An ind-object of a category $\mathcal{C}$ is a formal filtered colimit of objects of $C$. Here βformalβ means that the colimit is taken in the category of presheaves of $C$ (the free cocompletion of $X$). The category of ind-objects of $\mathcal{C}$ is written $ind$-$\mathcal{C}$ or $Ind(\mathcal{C})$.
Here, βindβ is short for βinductive systemβ, as in the inductive systems used to define directed colimits, and as contrasted with βproβ in the dual notion of pro-object corresponding to βprojective systemβ.
Recalling the nature of filtered colimits, this means that in particular chains of inclusions
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 βbig things in terms of small thingsβ also in another important sense: many large categories are actually (equivalent to) ind-categories of small categories. This means that, while 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 $C$ below). Such large categories equivalent to ind-categories are therefore called accessible categories.
There are several equivalent ways to define ind-objects.
One definition is to define the objects of $ind$-$C$ to be diagrams $F:D\to C$ where $D$ is a small filtered category.
The idea is to think of these diagrams as being the placeholder for the colimit over them (possibly non-existent in $C$). We identify an ordinary object of $C$ with the corresponding diagram $1\to C$. To see what the morphisms should be between $F:D\to C$ and $G:E\to C$, we stipulate that
Thus, we should have
Here
the first step is by assumption that each object is a suitable colimit;
the second by the fact that the contravariant Hom sends colimits to limits (see properties of colimit);
the third by the assumption that each object is a compact object;
the last by the assumption that the embedding is a full and faithful functor.
So then one defines
Recall the co-Yoneda lemma that every presheaf $X \in PSh(C)$ is a colimit over representable presheaves:
there is a functor $\alpha : D \to C$ (with $D$ possibly large) such that
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 representables, i.e. those for which
for $\mathcal{D}$ some filtered category.
Those for which $\mathcal{D}$ may be chosen to be $\mathbb{N}^{\leq}$, i.e. those that arise as sequential colimits, are also called strict ind-objects.
The functors $\mathcal{C}^{op}\to Set$ belonging to $Ind(\mathcal{C})$ under this definition β those which are filtered colimits of representables β have an equivalent characterization as the flat functors: those which βwould preserve all finite colimits if $\mathcal{C}$ had themβ. 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 accessible category β Definition.
Given that $[\mathcal{C}^{op},Set]$ is the free cocompletion of $\mathcal{C}$, $Ind(\mathcal{C})$ defined in this way is its βfree cocompletion under filtered colimits.β
To compare with the first definition, notice that indeed the formula for the hom-sets is reproduced:
Generally we have
by the fact that the hom-functor sends colimits to limits in its first argument (see properties at colimit).
By the Yoneda lemma this is
Using that colimits in $PSh(\mathcal{C})$ are computed objectwise (see again properties at colimit) this is
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).
For $C$ the category of finitely presented objects of some equationally defined structure, $ind\text{-}C$ is the category of all these structures.
The category Grp of groups is the ind-category of the category of finitely generated groups.
A formal scheme is an ind-object in schemes.
In the following we write $\underset{\longrightarrow}{\lim}^f$ for the βformal colimitsβ 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
If $\mathcal{C}$ is a locally small category then so is $Ind(\mathcal{C})$.
$Ind(\mathcal{C})$ admits small filtered colimits and the defining inclusion $Ind(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})$ commutes with these colimits.
(e.g. 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.
Let
$\mathcal{I}_1$ and $\mathcal{I}_2$ be two small filtered categories;
$\alpha_1 \colon \mathcal{I}_1 \longrightarrow \mathcal{C}$ and $\alpha_2 \colon \mathcal{I}_2 \longrightarrow \mathcal{C}$ be two functors;
$f \;\colon\; \underset{\longrightarrow}{\lim}^f \alpha_1 \longrightarrow \underset{\longrightarrow}{\lim}^f \alpha_2$
a morphism between their images in $Ind(\mathcal{C})$.
Then there exists
a small filtered category $K$
final functors $p_i \colon K \longrightarrow \mathcal{I}_i$
a natural transformation $\phi \;\colon\; \alpha_1 \circ p_1 \longrightarrow \alpha_2 \circ p_2$
such that the following diagram commutes
(e.g. KashiwaraSchapira 06, prop. 6.1.13, Artin-Mazur 69, appendix 3, prop. (3.1), corollary (3.2))
For each $f \colon A_1 \longrightarrow A_2$ a morphism in $Ind(\mathcal{C})$, then there exists
a small filtered category $\mathcal{I}$;
functors $\lapha_i \;\colon\; \mathcal{I} \to \mathcal{C}$ ($i \in \{1,2\}$);
a natural transformation $\phi \colon \alpha_1 \longrightarrow \alpha_2$
such that
(e.g. KashiwaraSchapira 06, corollary 6.1.14)
The canonical inclusion $y \;\colon\; \mathcal{C} \hookrightarrow Ind(\mathcal{C})$ (factoring the Yoneda embedding) is right exact.
(e.g. KashiwaraSchapira 06, corollary 6.1.6)
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.
(e.g. KashiwaraSchapira 06, corollary 6.1.17)
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.
(e.g. KashiwaraSchapira 06, prop. 6.1.18)
A functor $F \colon \mathcal{C}^{op} \to Set$ is in $Ind(\mathcal{C})$ (i.e. is a filtered colimit of representables) precisely if the comma category $(Y,const_F)$ (with $Y$ the Yoneda embedding) is filtered and cofinally small.
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 left exact and the comma category $(Y,F)$ (with $Y$ the Yoneda embedding) is cofinally small.
Ind-cocompletion is functorial β in fact an underlying 2-functor of a lax-idempotent 2-monad (KZ-monad). More in detail:
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
such that
Moreover,
this construction respects composition in that
if $F$ is a faithful functor or fully faithful functor, then so is $Ind(F)$, respectively.
(e.g. KashiwaraSchapira 06, prop.6.1.9-6.1.11)
Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two small categories. By prop. 9 the two projections out of their product category induce a functor of the form
This is an equivalence of categories.
(KashiwaraSchapira 06, prop. 6.1.12)
(KashiwaraSchapira 06, chapter 6.3)
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 $L$
which takes formal filtered colimits to actual filtered colimits in $\mathcal{C}$:
(KashiwaraSchapira 06, prop. 6.3.1)
Let $F \colon \mathcal{J} \longrightarrow \mathcal{C}$ be a functor such that
$F$ is a fully faithful functor;
$\mathcal{C}$ has all small filtered colimits;
for every object $J \in \mathcal{J}$ its image $F(J) \in \mathcal{C}$ is compact.
Then the composite
(with $Ind(F)$ from prop. 9 and $L$ from prop. 11) is a fully faithful functor.
(KashiwaraSchapira 06, prop. 6.3.4)
If $\mathcal{C}$ is a category such that
every object of $\mathcal{C}$ is a filtered colimit of compact objects;
$\mathcal{C}$ has all small filtered colimits
then the composite functor
(from prop. 12, where $\mathcal{C}_{cpt} \hookrightarrow \mathcal{C}$ is the full subcategory of compact objects) is an equivalence of categories.
(KashiwaraSchapira 06, corollary 6.3.5)
There is a notion of ind-object in an (β,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 universal Set-bundle $U : Set_* \to Set$ along $F : C^{op} \to Set$. This means that the forgetful functor $(Y,const_F) \to C$ is the fibration classified by $F$.
This is the starting point for the definition at ind-object in an (β,1)-category.
ind-object / ind-object in an (β,1)-category
Ind-categories were introduced in
and the dual notion of pro-object in
Ind-objects are discussed in
(in their dual guise as pro-objects)
Masaki Kashiwara, Pierre Schapira, section 6 of Categories and Sheaves , Grundlehren der mathematischen Wissenschaften 332 (2006)
Peter Johnstone, section VI.1 of Stone Spaces
They are discussed in relation to generalisations in
See also the remarks at the beginning of section 5.3 of