An ind-object of a category $\mathcal{C}$ is a formal filtered colimit of objects of $\mathcal{C}$. Here βformalβ 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, 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 $\mathcal{C}$ 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 $ind$-$\mathcal{C}$ to be diagrams $F:D\to \mathcal{C}$ 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 $\mathcal{C}$). 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
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 $X \in PSh(\mathcal{C})$ is a colimit over representable presheaves:
there is a functor $\alpha : D \to \mathcal{C}$ (with $D$ possibly large) such that
(with $Y$ the Yoneda embedding).
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 $D$ 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 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 FinDimVect 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 $ind-FinDimVect$ 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 $Y(V') : FinDimVect^{op} \to Set$ ($Y$ is the Yoneda embedding).
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.
The category Grp of groups is the ind-category of the category of finitely generated groups.
The category Set of sets is the ind-category of the category of finitely indexed sets.
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$\alpha_i \;\colon\; \mathcal{I} \to \mathcal{C}$ ($i \in \{1,2\}$);
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. 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)
Let $\mathcal{C}$ be a small category that has finite colimits and let $\mathcal{I}$ be a small category. Then the canonical functor
is an equivalence of categories.
(Henry 23)
(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. and $L$ from prop. ) 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. , 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 $F : \mathcal{C}^{op} \to Set$. This means that the 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 (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)
The relation between the Ind-completion and the ideal completion in order theory is discussed in section 1 of
See also
They are discussed in relation to generalisations in
See also the remarks at the beginning of section 5.3 of
Some relevant results are contained in:
Last revised on January 4, 2024 at 00:13:22. See the history of this page for a list of all contributions to it.