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\begin{document}

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\section*{ind-object}

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

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

[[!include 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{as_diagrams}{As diagrams}\dotfill \pageref*{as_diagrams} \linebreak
\noindent\hyperlink{as_filtered_colimits_of_representable_presheaves}{As filtered colimits of representable presheaves}\dotfill \pageref*{as_filtered_colimits_of_representable_presheaves} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_category_of_indobjects}{The category of ind-objects}\dotfill \pageref*{the_category_of_indobjects} \linebreak
\noindent\hyperlink{recognition_of_indobjects}{Recognition of Ind-objects}\dotfill \pageref*{recognition_of_indobjects} \linebreak
\noindent\hyperlink{functoriality}{Functoriality}\dotfill \pageref*{functoriality} \linebreak
\noindent\hyperlink{the_case_that__already_admits_filtered_colimits}{The case that $\mathcal{C}$ already admits filtered colimits}\dotfill \pageref*{the_case_that__already_admits_filtered_colimits} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{in_higher_category_theory}{In higher category theory}\dotfill \pageref*{in_higher_category_theory} \linebreak
\noindent\hyperlink{in_categories}{In $(\infty,1)$-categories}\dotfill \pageref*{in_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}

An \textbf{ind-object} of a [[category]] $\mathcal{C}$ is a \textbf{formal [[filtered colimit]]} of objects of $\mathcal{C}$. Here ``formal'' means that the colimit is taken in the [[category of presheaves]] of $\mathcal{C}$ (the [[free cocompletion]] of $\mathcal{C}$). The category of ind-objects of $\mathcal{C}$ is written $ind$-$\mathcal{C}$ or $Ind(\mathcal{C})$.

Here, ``ind'' is short for ``[[inductive system]]'', as in the inductive systems used to define [[directed colimits]], and as contrasted with ``pro'' in the dual notion of [[pro-object]] corresponding to ``projective system''.

Recalling the nature of [[filtered colimits]], this means that in particular chains of inclusions

\begin{displaymath}
c_1 \hookrightarrow c_2 \hookrightarrow c_3 \hookrightarrow c_4 \hookrightarrow \cdots
\end{displaymath}
of objects in $\mathcal{C}$ are regarded to converge to an object in $Ind(\mathcal{C})$, even if that object does not exist in $\mathcal{C}$ itself. Standard examples where ind-objects are relevant are categories $\mathcal{C}$ whose objects are finite in some sense, such as [[finite sets]] or [[finite dimensional vector spaces]]. Their ind-categories contain then also the infinite versions of these objects as limits of sequences of inclusions of finite objects of ever increasing size.

Moreover, ind-categories allow one to handle ``big things in terms of small things'' also in another important sense: many [[large category|large categories]] are actually ([[equivalence of categories|equivalent]] to) ind-categories of [[small category|small categories]]. This means that, while [[large category|large]], they are for all practical purposes controlled by a [[small category]] (see the description of the [[hom-set]] of $Ind(\mathcal{C})$ in terms of that of $\mathcal{C}$ below). Such [[large category|large categories]] equivalent to ind-categories are therefore called [[accessible category|accessible categories]].

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

There are several equivalent ways to define ind-objects.

\hypertarget{as_diagrams}{}\subsubsection*{{As diagrams}}\label{as_diagrams}

One definition is to define the objects of $ind$-$\mathcal{C}$ to be diagrams $F:D\to \mathcal{C}$ where $D$ is a [[small category|small]] [[filtered category|filtered]] category.\newline The idea is to think of these diagrams as being the placeholder for the [[colimit]] over them (possibly non-existent in $\mathcal{C}$). We identify an ordinary object of $\mathcal{C}$ with the corresponding diagram $1\to \mathcal{C}$. To see what the morphisms should be between $F:D\to \mathcal{C}$ and $G:E\to \mathcal{C}$, we stipulate that

\begin{enumerate}%
\item The embedding $\mathcal{C}\to ind$-$\mathcal{C}$ should be [[full and faithful functor|full and faithful]],
\item each diagram $F:D\to \mathcal{C}$ should be the colimit of itself (considered as a diagram in $ind$-$\mathcal{C}$ via the above embedding), and
\item the objects of $\mathcal{C}$ should be [[compact object|compact]] in $ind$-$\mathcal{C}$.

\end{enumerate}
Thus, we should have

\begin{displaymath}
\begin{aligned}
  ind\text{-}\mathcal{C}(F,G) &= ind\text{-}\mathcal{C}(colim_{d\in D} F d, colim_{e\in E} G   e)\\
   &\cong lim_{d\in D}\; ind\text{-}\mathcal{C}(F d, colim_{e\in E} G e)\\
   &\cong lim_{d\in D} colim_{e\in E}\; ind\text{-}\mathcal{C}C(F d, G e)\\
   &\cong lim_{d\in D} colim_{e\in E}\; \mathcal{C}(F d, G e)
\end{aligned}
\end{displaymath}
Here

\begin{itemize}%
\item the first step is by assumption that each object is a suitable colimit;


\item the second by the fact that the contravariant Hom sends colimits to limits (see properties of [[colimit]]);


\item the third by the assumption that each object is a [[compact object]];


\item the last by the assumption that the embedding is a [[full and faithful functor]].



\end{itemize}
So then one \emph{defines}

\begin{displaymath}
ind\text{-}\mathcal{C}(F,G) \coloneqq lim_{d\in D} colim_{e\in E}\; \mathcal{C}(F d, G e)
  \,.
\end{displaymath}
\hypertarget{as_filtered_colimits_of_representable_presheaves}{}\subsubsection*{{As filtered colimits of representable presheaves}}\label{as_filtered_colimits_of_representable_presheaves}

Recall the [[co-Yoneda lemma]] that every presheaf $X \in PSh(\mathcal{C})$ is a [[colimit]] over [[representable functor|representable presheaves]]:

there is a functor $\alpha : D \to \mathcal{C}$ (with $D$ possibly large) such that

\begin{displaymath}
X \simeq colim_{d \in D} Y(\alpha(d))
\end{displaymath}
(with $Y$ the [[Yoneda embedding]]).

\begin{defn}
\label{}\hypertarget{}{}
Let $Ind(\mathcal{C}) \subset PSh(\mathcal{C})$ be the [[full subcategory]] of the [[presheaf category]] $PSh(\mathcal{C}) = [\mathcal{C}^{op},Set]$ on those [[functors]]/[[presheaves]] which are [[filtered colimits]] of [[representable functor|representables]], i.e. those for which

\begin{displaymath}
X \simeq colim_{d \in D} Y(\alpha(d))
\end{displaymath}
for $D$ some [[filtered category]].

Those for which $D$ may be chosen to be $\mathbb{N}^{\leq}$, i.e. those that arise as [[sequential colimits]], are also called \emph{[[strict ind-objects]]}.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The functors $\mathcal{C}^{op}\to Set$ belonging to $Ind(\mathcal{C})$ under this definition --- those which are filtered colimits of representables --- have an equivalent characterization as the [[flat functors]]: those which ``would preserve all finite colimits if $\mathcal{C}$ had them''. In particular, if $\mathcal{C}$ has finite colimits, then $Ind(\mathcal{C})$ consists exactly of the finitely cocontinuous presheaves.

For more equivalent characterizations see at \emph{\href{accessible+category#definition}{accessible category -- Definition}}.

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Given that $[\mathcal{C}^{op},Set]$ is the [[free cocompletion]] of $\mathcal{C}$, $Ind(\mathcal{C})$ defined in this way is its ``free cocompletion under filtered colimits.''

\end{remark}
To compare with the first definition, notice that indeed the formula for the [[hom-sets]] is reproduced:

Generally we have

\begin{displaymath}
\begin{aligned}
     [\mathcal{C}^{op},Set](X,Y)
     & \simeq
     [\mathcal{C}^{op}, Set](colim_{d \in D} Y F d, colim_{d' \in D'} Y G d)
     \\
     & \simeq 
     lim_{d \in D} [\mathcal{C}^{op}, Set]( Y F d, colim_{d' \in D'} Y G d)
  \end{aligned}
\end{displaymath}
by the fact that the [[hom-functor]] sends [[colimits]] to [[limits]] in its first argument (see \emph{properties} at \emph{[[colimit]]}).

By the [[Yoneda lemma]] this is

\begin{displaymath}
\cdots \simeq 
   lim_{d \in D} (colim_{d' \in D'} Y G d')(F d)
  \,.
\end{displaymath}
Using that [[colimits]] in $PSh(\mathcal{C})$ are computed objectwise (see again \emph{properties} at \emph{[[colimit]]}) this is

\begin{displaymath}
\cdots \simeq 
   lim_{d \in D} colim_{d' \in D'} 
   \mathcal{C}(F d, G d')  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item Let [[FinVect]] be the category of [[finite-dimensional vector spaces]] (over some [[field]]). Let $V$ be an infinite-dimensional vector space. Then $V$ can be regarded as an object of $ind-FinVect$ as the colimit $colim_{V' \hookrightarrow V} Y(V')$ over the [[filtered category]] whose objects are inclusions $V' \hookrightarrow V$ of finite dimensional vector spaces $V'$ into $V$ of the representables $Y(V') : FinVect^{op} \to Set$ ($Y$ is the [[Yoneda embedding]]).


\item For $\mathcal{C}$ the category of finitely presented objects of some equationally defined structure, $ind\text{-}\mathcal{C}$ is the category of all these structures.

\begin{itemize}%
\item The category [[Grp]] of [[group]]s is the ind-category of the category of finitely generated groups.

\begin{itemize}%
\item The category [[Ab]] of [[abelian group]]s is the ind-category of the category of finitely generated abelian groups.

\end{itemize}


\end{itemize}

\item A [[formal scheme]] is an ind-object in [[schemes]].



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

In the following we write $\underset{\longrightarrow}{\lim}^f$ for the ``formal colimits'' defining ind-objects. I.e. if $\alpha \colon \mathcal{I} \to \mathcal{C}$ is a small [[diagram]] and with $i \colon Ind(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})$ the defining inclusion, then

\begin{displaymath}
\underset{\longrightarrow}{\lim}^f (\alpha)
  \;\coloneqq\;
  \underset{\longrightarrow}{\lim} (i \circ \alpha )
  \,.
\end{displaymath}
\hypertarget{the_category_of_indobjects}{}\subsubsection*{{The category of ind-objects}}\label{the_category_of_indobjects}

\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{C}$ is a [[locally small category]] then so is $Ind(\mathcal{C})$.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
$Ind(\mathcal{C})$ admits small [[filtered colimits]] and the defining inclusion $Ind(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})$ commutes with these colimits.

\end{prop}
(e.g. \hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, theorem 6.1.8})

The following says that morphisms between ind-objects are represented by [[natural transformation]] of the filtered diagrams that represent them, possibly up to restricting these diagrams first along a [[final functor]].

\begin{prop}
\label{MorphismsRepresentedByCofilteredSystemsOfMorphisms}\hypertarget{MorphismsRepresentedByCofilteredSystemsOfMorphisms}{}
Let

\begin{enumerate}%
\item $\mathcal{I}_1$ and $\mathcal{I}_2$ be two [[small category|small]] [[filtered categories]];


\item $\alpha_1 \colon \mathcal{I}_1 \longrightarrow \mathcal{C}$ and $\alpha_2 \colon \mathcal{I}_2 \longrightarrow \mathcal{C}$ be two [[functors]];


\item $f 
 \;\colon\;
\underset{\longrightarrow}{\lim}^f \alpha_1
 \longrightarrow
\underset{\longrightarrow}{\lim}^f \alpha_2$



\end{enumerate}
a [[morphism]] between their images in $Ind(\mathcal{C})$.

Then there exists

\begin{enumerate}%
\item a small [[filtered category]] $K$


\item [[final functors]] $p_i \colon K \longrightarrow \mathcal{I}_i$


\item a [[natural transformation]] $\phi \;\colon\; \alpha_1 \circ p_1  \longrightarrow \alpha_2 \circ p_2$



\end{enumerate}
such that the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    \underset{\longrightarrow}{\lim}^f(\alpha_1 \circ p_1)
      &\overset{\underset{\longrightarrow}{\lim}^f \phi}{\longrightarrow}&
    \underset{\longrightarrow}{\lim}^f(\alpha_2 \circ p_2)
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \underset{\longrightarrow}{\lim}^f \alpha_1
      &\underset{f}{\longrightarrow}&
    \underset{\longrightarrow}{\lim}^f \alpha_2
  }
  \,.
\end{displaymath}
\end{prop}
(e.g. \hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, prop. 6.1.13}, \hyperlink{ArtinMazur69}{Artin-Mazur 69, appendix 3, prop. (3.1), corollary (3.2)})

\begin{cor}
\label{}\hypertarget{}{}
For each $f \colon A_1 \longrightarrow A_2$ a [[morphism]] in $Ind(\mathcal{C})$, then there exists

\begin{enumerate}%
\item a [[small category|small]] [[filtered category]] $\mathcal{I}$;


\item [[functors]] $\alpha_i \;\colon\; \mathcal{I} \to \mathcal{C}$ ($i \in \{1,2\}$);


\item a [[natural transformation]] $\phi \colon \alpha_1 \longrightarrow \alpha_2$



\end{enumerate}
such that

\begin{displaymath}
\itexarray{
    A_1 &\overset{f}{\longrightarrow}& A_2
    \\
    {}^{\mathllap{\simeq}}\downarrow
      &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    \underset{\longrightarrow}{\lim}^f \alpha_1
      &\underset{\underset{\longrightarrow}{\lim}^f \phi }{\longrightarrow}&
    \underset{\longrightarrow}{\lim}^f \alpha_2
  }
  \,.
\end{displaymath}
\end{cor}
(e.g. \hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, corollary 6.1.14})

\begin{prop}
\label{}\hypertarget{}{}
The canonical inclusion $y \;\colon\; \mathcal{C} \hookrightarrow Ind(\mathcal{C})$ (factoring the [[Yoneda embedding]]) is [[exact functor|right exact]].

\end{prop}
(e.g. \hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, corollary 6.1.6})

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ have all [[finite limits]] or all [[small limits]]. Then also $Ind(\mathcal{C})$ has all finite or small limits, respectively, and the canonical functor $y \;\colon\; \mathcal{C} \longrightarrow Ind(\mathcal{C})$ preserves these, respectively.

\end{prop}
(e.g. \hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, corollary 6.1.17})

\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{C}$ has [[cokernels]], then so does $Ind(\mathcal{C})$.

If $\mathcal{C}$ has [[finite colimit]] [[coproducts]], then $Ind(\mathcal{C})$ has small [[coproducts]].

If $\mathcal{C}$ has all [[finite colimits]], then $Ind(\mathcal{C})$ has all small [[colimits]].

\end{prop}
(e.g. \hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, prop. 6.1.18})

\hypertarget{recognition_of_indobjects}{}\subsubsection*{{Recognition of Ind-objects}}\label{recognition_of_indobjects}

\begin{prop}
\label{}\hypertarget{}{}
A [[functor]] $F \colon \mathcal{C}^{op} \to Set$ is in $Ind(\mathcal{C})$ (i.e. is a [[filtered colimit]] of [[representable functor|representables]]) precisely if the [[comma category]] $(Y,const_F)$ (with $Y$ the [[Yoneda embedding]]) is [[filtered category|filtered]] and [[cofinally small category|cofinally small]].

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{C}$ admits finite [[colimits]], then $Ind(\mathcal{C})$ is the [[full subcategory]] of the [[presheaf]] category $PSh(\mathcal{C})$ consisting of those functors $F \colon \mathcal{C}^{op} \to Set$ such that $F$ is [[exact functor|left exact]] and the [[comma category]] $(Y,F)$ (with $Y$ the [[Yoneda embedding]]) is [[cofinally small category|cofinally small]].

\end{prop}
\hypertarget{functoriality}{}\subsubsection*{{Functoriality}}\label{functoriality}

Ind-cocompletion is functorial -- in fact an underlying 2-functor of a [[lax-idempotent 2-monad]] (KZ-monad). More in detail:

\begin{prop}
\label{FunctorialityOfInd}\hypertarget{FunctorialityOfInd}{}
Let $F \colon \mathcal{C}_1 \longrightarrow \mathcal{C}$ be a [[functor]]. Then there is a unique [[extension]] $Ind(F)$ of this functor to ind-objects, i.e. a [[commuting diagram]]

\begin{displaymath}
\itexarray{
    \mathcal{C}_1 &\overset{F}{\longrightarrow}& \mathcal{C}_2
    \\
    \downarrow
      &&
    \downarrow
    \\
    Ind(\mathcal{C}_1)
      &\underset{Ind(F)}{\longrightarrow}&
    Ind(\mathcal{C}_2)
  }
  \,,
\end{displaymath}
such that

\begin{displaymath}
Ind(F)( \underset{\longrightarrow}{\lim}^f \alpha )
  \simeq
  \underset{\longrightarrow}{\lim}^f ( F \circ \alpha ) 
  \,.
\end{displaymath}
Moreover,

\begin{enumerate}%
\item this construction respects [[composition]] in that

\begin{displaymath}
Ind(G \circ F ) \simeq Ind(G) \circ Ind(F)
\end{displaymath}

\item if $F$ is a [[faithful functor]] or [[fully faithful functor]], then so is $Ind(F)$, respectively.



\end{enumerate}
\end{prop}
(e.g. \hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, prop.6.1.9-6.1.11})

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two [[small categories]]. By prop. \ref{FunctorialityOfInd} the two [[projections]] out of their [[product category]] induce a functor of the form

\begin{displaymath}
Ind(\mathcal{C}_1 \times \mathcal{C}_2)
    \longrightarrow
  Ind(\mathcal{C}_1) \times Ind(\mathcal{C}_2)
  \,.
\end{displaymath}
This is an [[equivalence of categories]].

\end{prop}
(\hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, prop. 6.1.12})

\hypertarget{the_case_that__already_admits_filtered_colimits}{}\subsubsection*{{The case that $\mathcal{C}$ already admits filtered colimits}}\label{the_case_that__already_admits_filtered_colimits}

(\hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, chapter 6.3})

\begin{prop}
\label{ReflectionToYonedaEmbedding}\hypertarget{ReflectionToYonedaEmbedding}{}
Let $\mathcal{C}$ be a [[category]] which has all small [[filtered colimits]]. Then the canonical functor $\mathcal{C} \longrightarrow Ind(\mathcal{C})$ defines a [[reflective subcategory]], i.e. it is a [[fully faithful functor]] with a [[left adjoint]] $L$

\begin{displaymath}
\mathcal{C}
    \underoverset
     {\underset{}{\longrightarrow}}
     {\overset{L}{\longleftarrow}}
     {\bot}
  Ind(\mathcal{C})
\end{displaymath}
which takes formal filtered colimits to actual filtered colimits in $\mathcal{C}$:

\begin{displaymath}
L 
    \;\colon\; 
  \underset{\longrightarrow}{\lim}^f \alpha
    \mapsto
  \underset{\longrightarrow}{\lim} \alpha
\end{displaymath}
\end{prop}
(\hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, prop. 6.3.1})

\begin{prop}
\label{JFIsFullyFaithful}\hypertarget{JFIsFullyFaithful}{}
Let $F \colon \mathcal{J} \longrightarrow \mathcal{C}$ be a [[functor]] such that

\begin{enumerate}%
\item $F$ is a [[fully faithful functor]];


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


\item for every object $J \in \mathcal{J}$ its image $F(J) \in \mathcal{C}$ is [[compact object|compact]].



\end{enumerate}
Then the composite

\begin{displaymath}
Ind(\mathcal{J})
    \overset{Ind(F)}{\longrightarrow}
  Ind(\mathcal{C})
    \overset{L}{\longrightarrow}
  \mathcal{C}
\end{displaymath}
(with $Ind(F)$ from prop. \ref{FunctorialityOfInd} and $L$ from prop. \ref{ReflectionToYonedaEmbedding}) is a [[fully faithful functor]].

\end{prop}
(\hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, prop. 6.3.4})

\begin{prop}
\label{}\hypertarget{}{}
If $\mathcal{C}$ is a [[category]] such that

\begin{enumerate}%
\item every object of $\mathcal{C}$ is a [[filtered colimit]] of [[compact objects]];


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



\end{enumerate}
then the composite functor

\begin{displaymath}
Ind(\mathcal{C}_{cpt}) 
    \longrightarrow 
  Ind(\mathcal{C})
    \overset{L}{\longrightarrow}
  \mathcal{C}
\end{displaymath}
(from prop. \ref{JFIsFullyFaithful}, where $\mathcal{C}_{cpt} \hookrightarrow \mathcal{C}$ is the [[full subcategory]] of [[compact objects]]) is an [[equivalence of categories]].

\end{prop}
(\hyperlink{KashiwaraSchapira06}{KashiwaraSchapira 06, corollary 6.3.5})

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\begin{itemize}%
\item One important use of categories of ind-objects is in [[abelian sheaf]]-theory: for every [[small category|small]] [[abelian category]] $\mathcal{C}$ the category $ind\text{-}\mathcal{C}$ is a [[Grothendieck category]] and hence a good coefficient object for [[abelian sheaf cohomology]].

\end{itemize}
\hypertarget{in_higher_category_theory}{}\subsection*{{In higher category theory}}\label{in_higher_category_theory}

\hypertarget{in_categories}{}\subsubsection*{{In $(\infty,1)$-categories}}\label{in_categories}

There is a notion of [[ind-object in an (∞,1)-category]].

With regard to the third of the properties listed above, notice that the [[comma category]] $(Y,const_F)$ is the [[category of elements]] of $F$, i.e. the [[pullback]] of the [[generalized universal bundle|universal Set-bundle]] $U : Set_* \to Set$ along $F : \mathcal{C}^{op} \to Set$. This means that the [[stuff, structure, property|forgetful functor]] $(Y,const_F) \to \mathcal{C}$ is the fibration classified by $F$.

This is the starting point for the definition at [[ind-object in an (∞,1)-category]].

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

\begin{itemize}%
\item \textbf{ind-object} / [[ind-object in an (∞,1)-category]]


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


\item [[ind-pro-object]]


\item [[flat functor]]


\item [[KZ-monad]]


\item [[continuous category]]



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

Ind-categories were introduced in

\begin{itemize}%
\item [[Alexander Grothendieck]], [[Jean-Louis Verdier]] in [[SGA4]] Exp. 1 \href{http://sage.math.washington.edu/home/wstein/www/home/craigcitro/sga4/Grothendieck/SGA4/sga41.pdf}{pdf file}

\end{itemize}
and the dual notion of [[pro-object]] in

\begin{itemize}%
\item A. Grothendieck, \emph{Techniques de d\'e{}scente et th\'e{}or\`e{}mes d'existence en g\'e{}om\'e{}trie alg\'e{}brique, II: le th\'e{}or\`e{}me d'existence en th\'e{}orie formelle des modules}, Seminaire Bourbaki \textbf{195}, 1960, \href{http://archive.numdam.org/ARCHIVE/SB/SB_1958-1960__5_/SB_1958-1960__5__369_0/SB_1958-1960__5__369_0.pdf}{(pdf)}.

\end{itemize}
Ind-objects are discussed in

\begin{itemize}%
\item [[Michael Artin]], [[Barry Mazur]], appendix of \emph{\'E{}tale homotopy theory}, Lecture Notes in Maths. 100, Springer-Verlag, Berlin 1969.

\end{itemize}
(in their dual guise as [[pro-objects]])

\begin{itemize}%
\item [[Masaki Kashiwara]], [[Pierre Schapira]], section 6 of \emph{[[Categories and Sheaves]]} , Grundlehren der mathematischen Wissenschaften 332 (2006).

\end{itemize}
The relation between the Ind-completion and the ideal completion in order theory is discussed in section 1 of

\begin{itemize}%
\item [[Peter Johnstone]], [[André Joyal]], \emph{Continuous categories and exponentiable toposes} , JPAA \textbf{25} (1982) pp.255-296.

\end{itemize}
See also

\begin{itemize}%
\item [[Peter Johnstone]], section VI.1 of \emph{[[Stone Spaces]]}

\end{itemize}
They are discussed in relation to generalisations in

\begin{itemize}%
\item [[Jiří Adámek]], [[Francis Borceux]], [[Stephen Lack]], [[Jiri Rosicky|Ji\'i{} Rosický]], \emph{A classification of accessible categories,} Journal of Pure and Applied Algebra 175:7-30, 2002. \href{http://maths.mq.edu.au/~slack/papers/acc.html}{abstract}

\end{itemize}
See also the remarks at the beginning of section 5.3 of

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

\end{itemize}
[[!redirects ind-objects]] [[!redirects ind object]] [[!redirects ind-completion]] [[!redirects ind-cocompletion]]

[[!redirects Ind-objects]] [[!redirects Ind object]] [[!redirects Ind-completion]] [[!redirects Ind-cocompletion]]

[[!redirects category of ind-objects]] [[!redirects categories of ind-objects]]

[[!redirects category of Ind-objects]] [[!redirects categories of Ind-objects]]



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