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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),
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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2.2 Definition.Schubert, 1972
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 .
2.3 Definition.Schubert, 1972
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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2.13 Theorem. Deleanu, et al., 1974
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:
2.14 Theorem. Deleanu, et al., 1974
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.
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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.
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