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ind-object

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Idea

An ind-object of a category π’ž\mathcal{C} is a formal filtered colimit of objects of CC. Here β€œformal” means that the colimit is taken in the category of presheaves of CC (the free cocompletion of XX). The category of ind-objects of π’ž\mathcal{C} is written indind-π’ž\mathcal{C} or Ind(π’ž)Ind(\mathcal{C}).

Here, β€œind” is short for β€œinductive system”, as in the inductive systems used to define directed colimits, and as contrasted with β€œpro” in the dual notion of pro-object corresponding to β€œprojective system”.

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

c 1β†ͺc 2β†ͺc 3β†ͺc 4β†ͺβ‹― c_1 \hookrightarrow c_2 \hookrightarrow c_3 \hookrightarrow c_4 \hookrightarrow \cdots

of objects in π’ž\mathcal{C} are regarded to converge to an object in Ind(π’ž)Ind(\mathcal{C}), even if that object does not exist in π’ž\mathcal{C} itself. Standard examples where ind-objects are relevant are categories π’ž\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 β€œbig things in terms of small things” 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(π’ž)Ind(\mathcal{C}) in terms of that of CC below). Such large categories equivalent to ind-categories are therefore called accessible categories.

Definition

There are several equivalent ways to define ind-objects.

As diagrams

One definition is to define the objects of indind-CC to be diagrams F:D→CF:D\to C where DD 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 CC). We identify an ordinary object of CC with the corresponding diagram 1→C1\to C. To see what the morphisms should be between F:D→CF:D\to C and G:E→CG:E\to C, we stipulate that

  1. The embedding C→indC\to ind-CC should be full and faithful,
  2. each diagram F:D→CF:D\to C should be the colimit of itself (considered as a diagram in indind-CC via the above embedding), and
  3. the objects of CC should be compact in indind-CC.

Thus, we should have

ind-C(F,G) =ind-C(colim d∈DFd,colim e∈EGe) β‰…lim d∈Dind-C(Fd,colim e∈EGe) β‰…lim d∈Dcolim e∈Eind-C(Fd,Ge) β‰…lim d∈Dcolim e∈EC(Fd,Ge) \begin{aligned} ind\text{-}C(F,G) &= ind\text{-}C(colim_{d\in D} F d, colim_{e\in E} G e)\\ &\cong lim_{d\in D}\; ind\text{-}C(F d, colim_{e\in E} G e)\\ &\cong lim_{d\in D} colim_{e\in E}\; ind\text{-}C(F d, G e)\\ &\cong lim_{d\in D} colim_{e\in E}\; C(F d, G e) \end{aligned}

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

ind-C(F,G)≔lim d∈Dcolim e∈EC(Fd,Ge). ind\text{-}C(F,G) \coloneqq lim_{d\in D} colim_{e\in E}\; C(F d, G e) \,.

As filtered colimits of representable presheaves

Recall the co-Yoneda lemma that every presheaf X∈PSh(C)X \in PSh(C) is a colimit over representable presheaves:

there is a functor α:D→C\alpha : D \to C (with DD possibly large) such that

X≃colim d∈DY(Ξ±(d)). X \simeq colim_{d \in D} Y(\alpha(d)) \,.
Definition

Let Ind(π’ž)βŠ‚PSh(π’ž)Ind(\mathcal{C}) \subset PSh(\mathcal{C}) be the full subcategory of the presheaf category PSh(π’ž)=[π’ž op,Set]PSh(\mathcal{C}) = [\mathcal{C}^{op},Set] on those functors/presheaves which are filtered colimits of representables, i.e. those for which

X≃colim dβˆˆπ’ŸY(Ξ±(d)) X \simeq colim_{d \in \mathcal{D}} Y(\alpha(d))

for π’Ÿ\mathcal{D} some filtered category.

Those for which π’Ÿ\mathcal{D} may be chosen to be β„• ≀\mathbb{N}^{\leq}, i.e. those that arise as sequential colimits, are also called strict ind-objects.

Remark

The functors π’ž opβ†’Set\mathcal{C}^{op}\to Set belonging to Ind(π’ž)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 β€œwould preserve all finite colimits if π’ž\mathcal{C} had them”. In particular, if π’ž\mathcal{C} has finite colimits, then Ind(π’ž)Ind(\mathcal{C}) consists exactly of the finitely cocontinuous presheaves.

For more equivalent characterizations see at accessible category – Definition.

Remark

Given that [π’ž op,Set][\mathcal{C}^{op},Set] is the free cocompletion of π’ž\mathcal{C}, Ind(π’ž)Ind(\mathcal{C}) defined in this way is its β€œfree cocompletion under filtered colimits.”

To compare with the first definition, notice that indeed the formula for the hom-sets is reproduced:

Generally we have

[π’ž op,Set](X,Y) ≃[π’ž op,Set](colim dβˆˆπ’ŸYFd,colim dβ€²βˆˆπ’Ÿβ€²YGd) ≃lim dβˆˆπ’Ÿ[π’ž op,Set](YFd,colim dβ€²βˆˆπ’Ÿβ€²YGd) \begin{aligned} [\mathcal{C}^{op},Set](X,Y) & \simeq [\mathcal{C}^{op}, Set](colim_{d \in \mathcal{D}} Y F d, colim_{d' \in \mathcal{D}'} Y G d) \\ & \simeq lim_{d \in \mathcal{D}} [\mathcal{C}^{op}, Set]( Y F d, colim_{d' \in \mathcal{D}'} Y G d) \end{aligned}

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

⋯≃lim dβˆˆπ’Ÿ(colim dβ€²βˆˆπ’Ÿβ€²YGdβ€²)(Fd). \cdots \simeq lim_{d \in \mathcal{D}} (colim_{d' \in \mathcal{D}'} Y G d')(F d) \,.

Using that colimits in PSh(π’ž)PSh(\mathcal{C}) are computed objectwise (see again properties at colimit) this is

⋯≃lim d∈Dcolim dβ€²βˆˆπ’Ÿβ€²π’ž(Fd,Gdβ€²). \cdots \simeq lim_{d \in D} colim_{d' \in \mathcal{D}'} \mathcal{C}(F d, G d') \,.

Examples

  • Let FinVect be the category of finite-dimensional vector spaces (over some field). Let VV be an infinite-dimensional vector space. Then VV can be regarded as an object of indβˆ’FinVectind-FinVect as the colimit colim Vβ€²β†ͺVY(Vβ€²)colim_{V' \hookrightarrow V} Y(V') over the filtered category whose objects are inclusions Vβ€²β†ͺVV' \hookrightarrow V of finite dimensional vector spaces Vβ€²V' into VV of the representables Y(Vβ€²):FinVect opβ†’SetY(V') : FinVect^{op} \to Set (YY is the Yoneda embedding).

  • For CC the category of finitely presented objects of some equationally defined structure, ind-Cind\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.

      • The category Ab of abelian groups is the ind-category of the category of finitely generated abelian groups.
  • A formal scheme is an ind-object in schemes.

Properties

In the following we write lim⟢ f\underset{\longrightarrow}{\lim}^f for the β€œformal colimits” defining ind-objects. I.e. if Ξ±:β„β†’π’ž\alpha \colon \mathcal{I} \to \mathcal{C} is a small diagram and with i:Ind(π’ž)β†ͺPSh(π’ž)i \colon Ind(\mathcal{C}) \hookrightarrow PSh(\mathcal{C}) the defining inclusion, then

lim⟢ f(Ξ±)≔lim⟢(i∘α). \underset{\longrightarrow}{\lim}^f (\alpha) \;\coloneqq\; \underset{\longrightarrow}{\lim} (i \circ \alpha ) \,.

The category of ind-objects

Proposition

If π’ž\mathcal{C} is a locally small category then so is Ind(π’ž)Ind(\mathcal{C}).

Proposition

Ind(π’ž)Ind(\mathcal{C}) admits small filtered colimits and the defining inclusion Ind(π’ž)β†ͺPSh(π’ž)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.

Proposition

Let

  1. ℐ 1\mathcal{I}_1 and ℐ 2\mathcal{I}_2 be two small filtered categories;

  2. Ξ± 1:ℐ 1βŸΆπ’ž\alpha_1 \colon \mathcal{I}_1 \longrightarrow \mathcal{C} and Ξ± 2:ℐ 2βŸΆπ’ž\alpha_2 \colon \mathcal{I}_2 \longrightarrow \mathcal{C} be two functors;

  3. f:lim⟢ fα 1⟢lim⟢ fα 2f \;\colon\; \underset{\longrightarrow}{\lim}^f \alpha_1 \longrightarrow \underset{\longrightarrow}{\lim}^f \alpha_2

a morphism between their images in Ind(π’ž)Ind(\mathcal{C}).

Then there exists

  1. a small filtered category KK

  2. final functors p i:KβŸΆβ„ ip_i \colon K \longrightarrow \mathcal{I}_i

  3. a natural transformation Ο•:Ξ± 1∘p 1⟢α 2∘p 2\phi \;\colon\; \alpha_1 \circ p_1 \longrightarrow \alpha_2 \circ p_2

such that the following diagram commutes

lim⟢ f(Ξ± 1∘p 1) ⟢lim⟢ fΟ• lim⟢ f(Ξ± 2∘p 2) ≃↓ ↓ ≃ lim⟢ fΞ± 1 ⟢f lim⟢ fΞ± 2. \array{ \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 } \,.

(e.g. KashiwaraSchapira 06, prop. 6.1.13, Artin-Mazur 69, appendix 3, prop. (3.1), corollary (3.2))

Corollary

For each f:A 1⟢A 2f \colon A_1 \longrightarrow A_2 a morphism in Ind(π’ž)Ind(\mathcal{C}), then there exists

  1. a small filtered category ℐ\mathcal{I};

  2. functors lapha i:β„β†’π’ž\lapha_i \;\colon\; \mathcal{I} \to \mathcal{C} (i∈{1,2}i \in \{1,2\});

  3. a natural transformation Ο•:Ξ± 1⟢α 2\phi \colon \alpha_1 \longrightarrow \alpha_2

such that

A 1 ⟢f A 2 ≃↓ ↓ ≃ lim⟢ fΞ± 1 ⟢lim⟢ fΟ• lim⟢ fΞ± 2. \array{ 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 } \,.

(e.g. KashiwaraSchapira 06, corollary 6.1.14)

Proposition

The canonical inclusion y:π’žβ†ͺInd(π’ž)y \;\colon\; \mathcal{C} \hookrightarrow Ind(\mathcal{C}) (factoring the Yoneda embedding) is right exact.

(e.g. KashiwaraSchapira 06, corollary 6.1.6)

Proposition

Let π’ž\mathcal{C} have all finite limits or all small limits. Then also Ind(π’ž)Ind(\mathcal{C}) has all finite or small limits, respectively, and the canonical functor y:π’žβŸΆInd(π’ž)y \;\colon\; \mathcal{C} \longrightarrow Ind(\mathcal{C}) preserves these, respectively.

(e.g. KashiwaraSchapira 06, corollary 6.1.17)

Proposition

If π’ž\mathcal{C} has cokernels, then so does Ind(π’ž)Ind(\mathcal{C}).

If π’ž\mathcal{C} has finite colimit coproducts, then Ind(π’ž)Ind(\mathcal{C}) has small coproducts.

If π’ž\mathcal{C} has all finite colimits, then Ind(π’ž)Ind(\mathcal{C}) has all small colimits.

(e.g. KashiwaraSchapira 06, prop. 6.1.18)

Recognition of Ind-objects

Proposition

A functor F:π’ž opβ†’SetF \colon \mathcal{C}^{op} \to Set is in Ind(π’ž)Ind(\mathcal{C}) (i.e. is a filtered colimit of representables) precisely if the comma category (Y,const F)(Y,const_F) (with YY the Yoneda embedding) is filtered and cofinally small.

Proposition

If π’ž\mathcal{C} admits finite colimits, then Ind(π’ž)Ind(\mathcal{C}) is the full subcategory of the presheaf category PSh(π’ž)PSh(\mathcal{C}) consisting of those functors F:π’ž opβ†’SetF \colon \mathcal{C}^{op} \to Set such that FF is left exact and the comma category (Y,F)(Y,F) (with YY the Yoneda embedding) is cofinally small.

Functoriality

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

Proposition

Let F:π’ž 1βŸΆπ’žF \colon \mathcal{C}_1 \longrightarrow \mathcal{C} be a functor. Then there is a unique extension Ind(F)Ind(F) of this functor to ind-objects, i.e. a commuting diagram

π’ž 1 ⟢F π’ž 2 ↓ ↓ Ind(π’ž 1) ⟢Ind(F) Ind(π’ž 2), \array{ \mathcal{C}_1 &\overset{F}{\longrightarrow}& \mathcal{C}_2 \\ \downarrow && \downarrow \\ Ind(\mathcal{C}_1) &\underset{Ind(F)}{\longrightarrow}& Ind(\mathcal{C}_2) } \,,

such that

Ind(F)(lim⟢ fΞ±)≃lim⟢ f(F∘α). Ind(F)( \underset{\longrightarrow}{\lim}^f \alpha ) \simeq \underset{\longrightarrow}{\lim}^f ( F \circ \alpha ) \,.

Moreover,

  1. this construction respects composition in that

    Inf(G∘F)≃Ind(G)∘Ind(F) Inf(G \circ F ) \simeq Ind(G) \circ Ind(F)
  2. if FF is a faithful functor or fully faithful functor, then so is Ind(F)Ind(F), respectively.

(e.g. KashiwaraSchapira 06, prop.6.1.9-6.1.11)

Proposition

Let π’ž 1\mathcal{C}_1 and π’ž 2\mathcal{C}_2 be two small categories. By prop. 9 the two projections out of their product category induce a functor of the form

Ind(π’ž 1Γ—π’ž 2)⟢Ind(π’ž 1)Γ—Ind(π’ž 2). Ind(\mathcal{C}_1 \times \mathcal{C}_2) \longrightarrow Ind(\mathcal{C}_1) \times Ind(\mathcal{C}_2) \,.

This is an equivalence of categories.

(KashiwaraSchapira 06, prop. 6.1.12)

The case that π’ž\mathcal{C} already admits filtered colimits

(KashiwaraSchapira 06, chapter 6.3)

Proposition

Let π’ž\mathcal{C} be a category which has all small filtered colimits. Then the canonical functor π’žβŸΆInd(π’ž)\mathcal{C} \longrightarrow Ind(\mathcal{C}) defines a reflective subcategory, i.e. it is a fully faithful functor with a left adjoint LL

π’žβŠ₯⟢⟡LInd(π’ž) \mathcal{C} \underoverset {\underset{}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} Ind(\mathcal{C})

which takes formal filtered colimits to actual filtered colimits in π’ž\mathcal{C}:

L:lim⟢ fα↦lim⟢α L \;\colon\; \underset{\longrightarrow}{\lim}^f \alpha \mapsto \underset{\longrightarrow}{\lim} \alpha

(KashiwaraSchapira 06, prop. 6.3.1)

Proposition

Let F:π’₯βŸΆπ’žF \colon \mathcal{J} \longrightarrow \mathcal{C} be a functor such that

  1. FF is a fully faithful functor;

  2. π’ž\mathcal{C} has all small filtered colimits;

  3. for every object J∈π’₯J \in \mathcal{J} its image F(J)βˆˆπ’žF(J) \in \mathcal{C} is compact.

Then the composite

Ind(π’₯)⟢Ind(F)Ind(π’ž)⟢Lπ’ž Ind(\mathcal{J}) \overset{Ind(F)}{\longrightarrow} Ind(\mathcal{C}) \overset{L}{\longrightarrow} \mathcal{C}

(with Ind(F)Ind(F) from prop. 9 and LL from prop. 11) is a fully faithful functor.

(KashiwaraSchapira 06, prop. 6.3.4)

Proposition

If π’ž\mathcal{C} is a category such that

  1. every object of π’ž\mathcal{C} is a filtered colimit of compact objects;

  2. π’ž\mathcal{C} has all small filtered colimits

then the composite functor

Ind(π’ž cpt)⟢Ind(π’ž)⟢Lπ’ž Ind(\mathcal{C}_{cpt}) \longrightarrow Ind(\mathcal{C}) \overset{L}{\longrightarrow} \mathcal{C}

(from prop. 12, where π’ž cptβ†ͺπ’ž\mathcal{C}_{cpt} \hookrightarrow \mathcal{C} is the full subcategory of compact objects) is an equivalence of categories.

(KashiwaraSchapira 06, corollary 6.3.5)

Applications

In higher category theory

In (∞,1)(\infty,1)-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)(Y,const_F) is the category of elements of FF, i.e. the pullback of the universal Set-bundle U:Set *→SetU : Set_* \to Set along F:C op→SetF : C^{op} \to Set. This means that the forgetful functor (Y,const F)→C (Y,const_F) \to C is the fibration classified by FF.

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

References

Ind-categories were introduced in

and the dual notion of pro-object in

  • A. Grothendieck, Techniques de dΓ©scente et thΓ©orΓ¨mes d’existence en gΓ©omΓ©trie algΓ©brique, II: le thΓ©orΓ¨me d’existence en thΓ©orie formelle des modules, Seminaire Bourbaki 195, 1960, (pdf).

Ind-objects are discussed in

  • Michael Artin, Barry Mazur, appendix of Γ‰tale homotopy theory, Lecture Notes in Maths. 100, Springer-Verlag, Berlin 1969.

(in their dual guise as pro-objects)

They are discussed in relation to generalisations in

See also the remarks at the beginning of section 5.3 of

Revised on August 31, 2016 04:18:10 by Urs Schreiber (82.113.106.227)