A Categorical Construction of Minimal Model

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MATTER: International Journal of Science and Technology

ISSN 2454-5880

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48

A. Behera et al.

Special Issue, 2015, pp. 48-63

A CATEGORICAL CONSTRUCTION OF MINIMAL MODELA. Behera

Department of Mathematics, National Institute of Technology, Rourkela - 769008 (India),[email protected]

S. B. Choudhury

Department of Mathematics, National Institute of Technology, Rourkela - 769008 (India),

[email protected]

M. Routaray

Department of Mathematics, National Institute of Technology, Rourkela - 769008 (India),[email protected]

Abstract

Deleanu, Frei and Hilton have developed the notion of generalized Adams completion in a

categorical context; they have also suggested the dual notion, namely, Adams cocompletion of

an object in a category. The concept of rational homotopy theory was first characterized by

Quillen. In fact in rational homotopy theory Sullivan introduced the concept of minimal model.

In this note under a reasonable assumption, the minimal model of a 1-connected differential

graded algebra can be expressed as the Adams cocompletion of the differential graded algebra

with respect to a chosen set in the category of 1-connected differential graded algebras (in short

d.g.a.’s) over ℎ and d.g.a.-homomorphisms.

Keywords

Category of Fractions, Calculus of Right Fractions, Grothendieck Universe,

Adamscocompletion, Differential Graded Algebra, Minimal Model.

1.

Introduction

It is to be emphasized that many algebraic and geometrical constructions in Algebraic

Topology, Differential Topology, Differentiable Manifolds, Algebra, Analysis, Topology, etc.,

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can be viewed as Adams completions or cocompletions of objects in suitable categories, with

respect to carefully chosen sets of morphisms.

The notion of generalized completion (Adams completion) arose from a categorical

completion process suggested by Adams, 1973, 1975. Originally this was considered for

admissible categories and generalized homology (or cohomology) theories. Subsequently, this

notion has been considered in a more general framework by Deleanu, Frei & Hilton, 1974, where

an arbitrary category and an arbitrary set of morphisms of the category are considered; moreover

they have also suggested the dual notion, namely the cocompletion (Adams cocompletion) of an

object in a category.

The central idea of this note is to investigate a case showing how an algebraic

geometrical construction is characterized in terms of Adams cocompletion.

2. Adams completion

We recall the definitions of Grothendieck universe, category of fractions, calculus of

right fractions, Adams cocompletion and some characterizations of Adams cocompletion.

2.1 Definition.Schubert, 1972

A Grothendeick universe (or simply universe) is a collection of sets such that the

following axioms are satisfied:

U (1): If {: ∈ } is a family of sets belongingto , then⋃ ∈I is an element of .

U (2): If ∈ , then {} ∈ .

U (3): If ∈ and ∈ then ∈ .

U (4): If is a set belonging to, then , the power set of , is an element of .

U (5): If and are elements of , then { , }, the ordered pair , and ×

are elements of .

We fix a universe that contains ℕ the set of natural numbers (and so ℤ, ℚ, ℝ, ℂ .

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A category is said to be a small -category, being a fixed Grothendeick universe, if

the following conditions hold:

S(1) : The objects of form a set which is an element of .

S (2): For each pair , of objects of , the set Hom, is an element of .


Let be any arbitrary category and a set of morphisms of . A category of

fractions of with respect to is a category denoted by [−] together with a

functor ∶ → [−]having the following properties:

CF (1): For each ∈ , is an isomorphism in[ −].

CF(2) : is universal with respect to this property:If ∶ → is a functor such

thatis an isomorphism in, foreach ∈ ,then there exists a unique

functor ∶ [−] → such that = . Thus we have the following

commutative diagram:

→ [−]

↓ ↙

F igure 1

2.4 Note.

For the explicit construction of the category [ −], we refer to Schubert, 1972. We

content ourselves merely with the observation that the objects of [ −] are same as those of

and in the case when admits a calculus of left (right) fractions, the category [ −] can

be described very nicely Gabriel &Zisman, 1967, Schubert, 1972.

2.5 Definition. Schubert, 1972

A family of morphisms in a category is saidto admit a calculus of right fractions if

(a) any diagram



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↓

→

F igure 2

inwith ∈ can be completed to a diagram

→

↓ ↓

→

F igure 3

with ∈ and = ,

(b) given

→ →

→

→ with ∈ and = ,there is a morphism ∶

→ in such that = .

A simple characterization for a family to admit a calculus of right fractions is the

following.

2.6 Theorem. Deleanu, et al., 1974

Let be a closed family of morphisms of satisfying

(a) if ∈ and ∈ , then ∈ ,

(b) any diagram

⦁

↓

⦁

→ ⦁

F igure 4

in with ∈ , can be embedded in a weak pull-back diagram



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⦁

→ ⦁

↓ ↓

⦁

→ ⦁

F igure 5

with ∈ .

Then admits a calculus of right fractions.

2.7 Remark.

There are some set-theoretic difficulties in constructing the category[−]; these

difficulties may be overcome by making some mild hypotheses and using Grothendeick

universes. Precisely speaking, the main logical difficulty involved in the construction of a

category of fractions and its use, arises from the fact that if the category belongs to a

particular universe, the category[−] would, in general belongs to a higher universe Schubert,

1972. In most applications, however, it is necessary that we remain within the given initial

universe. This logical difficulty can be overcome by making some kind of assumptions which

would ensure that the category of fractions remains within the same universe Deleanu, 1975.

Also the following theorem shows that if admits a calculus of left (right) fractions, then the

category of fractions [−] remains within the same universe as to the universe to which the

category belongs.

2.8 Theorem. Nanda, 1980

Let be a small -category and a set of morphisms of that admits a calculus

of left (right) fractions. Then [−] is a small -category.

2.9 Definition. Deleanu, et al., 1974

Let be an arbitrary category anda set of morphisms of . Let [−] denote the

category of fractions of with respectand : → [−] be the canonical functor. Let

denote the category of sets and functions. Then for a given object of , [−], ∶ →



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defines a covariant functor. If this functor is representable by an object of ,

i.e.,[−], ≅ , .Then is called the ( generalized ) Adams cocompletionof with

respect to the set of morphisms or simply the -cocompletion of . We shall often refer to

as the cocompletion of Deleanu, et al., 1974.

We recall some results on the existence of Adams cocompletion. We state

Deleanu’stheoremDeleanu, 1975 that under certain conditions, global Adams cocompletion of an

object always exists.

2.10 Theorem. Deleanu, 1975

Let be a complete small -category (is a fixed Grothendeick universe) and a set of

morphisms of that admits a calculus of right fractions. Suppose that the following compatibility

condition with product is satisfied : if each ∶ → , ∈ , is an element of , then

∏ ∈ ∶ ∏ →∈ ∏ ∈ is an element of .Then every object of has an Adams

cocompletion with respect to the set of morphisms.

The concept of Adams cocompletion can be characterized in terms of a couniversal

property.

2.11 Definition. Deleanu, et al., 1974

Given aset of morphisms of , we define ,̅ the saturation of as the set of all

morphisms in such that is an isomorphism in [−]. is said to be saturated if

= .̅

2.12 Proposition. Deleanu, et al., 1974

A family of morphisms of is saturated if and only if thereexists a factor ∶ →

such that

is the collection of morphisms

such that

is invertible.

Deleanu, Frei and Hilton have shown that if the set of morphisms is saturated then the

Adams cocompletion of a space is characterized by a certain couniversal property.



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Let be a saturated family of morphisms of admitting a calculus of right fractions.

Then an object of is the -cocompletion of the object with respect to if and only

if there exists a morphism ∶

→ in which is couniversal with respect to morphisms of

: given a morphism ∶ → in there exists a unique morphism ∶ → in such

that = . In other words, the following diagram is commutative:

→

↓ ↗

Figure 6

For most of the application it is essential that the morphism ∶ → has to be in

; this is the case when is saturated and the result is as follows:


Let be a saturated family of morphisms of and let every object of admit an

-cocompletion. Then the morphism ∶ → belongs to and is universal for

morphisms to -cocomplete objects and couniversal for morphisms in .

3. The category

Let be the category of 1-connected differential graded algebras over ℚ (in short

d.g.a.) and d.g.a.-homomorphisms. Let denote the set of all d.g.a.-epimorphisms in

which induce homology isomorphisms in all dimensions. The following results will be

required in the sequel.

3.1 Proposition.

Is saturated.

Proof.The proof is evident from Proposition 2.12. ∎



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3.2Proposition.

admits a calculus of right fractions.

Proof.Clearly, is a closed family of morphisms of the category . We shall verify

conditions () and () of Theorem 2.6. Let, ∈ . We show that if ∈ and ∈ ,

then ∈ . Clearly is an epimorphism.We have∗ = ∗∗and ∗ are both homology

isomorphisms implying∗ is a homology isomorphism. Thus ∈ . Hence condition () of

Theorem 2.6 holds.

To prove condition () of Theorem 2.6 consider the diagram

↓

→

Figure 7

In with ∈ .We assert that the above diagram can be completed to a weak pull-back

diagram

→

↓ ↓

→

Figure 8

Inwith ∈ . Since , andare in we write = Σ≥ , = Σ≥ , =

Σ≥ , = Σ≥ , = Σ≥ and ∶ → , ∶ → ,are d.g.a.-homomorphisms.

Let = {, ∈ × ∶ = } ⊂ × .

We have to show that = Σ≥ is a differential graded algebra. Let ∶ → and ∶ → be the usual projections. Let = Σ≥ and = Σ≥ . Clearly the

above diagram is commutative. It is required to show that is a d.g.a..Define a multiplication

in in the following way: , ∙ ′, ′ = ′, ′,where, ∈ , ′, ′ ∈ .Let

= Σ≥ , ∶ → +and = Σ≥ , ∶ → +.Define ∶ →



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+ by the rule , = , ,

, ∈ .Let = Σ≥ .Since, = (, ) = 0,0for all

, ∈ we have that is a differential. Next we show that is a derivation: For

, ∈ and2, 2 ∈ ,, ∙ 2, 2 = 2, 2 = 2, 2 =

∙ 2 1 ∙ 2, ∙ 2 1 ∙ 2 = ∙ 2 ,

∙ 2 1 ∙ 2, 1 ∙ 2 = ,

∙ 2, 2 (1, 1 ∙ 2, 2 = , ∙ 2, 2

1, ∙ 2, 2.

Thus becomes a d.g.a..

We show that is 1-connected, i.e., ≅ ℚand ≅ 0. We have =

⁄ = = {, ∈ × ∶ = }.Let 1 ∈ and

1 ∈ . Then1, 1 = 1, 1 = 0implies that1, 1 ∈ . =

≅ ℚimplies that = ℚ1.Similarly, = ≅ ℚimplies that =

ℚ1 .Thus , ∈ = ⊂ × if and only if = 1 and = 1 for

some ∈ ℚ. Thus ≅ ℚ.

In order to show ≅ 0, let, ∈ . This implies that ∈ , ∈

and = . Sinc is 1-connected we have ≅ 0, i.e., = ⁄ ;

hence = ′, ′ ∈ . Similarly sinceis 1-connected we have ≅ 0,

i.e., = ⁄ ;hence = ′, ′ ∈ . Now = ,i.e., ′ =

′.This gives ′ = ′,i.e.,′ ′ = 0.Thus ′

′ ∈ .But ∈ . Hence ∗ ∶ → is an isomorphism, i.e., ∶

→ is an isomorphism. Hence there exists an element ̃ ∈ such that

̃ = ′ ′.Moreover ′, ̃ ′ = ′, ̃

′)= ′,

0 ′)= ′,

′ = , showing thata,c ∈ . Thus ≅ 0.

Clearly is a d.g.a.-epimorphism. We show that∗ ∶ ∗ → ∗ is an

isomorphism. First we show that ∗ ∶ ∗ → ∗ is a monomorphism. The hollowing

commutative diagram will be used in the sequel.



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⋮ ⋮ ⋮↓ ↓ ↓

−2 −2

−2

−2

↓ −2

↓ ↓ −2

− −

−

− ↓ − ↓ ↓ −

→

←

↓ ↓ ↓⋮ ⋮ ⋮

Figure 9

Since ∶ → is the usual projection, we have , = for every , ∈ .

Hence the algebra homomorphism ∗ ∶ → is given by ∗[, ] = [, ] =

[]for [, ] ∈ . We note that =

⁄ ⊂ × × ⁄ .Hence

= ̅ × ̅ ̅ × ̅⁄

for some ̅ ⊂ and ̅ ⊂ . For any [, ] ∈ we have [, ] = ,

= , ̅ × ̅,, ∈ ⊂ . Then, − ′, ′ ∈

, , for every − ′, ′ ∈ where ′, ′ ∈ − ⊂ , i.e., ,

− ′, ′ = , − ′, − ′ ∈ , ̅ × ̅, for

every− ′, ′ = − ′, − ′ ∈ ̅ × ̅. Thus − ′,

− ′ ∈ , ̅ × ̅, i.e.,[ − ′, −

′]= [, ] ∈ .

We note that [] = [ − ′ ]and[] = [ − ′].

Now let[, ], [2, 2] ∈ and assume that ∗[, ] = ∗[2, 2]; this

gives[] = [2], i.e. [ − ′] = [2 − ′].

Since, , 2, 2, − ′, − ′ ∈ , we have = ), 2 = 2)

and − ′ = − ′). So − ′ = −

′) and 2

− ′ = 2 − ′). Therefore, from the above,∗[, ] = ∗[2, 2]

gives ∗[ − ′] = ∗[ − ′], i.e.,[ − ′ ] = [ 2



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− ′];this gives [ − ′] = [2 −

′], i.e., ∗[ − ′] =

∗[2 − ′]. Since ∗ is an isomorphism we have [ −

′] = [2 − ′].

Hence we have[ − ′], [ − ′]= [2 − ′], [2

−

′

];we apply the

isomorphism∗ ∶ ̅ ̅⁄ × ̅ ̅⁄ → ̅ × ̅ ̅ × ̅⁄ to

the above to get∗[ − ′], [ − ′] = ∗[2 − ′], [2

− ′], i.e.,[ − ′, −

′] = [2 − ′, 2 − ′]. Thus

[, ] = [2, 2], showing that ∗ ∶ ∗ → ∗ is a monomorphism.

Next we show that ∗ ∶ ∗ → ∗ is anepimorphism.Let[] ∈ ∗ be

arbitrary. Then ∈ . Since is an epimorphism = for some ∈ .

Hence , ∈ . Then∗[, ] = [, ] = []showing ∗ is an epimorphism. Since

is an epimorphism and ∗ is an isomorphism we conclude that ∈ .

Next for any d.g.a. = Σ≥

and d.g.a.-homomorphisms = { ∶ → }and =

{ ∶ → }in, let the following diagram

→

↓ ↓

→

Figure 10

commute, i.e., = . Consider the diagram

↘

↘ ℎ

→

↘ ↓ ↓

→

Figure 11

Define ℎ = { ℎ ∶ → } by the ruleℎ = (, )for ∈ . Clearly ℎ is

well defined and is a d.g.a. homomorhism. Now for any ∈ ,ℎ = (, ) =



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andℎ = (, ) = , i.e.,ℎ = and ℎ = . This completes the proof of

Proposition 3.2.∎

3.3 Proposition.

If each ∶ → , ∈ ,is an element of , where the index set is an element of ,

then ∏ ∈ ∶ ∏ →∈ ∏ ∈ is an element of .

Proof.The proof is trivial. ∎

The following result can be obtained from the above discussion.

3.4 Proposition. The categoryis complete.

From Propositions 3.1- 3.4, it follows that the conditions of Theorem 2.10 are fulfilled

and by the use of Theorem 2.13, we obtain the following result.

3.5 Theorem.

Every object of the category has an Adams cocompletion with respect to the

set of morphisms and there exists a morphism ∶ → in which is couniversal withrespect to the morphisms in, that is, given a morphism ∶ → in there exists a unique

morphismt ∶ AS → Bsuch that s t = e. In other words the following diagram is commutative:

→

↓ ↗

Figure 12

3.

Minimal model

We recall the following algebraic preliminaries.



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4.1 Definition.Deschner, 1975, Wu, 1980

A d.g.a. is called a minimal algebra if it satisfies the following properties:

is free as a graded algebra.

has decomposable differentials.

= ℚ, = 0.

has homology of finite type, i.e., for each, is a finite dimensionalvector

space.

Let ℳ be the full subcategory of the category consisting of all minimal algebras

and all d.g.a.-maps between them.

4.2 Definition. Deschner, 1975, Wu, 1980

Let be a simply connected d.g.a..A d.g.a. = is called a minimal model of if the

following conditions hold:

(i) ∈ ℳ.

(ii) Thereis a d.g.a.-map ∶ → which induces an isomorphism on homology,

i.e.,∗ ∶ ∗≅→ ∗.

Henceforth we assume that the d.g.a.-map ∶ → is a d.g.a.-epimorphism.

4.3 Theorem.Deschner, 1975, Wu, 1980

LetA be a simply connected d.g.a. andM be its minimal model. The mapρ ∶ M →

Ahas couniversal property, i.e., for any d.g.a. Zand d.g.a.-mapφ ∶ Z → A,there exists a d.g.a.-

map θ ∶ M → Zsuch that ρ ≃ φθ; furthermore if the d.g.a.-map φ ∶ Z → Ais an

epimorphism thenρ = φθ, i.e., the following diagram is commutative:

→

↓ ↗

F igure 13



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5. The result

We show that under a reasonable assumption,the minimal model of a 1-connected d.g.a.

can be expressed as the Adams cocompletion of the d.g.a. with respect to the chosen set of d.g.a.-

maps.

5.1 Theorem. ≅ .

Proof.Let ∶ → be the map as in Theorem 3.5 and ∶ → be the d.g.a.-map as in

Theorem 4.3. Since the d.g.a.-map ∶ → is a d.g.a.-epimorphism, by the couniversal

property of there exists a d.g.a.-map ∶ → such that = .

→

↓ ↗

F igure 14

By the couniversal property of there exists a d.g.a.-map ∶ → such that = .

→

↓ ↗

F igure 15

Consider the diagram



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→

↓

1

↓ ↗

↓

F igure 16

Thus we have = = .By the uniqueness condition of the couniversal property

of (Theorem 3.5), we conclude that = 1.Next consider the diagram

→

↓

1 ↓ ↗

↓

F igure 17

Thus we have = = . By the couniversal oroperty of (Theorem 4.3), we

conclude that = 1.Thus ≅ . This completes the proof of Theorem 5.1.

REFERENCES

Adams, J. F. (1973). Idempotent Functors in Homotopy Theory: Manifolds, Conf.

Adams J. F. (1975).Localization and Completion, Lecture Notes in Mathematics.Univ. of

Chicago.

Deleanu A., Frei A. & Hilton P. J. (1974). Generalized Adams completion.Cahiers de Top.et

Geom. Diff., 15, 61-82.



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Deleanu A. (1975). Existence of the Adams completion for objects of cocomplete categories. J.

Pure and Appl. Alg., 6, 31-39.

Deschner A. J. (1976). Sullivan’s Theory of Minimal Models, Thesis.Univ. of British Columbia.

Gabriel P. &Zisman M. (1967).Calculus of Fractions and Homotopy Theory.Springer-

Verlag, New York.

Halperin S. (1977, 1981). Lectures on Minimal Models.Publ. U. E. R. de Math ́ matiques, Univ.

de Lille I.

Lane S. Mac (1971). Categories for the working Mathematicians.Springer-Verlag, New York.

Nanda S. (1980). A note on the universe of a category of fractions.Canad.Math. Bull., 23(4),

425-427.

Schubert H. (1972). Categories. Springer Verlag, New York.

Wu wen-tsun (1980).Rational Homotopy Type.LNM 1264, Springer-Verlag.

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A Categorical Construction of Minimal Model