective generator.

\end{utheorem}
\begin{proof}
This theorem, minus the explicit description of $R$, can be found as Exercise F on page 103 of Peter Freyd's book \href{http://www.emis.de/journals/TAC/reprints/articles/3/tr3.pdf#page=132}{Abelian Categories}. The first part of this theorem can also be found as Prop. 2.1.7. of Victor Ginzburg's \href{http://arxiv.org/PS_cache/math/pdf/0506/0506603v1.pdf#page=4}{Lectures on noncommutative geometry}. Conversely, it is easy to see that $R$ is a compact projective generator of $R Mod$.

\end{proof}
Zoran: While Ginzburg's reference is surely a worthy to look at, it would be better not to give false impression that this [[reconstruction theorem]] is due Ginzburg or at all new. It is rather a classical and well know fact probably from early 1960s, essentially small strengthening of a variant of a circle of abelian reconstruction theorems including the \href{http://myyn.org/m/article/gabriel-popescu-theorem-for-ab5-categories}{Gabriel-Popescu theorem}(probably our variant could be read off from classical algera book by Faith for example, or Popescu's book on abelian categories, in any case it is well known in [[noncommutative algebraic geometry]]). In fact for this fact, if I think better, the reconstruction belongs usually to expositions which treat classical Morita theory for rings.

A triangulated category is \textbf{compactly generated} if it is generated (see [[generator]]) by a \emph{set} of compact objects.

The notion can be modified for categories [[enriched category|enriched]] over a [[closed monoidal category]] (compare to the notions of finite and/or rigid objects in various contexts).

Compact objects in the derived categories of quasicoherent sheaves over a scheme are called [[perfect complexes]]. Any [[compact object]] in the [[category of modules]] over a perfect ring is finitely generated as a module.

In non-additive contexts, the above definition is not right. For instance, with this definition a [[topological space]] would be compact iff it is [[connected space|connected]]. In general one should expect to instead preserve filtered colimits, as above.

\hypertarget{CompactObjectsInTop}{}\subsubsection*{{Compact objects in $Top$}}\label{CompactObjectsInTop}

Recall the above example of [[compact space|compact topological spaces]]. Notice that the statement which one might expect, that a topological space $X$ is [[compact space|compact]] if it is a compact object in [[Top]], is not quite right in general.

A counterexample is given for instance on page 49 of [[Mark Hovey|Hovey]]`s \emph{Model Categories}, which itself was corrected by Don Stanley (see the \href{http://hopf.math.purdue.edu/Hovey/model-err.pdf}{errata} of that book). See also the blog discussion \href{http://golem.ph.utexas.edu/category/2009/05/journal_club_geometric_infinit_3.html#c023790}{here}.

Namely, the two-element set with the [[indiscrete topology]] is a compact space $X$ for which

\begin{displaymath}
Hom(X, -): Top \rightarrow Top
\end{displaymath}
doesn't preserve filtered colimits, in fact not even [[sequential colimit|colimits of sequences]] (functors out of the [[poset|ordered set]] of [[natural numbers]]).

For example, consider the sequence of spaces

\begin{displaymath}
X_n=[n,\infty) \times \{0,1\}
\end{displaymath}
where the [[open sets]] are of the form

\begin{displaymath}
[n, \infty] \times \{0\} \cup [m,\infty) \times \{1\}
\end{displaymath}
(where $m \geq n$), plus the empty set. Define $X_n \rightarrow X_{n+1}$ so that it sends a pair $(k, \epsilon)$ to itself if $k \gt n$, and $(n,\epsilon)$ to $(n+1,\epsilon)$. This defines a functor

\begin{displaymath}
F: \mathbb{N} \rightarrow Top
\end{displaymath}
The colimit $X_\infty$ of this sequence is the two-element set $\{0,1\}$ with the indiscrete topology. However, the identity map on this space does not factor through any of the canonical maps $X_n \rightarrow X_\infty$. It follows that the comparison map

\begin{displaymath}
colim_n Hom(X_\infty, X_n) \rightarrow Hom(X_\infty, X_\infty)
\end{displaymath}
is not surjective, and therefore not an isomorphism.

\emph{[[Todd Trimble|Todd]]} (posted from n-category cafe): I don't know if the story is any different for $X$ compact \emph{Hausdorff}, but it could be worth considering.

But with a bit of care on the assumptions, similar results do hold:

If $Y$ is compact, then $hom(Y,-)$ preserves colimits of functors mapping out of [[limit ordinals]], provided that the arrows of the cocone diagram,

\begin{displaymath}
X_\alpha \rightarrow X_\beta,
\end{displaymath}
are [[closed map|closed]] inclusions of $T_1$[[separation axiom|-spaces]]. (This applies for example to the sequence of inclusions of n-skeleta in a [[CW-complex]]. Taking $Y=S_k$, this has obvious desirable consequences for the functor $\pi_k$.)

This example is discussed on page 50 of Hovey's book.

Hovey wants this result in view of a [[small object argument]] on the way to proving that $Top$ is a [[model category]]. See \href{classical+model+structure+on+topological+spaces#CompactSubsetsAreSmallInCellComplexes}{this lemma} at \emph{[[classical model structure on topological spaces]]}.

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

\begin{itemize}%
\item [[compact element]] (in a [[(0,1)-category]])


\item \textbf{compact object}


\item [[compact topos]]


\item [[compact object in an (∞,1)-category]]


\item [[small object]], [[small object argument]]


\item [[finitely generated object]]


\item [[locally presentable category]], [[accessible category]]


\item [[compactly generated (∞,1)-category]]



\end{itemize}
[[!include finite objects -- table]]

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

Compact objects are discussed under the term ``finitely presentable'' or ``finitely-presentable'' objects for instance in

\begin{itemize}%
\item [[Jiří Adámek]], [[Jiří Rosický]], \emph{[[Locally Presentable and Accessible Categories]]}, Cambridge University Press in the London Mathematical Society Lecture Note Series, number 189, (1994)


\item [[Michael Makkai]], [[Robert Paré]], \emph{Accessible categories: The foundations of categorical model theory} Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.


\item [[Masaki Kashiwara]], [[Pierre Schapira]], around Definition 6.3.3 of \emph{[[Categories and Sheaves]]};


\item [[Peter Johnstone]], \emph{[[Stone Spaces]]}, Definition VI.1.8;


\item [[Peter Johnstone]], the \emph{[[Elephant]]}, D2.3.1.



\end{itemize}
For the pages quoted in the context of the discussion of compact objects in [[Top]] see

\begin{itemize}%
\item [[Mark Hovey]], \emph{Model categories}.

\end{itemize}
For the general definition with an eye towards the definition of [[compact object in an (infinity,1)-category]] see section A.1.1 section 5.3.4 of

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

\end{itemize}
[[!redirects compact object]] [[!redirects compact objects]]

[[!redirects finitely presentable object]] [[!redirects finitely presentable objects]]

[[!redirects finitely-presentable object]] [[!redirects finitely-presentable objects]]

[[!redirects presentable object]] [[!redirects presentable objects]]



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