On Adams Completion and Cocompletion

Transcription

1 On Adam Completion and Cocompletion Mitali Routaray Department of Mathematic National Intitute of Technology Rourkela

2 On Adam Completion and Cocompletion Diertation ubmitted in partial fulfillment of the requirement of the degree of Doctor of Philoophy in Mathematic by Mitali Routaray (Roll Number: 512ma302) baed on reearch carried out under the uperviion of Prof. Akrur Behera [MA] July, 2016 Department of Mathematic National Intitute of Technology Rourkela

3 Department of Mathematic National Intitute of Technology Rourkela July 25, 2016 Certificate of Examination Roll Number: 512ma302 Name: Mitali Routaray Title of Diertation: On Adam Completion and Cocompletion We the below igned, after checking the diertation mentioned above and the official record book () of the tudent, hereby tate our approval of the diertation ubmitted in partial fulfillment of the requirement of the degree of Doctor of Philoophy in Mathematic at National Intitute of Technology Rourkela. We are atified with the volume, quality, correctne, and originality of the work. Prof. Akrur Behera [MA] Principal Supervior Prof. D.P Mohapatra [CS] Member, DSC Prof. J. Mohapatra [MA] Member, DSC Prof. K.C. Pati [MA] Member, DSC External Examiner Prof. S. Chakraverty [MA] Chairperon, DSC Prof. K.C Pati [MA] Head of the Department

4 Department of Mathematic National Intitute of Technology Rourkela Prof. Akrur Behera [MA] July 25, 2016 Supervior Certificate Thi i to certify that the work preented in the diertation entitled On Adam Completion and Cocompletion ubmitted by Mitali Routaray, Roll Number 512ma302, i a record of original reearch carried out by her under my uperviion in partial fulfillment of the requirement of the degree of Doctor of Philoophy in Mathematic. Neither thi diertation nor any part of it ha been ubmitted earlier for any degree or diploma to any intitute or univerity in India or abroad. Prof. Akrur Behera [MA]

5 Declaration of Originality I, Mitali Routaray, Roll Number 512ma302 hereby declare that thi diertation entitled On Adam Completion and Cocompletion preent my original work carried out a a doctoral tudent of NIT Rourkela and to the bet of my knowledge, contain no material previouly publihed or written by another peron, nor any material preented by me for the award of any degree or diploma of NIT Rourkela or any other intitution. Any contribution made to thi reearch by other, with whom I have worked at NIT Rourkela or elewhere, i explicitly acknowledged in the diertation. Work of other author cited in thi diertation have been duly acknowledged under the ection Reference or Bibliography. I have alo ubmitted my original reearch record to the crutiny committee for evaluation of my diertation. I am fully aware that in cae of any non-compliance detected in future, the Senate of NIT Rourkela may withdraw the degree awarded to me on the bai of the preent diertation. July 25, 2016 NIT Rourkela Mitali Routaray

6 Acknowledgment Firt and foremot, I would like to expre my incere gratitude to my upervior Prof. Akrur Behera for hi continuou upport and inightful guidance throughout my thei work. I have greatly benefited from hi patience and many a time he ha litened to my naive idea carefully and corrected my mitake. In all my need, I alway found him a a kind and helpful peron. I am indebted to him for all the care and help that I got from him during my Ph.D. time. I thank the Director, National Intitute of Technology, Rourkela, for permitting me to avail the neceary facilitie of the Intitute for the completion of thi work. I would like to thank my intitute National Intitute of Technology, Rourkela for providing me a vibrant reearch environment. I have enjoyed excellent computer facility, a well managed library and a clean and well furnihed hotel at my intitute. A pecial word of thank to my friend Snigdha Bharati Choudhury and Prakah Kumar Sahu, Ph.D. Scholar, Department of Mathematic, National Intitute of Technology Rourkela for their moral upport, helpful pirit and encouragement which rejuvenated my vigor for reearch and motivated me to have achievement beyond my own expectation. Lat, but not the leat, I feel pleaed and privileged to fulfill my parent and uncle (Bidhubhuan Routaray) ambition and I am greatly indebted to them for bearing the inconvenience during my thei work. I thank my brother (Chanakya Routaray) and iter (Punyatoya Routaray) for their unconditional love and emotional upport, epecially during the time of difficultie. July 25, 2016 NIT Rourkela Mitali Routaray Roll Number: 512ma302

7 Abtract The minimal model of a 1-connected differential graded Lie algebra i obtained a the Adam cocompletion of the differential graded Lie algebra with repect to a choen et of morphim in the category of 1-connected differential graded Lie algebra (d.g.l.a. ) over the field of rational and d.g.l.a.-homomorphim. The Potnikov-like approimation of a module i obtained a the Adam completion of the pace with the help of a uitable et of morphim in the category of ome pecific module and module homomorphim. The Cartan-Whitehead decompoition of topological G-module i obtained a the Adam cocompletion of the pace with repect to uitable et of morphim. Potnikov-like approximation i obtained for a topological G-module, in term of Adam completion with repect to a uitable et of morphim, uing cohomology theory of topological G-module. The ring of fraction of the algebra of all bounded linear operator on a eparable infinite dimenional Banach pace i iomorphic to the Adam completion of the algebra with repect to a carefully choen et of morphim in the category of eparable infinite dimenional Banach pace and bounded linear norm preerving operator of norm at mot 1. The nth tenor algebra and ymmetric algebra are each iomorphic to the Adam completion of the algebra. The exterior algebra and Clifford algebra are each iomorphic to the Adam completion of the algebra with repect to a choen et of morphim in the category of module and module homomorphim. Keyword: Grothendieck univere; Adam completion; Adam cocompletion; Minimal model; G-module; Tenor algebra; Symmetric algebra; Exterior algebra; Clifford algebra.

8 Content Certificate of Examination Supervior Certificate Declaration of Originality Acknowledgment Abtract ii iii iv v vi 0 Introduction 1 1 Pre-Requiite Category of fraction Calculu of left (right) fraction Adam completion and cocompletion Exitence theorem Couniveral property A Serre cla C of module A Categorical Contruction of Minimal Model of Lie Algebra Minimal model The category A The main reult Homotopy Approximation of Module Homotopy of Module The category M Exitence of Adam completion in M A Potnikov-like approximation Topological G-Module and Adam cocompletion Topological G-module The category G Exitence of Adam completion in G Cartan-Whitehead-like tower vii

9 5 Cohomology Decompoition of Topological G-Module The category M Exitence of Adam Completion in M Ring of Fraction a Adam Completion Ring of fraction The category B The main reult A Categorical Study of Symmetric and Tenor Algebra Tenor algebra The Category A Tenor algebra a Adam completion Symmetric algebra The category M Symmetric algebra a Adam completion Exterior Algebra and Clifford Algebra a Adam Completion Exterior algebra The category A Exterior algebra a Adam completion Clifford algebra The category A Clifford algebra a Adam completion Reference 94 Diemination 97 viii

10 Chapter 0 Introduction The concept of the Adam completion wa propoed by J. F. Adam [1 4]; in fact thi idea firt aroe with repect to the problem of tability. It characterization and propertie were clearly categorical in nature. However, only in later work by Deleanu, Frei and Hilton the theory wa freed from it topological bound. The greatet difficulty, in dealing with the Adam completion from the categorical point of view (hence in general), lie in it et theoretical apect. In fact category of fraction, which play a baic role here, i not alway well defined, ince there i no guarantee that the collection of morphim between any two of it object i a et. It i well known that the uual et theory, a decribed by Zermelo and Fraenkel [5, 6], when ued without extreme rigor lead very eaily to ome incoherent reult. The mot famou of thoe i the Ruell paradox, which implie that the et of all the et i not a et. To avoid thoe difficultie we will work in the logical framework of univere of Grothendieck. The firt tep in thi direction i to forget the exitence of primitive, i.e., indiviible, element and to conider any et a a collection of other et, where the collection can even be empty or conit of a ingle element. With thi agreement Grothendieck univere i defined in [7]. Thi thei doe not attempt to make a tudy of et theory; however the concept of univere i eential ince their ue eem to be unavaoidable in ome categorical contruction, in particular in the contruction of category of fraction. It i a firmly etablihed fact that the collection of object of a category need not be a et, but the logical contradiction which i at the bai of the Ruell paradox work alo in thi cae, o that the category of all categorie cannot be conidered a a category. Neverthele many time it i very ueful to conider thi or other kind of tructure which preent the ame difficulty. Thee difficultie may be overcome by making ome mild hypothee and uing Grothendieck univere [7]. Preciely peaking if we tart with a category belonging to a certain Grothendieck univere then the category of fraction with repect to a et of morphim of the category 1

11 Introduction belong to a higher univere [7]. We note that the cae in which we are intereted, will not preent uch difficulty. However, Nanda [8] ha proved that if the et of morphim admit a calculu of left (right) fraction then the category of fraction with repect to the et of morphim of the category belong to the ame univere a to the univere that the category belong. Alo if the et of morphim of the category admit a calculu of left (right) fraction then the category of fraction can be decribed nicely; thi explicit contruction i given in [7]. The central idea of thi thei i to invetigate ome cae howing how ome algebraic and geometrical contruction are characterized in term of Adam completion or cocompletion. We will deal with uch cae involving the concept of calculu of left (right) fraction. In fact in each of the characterization that we have undertaken in our tudy, the et of morphim of the category ha to admit either calculu of left fraction or calculu of right fraction. In Chapter 1, we recall the definition of Grothendieck univere, category of fraction, calculu of left (right) fraction [7] and generalized Adam completion (cocompletion) [9]. We tate ome reult on the exitence of global Adam completion (cocompletion) of an object in a cocomplete (complete) category with repect to a et of morphim in the category [9]. Deleanu, Frei and Hilton [9] have hown that if the et of morphim in the category i aturated then the Adam completion (cocompletion) of an object i characterized by a certain couniveral property. We tate a tronger verion of thi reult proved by Behera and Nanda [10] where the aturation aumption on the et of morphim i dropped. We alo tate Behera and Nanda reult [10] that the canonical map from an object to it Adam completion (from Adam cocompletion to the object) i an element of the et of morphim under very moderate aumption. Thee two reult are fairly general in nature and applicable to mot cae of interet. The concept of rational homotopy theory wa firt characterized by Quillen. In fact in rational homotopy theory Sullivan introduced the concept of minimal model. In Chapter 2, a categorical contruction of minimal model of lie algebra i preented. In fact we prove that the minimal model of a 1-connected differential graded Lie algebra can be expreed a the Adam cocompletion of the differential graded Lie algebra with repect to a choen et of morphim in the category of 1-connected differential graded Lie algebra (in hort d.g.l.a. ) over the field of rational and d.g.l.a.-homomorphim. Behera and Nanda have tudied Potnikov approximation of a pace, by introducing a Serre cla C of abelian group. They have obtained the mod-c Potnikov approximation of a 1-connected baed CW -complex, with the help of a uitable et of morphim in 2

12 the category of 1-connected baed CW -complexe and baed map. In Chapter 3, we have obtained the Potnikov-like approimation of a module, where the different tage of the approximation are hown to be the Adam completion of the module, with the help of a uitable et of morphim in the category of ome pecific module and module homomorphim. It i known that the different tage of the Cartan-Whitehead decompoition of a 0-connected pace can be obtained a the Adam cocompletion of the pace with repect to uitable et of morphim [10]. In Chapter 4, Cartan-Whitehead decompoition i obtained for topological G-module. In Chapter 5, we tudy the dual of the decompoition of a topological G-module obtained in Chapter 4. In fact, the central idea of thi chapter i to obtain a Potnikov-like tower of a topological G-module, uing the cohomology theory of topological G-module. In Chapter 6, it i hown that ring of fraction of B(H), the algebra of all bounded linear operator on a eparable infinite dimenional Hilbert pace H i iomorphic to the Adam completion of B(H) with repect to a choen et of morphim in a uitable category. In thi chapter, we how that the ring of fraction of the algebra of all bounded linear operator on a eparable infinite dimenional Banach pace i iomorphic to the Adam completion of the algebra with repect to a carefully choen et of morphim in the category of eparable infinite dimenional Banach pace and bounded linear norm preerving operator of norm at mot 1. Chapter 7 i devoted to categorical tudy of tenor algebra and ymmetric algebra. The purpoe here i to obtain the tenor algebra and ymmetric algebra in term of Adam completion. Under ome reaonable aumption, we how that given an algebra, it nth tenor algebra and ymmetric algebra are each iomorphic to the Adam completion of the algebra. In Chapter 8, we obtain that given an algebra, it exterior algebra and Clifford algebra are each iomorphic to the Adam completion of the algebra with repect to a choen et of morphim in the category of module and module homomorphim.

13 Chapter 1 Pre-Requiite In thi chapter we recall the definition of Adam completion (cocompletion) and ome known reult on the exitence of global Adam completion (cocompletion) of an object in a category C with repect to a family of morphim S in C. A characterization of Adam completion (cocompletion) in term of it couniveral property proved by Deleanu, Frei and Hilton i recalled. We alo decribe a tronger verion of thi reult proved by Behera and Nanda [11]. We alo tate Behera and Nanda reult [11] that the canonical map from an object to it Adam completion i an element of the et of morphim under very moderate aumption. Thi chapter erve a the bae and background for the tudy of ubequent chapter and we hall keep on referring back to it a and when required. 1.1 Category of fraction In thi ection we recall the abtract definition of category of fraction and ome other related definition. We tart with univere. Definition ([7], p. 266) A Grothendeick univere (or imply univere) i a collection U of et uch that the following axiom are atified: U(1): If {X i : i I} i a family of et belonging to U then i I X i i an element of U. U(2): If x U then {x} U. U(3): If x X and X U then x U. U(4): If X i a et belonging to U then P (X), the power et of X i an element of U. U(5): If X and Y are element of U then {X, Y }, the ordered pair (X, Y ) and X Y are element of U. We fix a univere U that contain N, the et of natural number (and o Z, Q, R, C). Definition ([7], p. 267) A category C i aid to be a mall U -category, U being a fixed Grothendeick univere, if the following condition hold: S(1): The object of C form a et which i an element of U. S(2): For each pair (X, Y ) of object of C, the et C (X, Y ) i an element of U. 4

14 Chapter 1 Pre-Requiite Definition ([7], p. 269) Let C be any arbitrary category and S a et of morphim of C. A category of fraction of C with repect to S i a category denoted by C [S 1 ] together with a functor F S : C C [S 1 ] having the following propertie: CF(1): For each S, F S () i an iomorphim in C [S 1 ]. CF(2): F S i univeral with repect to thi property : if G : C D i a functor uch that G() i an iomorphim in D, for each S, then there exit a unique functor H : C [S 1 ] D uch that G = HF S. Thu we have the following commutative diagram: G C D F S H C [S 1 ] Reamrk For the explicit contruction of the category C [S 1 ], we refer to [7]. We content ourelve merely with the obervation that the object of C [S 1 ] are ame a thoe of C and in the cae when S admit a calculu of left (right) fraction, the category C [S 1 ] can be decribed very nicely [7, 12]. 1.2 Calculu of left (right) fraction A dicued in [7], for contructing the category of fraction, the notion of calculu of left (right) fraction play a very crucial role. Definition ([7], p. 258) A family of morphim S in the category C i aid to admit a calculu of left fraction if (a) S i cloed under finite compoition and contain identitie of C, (b) any diagram X Y f Z in C with S can be completed to a diagram 5

15 Chapter 1 Pre-Requiite X Y f g Z t W with t S and tf = g, (c) given f t X Y Z W g with S and f = g, there i a morphim t : Z W in S uch that tf = tg. A imple characterization for a family of morphim S to admit a calculu of left fraction i the following. Theorem ([9], Theorem 1.3, p. 67) Let S be a cloed family of morphim of atifying C (a) if uv S and v S, then u S, (b) every diagram f in C with S can be embedded in a weak puh-out diagram f g t with t S. Then S admit a calculu of left fraction. The notion of a et of morphim admitting a calculu of right fraction i defined dually. 6

16 Chapter 1 Pre-Requiite Definition ([7], p. 267) A family S of morphim in a category C i aid to admit a calculu of right fraction if (a) S i cloed under finite compoition and contain identitie of C, (b) any diagram X f Z Y in C with S can be completed to a diagram g W Z t X Y f with t S and ft = g, (c) given f t W X Y Z g with S and f = g, there i a morphim t : W X in S uch that ft = gt. The analog of Theorem follow immediately by duality. Theorem ([9], Theorem 1.3, p. 70) Let S be a cloed family of morphim of atifying C (a) if vu S and v S, then u S, (b) any diagram f 7

17 Chapter 1 Pre-Requiite in C with S, can be embedded in a weak pull-back diagram t g f with t S. Then S admit a calculu of right fraction. Reamrk There are ome et-theoretic difficultie in contructing the category C [S 1 ]; thee difficultie may be overcome by making ome mild hypothee and uing Grothendeick univere. Preciely peaking, the main logical difficulty involved in the contruction of a category of fraction and it ue, arie from the fact that if the category C belong to a particular univere, the category C [S 1 ] would, in general belong to a higher univere ([7], Propoition ). In mot application, however, it i neceary that we remain within the given initial univere. Thi logical difficulty can be overcome by making ome kind of aumption which would enure that the category of fraction remain within the ame univere [13 15]. Alo the following theorem (Theorem 1.2.6) how that if S admit a calculu of left (right) fraction, then the category of fraction C [S 1 ] remain within the ame univere a to the univere to which the category C belong. The following reult will be ued in our tudy. Theorem [8] Let C be a mall U -category and S a et of morphim of admit a calculu of left (right) fraction. Then C [S 1 ] i a mall U -category. C that 1.3 Adam completion and cocompletion Sullivan introduced the concept of localization [16]. Boufield introduced the concept of localization in categorie [17]. Both the contruction are applicable to many cae of intert. Sullivan contruction i neat and concrete. Boufield contruction i general and categorical. Several author have worked on both the contruction [18]. The notion of generalized completion (Adam completion) aroe from a categorical completion proce uggeted by Adam [1, 2]. Originally thi wa conidered for admiible categorie and generalized homology (or cohomology) theorie. Subequently, thi notion ha been conidered in a more general framework by Deleanu, Frei and Hilton [9], where an arbitrary category and an arbitrary et of morphim of the category are conidered; moreover they have alo uggeted the dual notion, namely the cocompletion (Adam cocompletion) of an object in a category. We recall the definition of Adam completion and cocompletion. 8

18 Chapter 1 Pre-Requiite Definition [9] Let C be an arbitrary category and S a et of morphim of C [S 1 ] denote the category of fraction of C with repect to S and C. Let F : C C [S 1 ] be the canonical functor. Let S denote the category of et and function. Then for a given object Y of C, C [S 1 ](-, Y ) : C S define a contravariant functor. If thi functor i repreentable by an object Y S of C, i.e., C [S 1 ](-, Y ) = C (-, Y S ) then Y S i called the (generalized) Adam completion of Y with repect to the et of morphim S or imply the S-completion of Y. We hall often refer to Y S a the completion of Y. The above definition can be dualized a follow: Definition [9] Let C be an arbitrary category and S a et of morphim of C [S 1 ] denote the category of fraction of C with repect S and C. Let F : C C [S 1 ] be the canonical functor. Let S denote the category of et and function. Then for a given object Y of C, C [S 1 ](Y, -) : C S define a covariant functor. If thi functor i repreentable by an object Y S of C, i.e., C [S 1 ](Y, -) = C (Y S, -) then Y S i called the (generalized) Adam cocompletion of Y with repect to the et of morphim S or imply the S-cocompletion of Y. We hall often refer to Y S a the cocompletion of Y. 1.4 Exitence theorem We recall ome reult on the exitence of Adam completion and cocompletion. We tate Deleanu theorem [15] that under certain condition, global Adam completion of an object alway exit. Theorem ([15], Theorem 1; [8], Theorem 1) Let C be a cocomplete mall U -category (U i a fixed Grothendeick univere) and S a et of morphim of C that admit a calculu of left fraction. Suppoe that the following compatibility condition with coproduct i atified. 9

19 Chapter 1 Pre-Requiite (C) If each i : X i Y i, i I, i an element of S, where the index et I i an element of U, then i an element of S. i : X i Y i i I i I i I Then every object X of C ha an Adam completion X S with repect to the et of morphim S. Reamrk Deleanu theorem quoted above ha an extra condition to enure that C [S 1 ] i again a mall U -category; in view of Theorem the extra condition i not neceary. Theorem can be dualized a follow. Theorem ([8], Theorem 2 ) Let C be a complete mall U -category (U i a fixed Grothendeick univere) and S a et of morphim of C that admit a calculu of right fraction. Suppoe that the following compatibility condition with product i atified. (P) If each i : X i Y i, i I, i an element of S, where the index et I i an element of U, then i an element of S. i : X i Y i i I i I i I Then every object X of morphim S. C ha an Adam cocompletion X S with repect to the et of We will recall ome more reult on the exitence of Adam completion and cocompletion in the relevant chapter. 1.5 Couniveral property Deleanu, Frei and Hilton have developed characterization of Adam completion and cocompletion in term of a couniveral property. Definition [9] Given a et S of morphim of C, we define S, the aturation of S a the et of all morphim u in C uch that F (u) i an iomorphim in C [S 1 ]. S i aid to be aturated if S = S. Theorem ( [9], Propoition 1.1, p. 63) A family S of morphim of C i aturated if and only if there exit a functor F : C D uch that S i the collection of all morphim f uch that F (f) i invertible. Deleanu, Frei and Hilton have hown that if the et of morphim S i aturated then the Adam completion of a pace i characterized by a certain couniveral property. 10

20 Chapter 1 Pre-Requiite Theorem ([9], Theorem 1.2, p. 63) Let S be a aturated family of morphim of C admitting a calculu of left fraction. Then an object Y S of C i the S-completion of the object Y with repect to S if and only if there exit a morphim e : Y Y S in S which i couniveral with repect to morphim of S : given a morphim : Y Z in S there exit a unique morphim t : Z Y S in S uch that t = e. In other word, the following diagram i commutative: e Y Y S Z t Theorem can be dualized a follow. Theorem ([9], Theorem 1.4, p. 68) Let S be a aturated family of morphim of C admitting a calculu of right fraction. Then an object Y S of C i the S-cocompletion of the object Y with repect to S if and only if there exit a morphim e : Y S Y in S which i couniveral with repect to morphim of S : given a morphim : Z Y in S there exit a unique morphim t : Y S Z in S uch that t = e. In other word, the following diagram i commutative : Y S e Y t Z In mot of application, however, the et of morphim S i not aturated. The following i a tronger verion of Deleanu, Frei and Hilton characterization of Adam completion in term of a couniveral property. Theorem ([11], Theorem 1.2, p. 528) Let S be a et of morphim of C admitting a calculu of left fraction. Then an object Y S of C i the S-completion of the object Y with repect to S if and only if there exit a morphim e : Y Y S in S which i couniveral with repect to morphim of S: given a morphim : Y Z in S there exit a unique morphim t : Z Y S in S uch that t = e. In other word, the following diagram i commutative : e Y Y S Z t 11

21 Chapter 1 Pre-Requiite Theorem can be dualized a follow. Theorem ([10], Propoition 1.1, p. 224) Let S be a et of morphim of C admitting a calculu of right fraction. Then an object Y S of C i the S-cocompletion of the object Y with repect to S if and only if there exit a morphim e : Y S Y in S which i couniveral with repect to morphim of S : given a morphim : Z Y in S there exit a unique morphim t : Y S Z in S uch that t = e. In other word, the following diagram i commutative : Y S e Y t Z For mot of the application it i eential that the morphim e : Y Y S (e : Y S Y ) ha to be in S; thi i the cae when S i aturated and the reult are a follow: Theorem ([9], Theorem 2.9, p. 76) Let S be a aturated family of morphim of C and let every object of C admit an S-completion. Then the morphim e : Y Y S belong to S and i univeral for morphim to S-complete object and couniveral for morphim in S. The above reult can be dualized a follow. Theorem ([9], dual of Theorem 2.9, p. 76) Let S be a aturated family of morphim of C and let every object of C admit an S-cocompletion. Then the morphim e : Y S Y belong to S and i univeral for morphim to S-cocomplete object and couniveral for morphim in S. However, in many cae of practical interet S i not aturated. The following reult how that under ome extra condition on S, the morphim e : Y Y S (e : Y S Y ) alway belong to S. Theorem ([11], Theorem 1.3, p. 533) Let S be a et of morphim in a category C admitting a calculu of left fraction. Let e : Y Y S be the canonical morphim a defined in Theorem 1.5.5, where Y S i the S-completion of Y. Furthermore, let S 1 and S 2 be et of morphim in the category C which have the following propertie : (a) S 1 and S 2 are cloed under compoition, (b) fg S 1 implie that g S 1, (c) fg S 2 implie that f S 2, (d) S = S 1 S 2. 12

22 Chapter 1 Pre-Requiite Then e S. Theorem can be dualized a follow: Theorem ([11], dual of Theorem 1.3, p. 533) Let S be a et of morphim in a category C admitting a calculu of right fraction. Let e : Y S Y be the canonical morphim a defined in Theorem 1.5.6, where Y S i the S-cocompletion of Y. Furthermore, let S 1 and S 2 be et of morphim in the category C which have the following propertie : (a) S 1 and S 2 are cloed under compoition, (b) fg S 1 implie that g S 1, (c) fg S 2 implie that f S 2, (d) S = S 1 S 2. Then e S. 1.6 A Serre cla C of module We collect ome relevant definition and theorem involving Serre clae of module [19]. Definition [19] A nonempty cla C of module i called a Serre cla of module if and only if whenever the three-term equence A B C of module i exact and A, C C then B C. Definition [19] Let C be a Serrre cla of module and A, B C. A homomorphim f : A B (a) i a C-monomorphim if ker f C. (b) i a C-epimorphim if coker f C. (c) i a C-iomorphim if it i both a C-monomorphim and C-epimorphim. Theorem [20] Let A, B C and f : A B and g : B C be two homomorphim. Then the following tatement are true. (a) If gf i C-monic, then o i f. (b) If gf i C-epic, then o i g. Theorem (Five lemma of module) [21] Let 13

23 M 1 M 2 M 3 M 4 M 5 α β γ δ ϵ N 1 N 2 N 3 N 4 N 5 be a row exact commutative diagram of R-module and R-module homomorphim where R i a ring. Then the following hold : (a) If α i an epimorphim and β, and δ are monomorphim, then γ i a monomorphim. (b) If ϵ i a monomorphim and β, and δ are epimorphim, then γ i an epimorphim. (c) If α, β, δ and ϵ are iomorphim, then γ i an iomorphim.

24 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra Quillen ha etablihed algebraic model for rational homotopy theory [22]. In fact Sullivan introduced the concept of minimal model [23] in rational homotopy theory. It may be noted that there are three baic contruction in literature which relate a 1-connected topological pace X to a differential graded algebra. Adam and Hilton [24] contructed a chain algebra (with integer coefficient) for the loop pace ΩX, a pecial verion of which i Adam cobra contruction [25]. Later Quillen [22] aociated a differential graded rational Lie algebra L(X) to the pace X and Sullivan [23, 26] uing implical differential form with rational coefficient, obtained a differential graded commutative cochain algebra for X. For thee cochain algebra, Sullivan introduced the notion of minimal model, which correpond to the Potnikov decompoition of a pace. The aim of thi chapter i to invetigate a cae howing how thi algebraic contruction i characterized in term of Adam cocompletion. Baue and Lemaire [27] have contructed minimal model for chain algebra (over any field) and for rational differential graded Lie algebra. In thi chapter, we prove that the minimal model of a imply connected differential graded Lie algebra i characterized in term of Adam cocompletion. 2.1 Minimal model We recall the following algebraic preliminarie. Definition [28, 29] Let Q be the et of rational number. By a graded algebra A over Q we mean a graded Q vector pace together with (a) an aociative multiplication A = A n n 0 15

25 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra µ : A A A which i graded (µ(a n A m ) A n+m ) and (b) graded commutative a b = ( 1) nm b a, a A n b B m. We alo aume, unle otherwie tated, that A ha an identity element 1 A 0. The element of A n are aid to be homogeneou of degree n (or dimenion n). Definition [28, 29] A differential graded algebra (or d.g.a) i a graded algebra A, together with a differential d, of degree +1, which i a derivation. Thi mean that for each n there i a vector pace homomorphim d = d n : A n A n+1 atifying (a) d d = 0 (differential) and (b) d(a b) = d(a) b + ( 1) n a d(b) for a A n (derivation). Definition [29] Let (A, d) be a d.g.a. Let Z n (A) = {d n : A n A n+1 } = cycle A n, B n = {d n 1 : A n 1 A n } = boundarie A n, Z (A) = Z n (A), n 0 B (A) = B n (A). n 0 A d 2 = 0, we have B n (A) Z n (A). The nth homology pace of A i defined to be the quotient vector pace H n (A) = Z n (A)/B n (A) H (A) = H n (A) = Z (A)/B (A) n 0 i a graded algebra, called the homology algebra of A. Definition [29] A d.g.a. A i aid to be connected if H 0 (A) = Q and that A i imply connected (1-connected) if it i connected and H 1 (A) = 0. A i called n-connected if 16

26 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra (a) H 0 (A 0 ) = Q and (b) H p (A n ) = 0, 1 p n. A i aid to be of finite type if for each n, H n (A) i a vector pace of finite dimenional over Q. Definition [29, 30] Let A and B be graded algebra over Q. A function f : A B i called a homomorphim if it preerve all algebraic tructure, that i, (a) f(a n ) B n, (b) f(a + b) = f(a) + f(b), (c) f(a b) = f(a) f(b). We aume that f(1) = 1. If A and B are d.g.a. it i required alo that f n commute with differential, i.e., f n+1 d A n = d B n f n A n d A n A n+1 f n f n+1 B n d B n B n+1 If f : A B i a d.g.a. homomorphim then f induce a map f : H (A) H (B) defined by the rule f ([z]) = [f(z)], where [z] denote the homology cla of the element z Z (A). Clearly f i a homomorphim of graded algebra. Let DG A denote the category of differential graded algebra and differential graded algebra homomorphim. Definition [29] Let h : A A be a map of Lie algebra. Then h i a weak iomorphim if and only if H (h) : H (A) H (A ) i an iomorphim. Propoition [27] Let H (M) be the homology of the differential graded vector pace M. A weak iomorphim f : M N i differential graded map uch that H (f) : H (M) H (N) i an iomorphim. 17

27 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra Definition [28, 29] A differential graded algebra M i called a minimal algebra if and only if it atifie the following proprietie : (a) M i free a a graded algebra, (b) M ha a decompoable differential, (c) M 0 = Q, M 1 = 0, (d) M ha homology of finite type, that i, for each n, H n (M) i a finite dimenional vector pace over Q. Let M be the full ubcategory of the category DG A coniting of all minimal algebra and all differential graded algebra map between them. Definition [28, 29] If A i imple connected differential graded algebra. A differential graded algebra M = M A i called a minimal model of A if the following condition hold : (a) M M, (b) there i a d.g.a. map ρ : M A which induce an iomorphim on homology, i.e., ρ : H (M) = H (A). Definition [31] A differential graded Lie algebra i the data of a differential graded vector pace (L, d) together a with a bilinear map [-, -] : L L L atifying the following propertie: (a) [-, -] i homogeneou kew ymmetric; thi mean [L i, L j ] L i+j and [a, b] + ( 1)ā b+1 [b, a] = 0 for every a, b homogeneou. (b) Every a, b, c homogeneou atifie the Jacobi identity [a, [b, c]] = [[a, b], c] + ( 1)ā b[b, [a, c]]. (c) d(l i ) L i+1, d d = 0 and d[a, b] = [d(a), b] + ( 1)ā[a, d(b)]. The map d i called the differential of d. 18

28 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra Definition [30, 31] Given two Lie algebra L and L, their direct um i the Lie algebra coniting of the vector pace L L, of the pair (x, x ), x L, x L with the operation [(x, x ), (y, y )] = ([x, y], [x, y ]), x, y L and x, y L. Definition [31] A morphim of differential graded Lie algebra i a graded linear map f : L L that commute with bracket and the differential, i.e., f[x, y] L = [f(x), f(y)] L and f(d L x) = d L f(x). Theorem ([27], p.226, Propoition 1.4) Let L be a imply connected differential graded Lie algebra over Q and M = M L be a minimal model for L. Then the map h : M L induce weak iomorphim and h ha the following couniveral property: for any 1-connected differential graded Lie algebra L and differential graded Lie algebra map f : L L which induce a weak iomorphim, there exit a differential graded Lie algebra map g : M L uch that the diagram commute up to Lie algebra homotopy fg h and g i unique up to Lie algebra homotopy. g M L h f L 2.2 The category A Let U be a fixed Grothendieck univere. Let A be the category of 1-connected differential graded Lie algebra over Q (in hort d.g.l.a. ) and differential graded Lie algebra homomorphim where every element of A i an element of U. 19

29 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra Let S denote the et of all differential graded Lie algebra homomorphim in A which induce weak iomorphim in all dimenion. We prove the following reult. Propoition S i aturated. Proof. The proof i evident from Theorem Next we how that the et of morphim S of the category A admit a calculu of right fraction. Propoition S admit a calculu of right fraction. Proof. Clearly, S i a cloed family of morphim of the category A. We hall verify condition (a) and (b) of Theorem Let u, v S. We how that if vu S and v S, then u S. We have (vu) = v u and v are both homology iomorphim implying u i a homology iomorphim. Thu u S. Hence condition (a) of Theorem hold. To prove condition (b) of Theorem conider the diagram A f C B with S. We aert that the above diagram can be completed to a weak pull-back diagram g D C t A B f with t S. Since A, B and C are in A we write A = A n, B = B n, C = C n, n 0 n 0 n 0 and where f = f n, = n n 0 n 0 f n : A n B n, n : C n B n 20

30 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra are differential graded Lie algebra homomorphim. Let D n = {[(a, c), (a, c )] = ([a, a ], [c, c ]) A n C n : f n [a, a ] = n [c, c ]} A n C n where a, a A n and c, c C n. We have to how that D = D n i a differential graded n 0 Lie algebra. Let t n : D n A n be defined by t n ([a, a ], [c, c ]) = [a, a ] and g n : D n C n be defined by g n ([a, a ], [c, c ]) = [c, c ]. Clearly, t n and g n are differential graded Lie algebra homomorphim and the above diagram i commutative. Let (a, c) D n, (a, c ) D m, d A n : A n A n+1, d A = d A n n 0 and d C n : C n C n+1, d C = d C n. n 0 Define d D n : D n D n+1 by the rule d D n [(a, c), (a, c )] = d D n ([a, a ], [c, c ]) = (d A n [a, a ], d C n [c, c ]). Let d D = d D n. For (a, c) D n, (a, c ) D m, n 0 [(a, c), (a, c )] = ([a, a ], [c, c ]) = ( ( 1) nm+1 [a, a], ( 1) nm+1 [c, c] ) = ( 1) nm+1 ([a, a], [c, c]) = ( 1) nm+1 [(a, c ), (a, c)]. Thu we get [(a, c), (a, c )] + ( 1) nm+1 [(a, c ), (a, c)] = 0. Next we have to how that [a, [b, c]] = [[a, b], c] + ( 1)ā b[b, [a, c]]. 21

31 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra Since D n+m i a Lie algebra, it atifie the Jacobi property, i.e., [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0. where [a, [b, c]], [b, [c, a]] and [c, [a, b]] D n+m. We have [a, [b, c]] = [b, [c, a]] [c, [a, b]] = [b, [c, a]] + [[a, b], c] = [[a, b], c] + ( 1) nm [b, [a, c]]. We how that d D i a differential. We have d D [(a, c), (a, c )] = d D ([a, a ], [c, c ]) = (d A [a, a ], d C [c, c ]) = ([d A a, a ] + ( 1) n [a, d A a ], [d C c, c ] + ( 1) n [c, d C c ]) = ([d A a, a ], [d C c, c ]) + ( 1) n ([a, d A a ], [c, d C c ]) = [(d A a, d C c), (a, c )] + ( 1) n [(a, c)(d A a, d C c )] = [d D (a, c), (a, c )] + ( 1) n [(a, c), d D (a, c )]. Thu D become a differential graded Lie algebra. Next we have to how that D i 1-connected, i.e., H 0 (D) = Q and H 1 (D) = 0. Firt we how that H 0 (D) = Q. We have H 0 (D) = Z 0 (D)/B 0 (D) = Z 0 (D) = {([a, a ], [c, c ]) Z 0 (A) Z 0 (C) : f 0 [a, a ] = 0 [c, c ]}. Let [1 A, 1 A ] A 0, [1 C, 1 C ] C 0. Then d D 0 ([1 A, 1 A], [1 C, 1 C]) = (d A 0 [1 A, 1 A], d C 0 [1 C, 1 C]) = (0, 0) implie that ([1 A, 1 A ], [1 C, 1 C ]) Z 0(D). Since A and C are 1-connected, we have H 0 (A) = Z 0 (A) = Q = Q[1 A, 1 A ] 22

32 Chapter 2 and A Categorical Contruction of Minimal Model of Lie Algebra H 0 (C) = Z 0 (C) = Q = Q[1 C, 1 C ]. Thu ([a, a ], [c, c ]) H 0 (D) = Z 0 (D) Z 0 (A) Z 0 (C) if and only if and [a, a ] = r[1 A, 1 A ] [c, c ] = r[1 C, 1 C ] for ome r Q. Now we get H 0 (D) = Q. Next we have to how that H 1 (D) = 0. Let ([a, a ], [c, c ]) Z 1 (D). Thi implie that and [a, a ] Z 1 (A), [c, c ] Z 1 (C) f 1 [a, a ] = 1 [c, c ]. Since A i 1-connected, we have H 1 (A) = 0,.., Z 1 (A)/B 1 (A) = B 1 (A); hence [a, a ] = d A 0 [ã, â] where [ã, â] A 0. Similarly ince C i 1-connected we have H 1 (C) = 0,.., Z 1 (C)/B 1 (C) = B 1 (C); hence where [ c, ĉ] C 0. Now i.e., [c, c ] = d C 0 [ c, ĉ] f 1 [a, a ] = 1 [c, c ], f 1 d A 0 [ã, â] = 1 d C 0 [ c, ĉ]. 23

33 Chapter 2 Thi give i.e., howing Thu A Categorical Contruction of Minimal Model of Lie Algebra d B 0 f 0 [ã, â] = d B 0 0 [ c, ĉ], d B 0 (f 0 [ã, â] 0 [ c, ĉ]) = 0 f 0 [ã, â] 0 [ c, ĉ]) d B 0. [f 0 [ã, â] 0 [ c, ĉ]] H 0 (B). But 0 S. Hence 0 : H 0 (C) H 0 (B) i an iomorphim. Hence there exit an element [ c, c] H 0 (C) uch that 0 [ c, c] = f 0 [ã, â] 0 [ c, ĉ] Thu i.e., So 0 [ c, c] f 0 [ã, â] 0 [ c, ĉ] B 0 (B) = 0, f 0 [ã, â] = 0 ([ c, ĉ] + [ c, c]). ([ã, â], [ c, ĉ] + [ c, c]) D 0. Moreover d D 0 (([a, â], [ c, c]) + [c, c]) = (d A 0 [a, â], d C 0 ([ c, c] + [c, ĉ])) = (d A 0 [a, â], (d C 0 [ c, c] + d C 0 [c, ĉ])) = (d A 0 [a, â], 0 + d C 0 [c, ĉ]) = (d A 0 [a, â], d C 0 [c, ĉ]) = ([a, a ], [c, c ]) howing that ([a, a ], [c, c ]) B 1 (D). Thu H 1 (D) = 0. Next we have to hown that t S, i.e., t : H (D) H (A) i an iomorphim. Let 24

34 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra F = ker g. Then we have the following commutative diagram [32] F F g D C t A B f We conider the exact homology equence H n 1 (C) H n (F ) H n (D) t H n 1 (B) H n (F ) H n (A) H n (C) H n+1 (F ) t S. H n (B) H n+1 (F ) Clearly, the above diagram i commutative. By Five lemma t i an iomorphim. So Next for any differential graded Lie algebra E = algebra homomorphim and in A, let the following diagram u = u n : E A n 0 v = v n : E C n 0 E n and differential graded Lie n 0 v E C u A B f commute, i.e., fu = v. For any d.g.l.a E and d.g.l.a. homomorphim u : E A and 25

35 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra v : E C with fu = v. Conider the diagram E h u v g D t A f C B Define h : E D by the rule h[x, y] = ([ux, uy], [vx, vy]) for [x, y] E. Clearly, h i well defined and alo differential graded Lie algebra homomorphim. Next we how that the two triangle are commutative, i.e., th = u and gh = v. For any [x, y] E, th[x, y] = t([ux, uy], [vx, vy]) = [ux, uy] = u[x, y] and gh[x, y] = g([ux, uy], [vx, vy]) = [vx, vy] = v[x, y], i.e., th = u and gh = v. Propoition Let { i : L i L i, i I} be a ubet of S; then i : L i L i i I i I i I i an element of S, where the index et I i an element of U. Proof. The proof i trivial. The following reult can be obtained from the above dicuion. Propoition The category A i complete. From Propoition 2.2.2, and 2.2.4, it follow that the condition of Theorem are fulfilled and by the ue of Theorem 1.5.4, we obtain the following reult. Theorem Every object L of the category A ha an Adam cocompletion L S with repect to the et of morphim S and there exit a morphim e : L S L in S which i couniveral with repect to the morphim in S, that i, given a morphim : L L in 26

36 Chapter 2 A Categorical Contruction of Minimal Model of Lie Algebra S there exit a unique morphim t : L S L in S uch that t = e. In other word the following diagram i commutative: L S e L t L 2.3 The main reult We how that the minimal model of a 1-connected differential graded Lie algebra can be expreed a the Adam cocompletion of the d.g.l.a. with repect to the choen et of differential graded Lie algebra map. Theorem M L L S. Proof. Let e : L S L be the map a in Theorem and h : M L M be a differential graded Lie algebra map a in Theorem By the couniveral property of e there exit a d.g.l.a map θ : L S M L uch that e = hθ L S e L θ h M L By the couniveral property of h (Theorem ) there exit a differential graded Lie algebra map φ : M L L S uch that eφ = h. M L φ h e L L S 27

37 Conider the following diagram: θ L S e L 1 LS M L φ e L S We have eφθ = hθ = e. By the uniquene condition of the couniveral property of e (Theorem 2.2.5), we conclude that φθ = 1 LS. Next conider the following diagram: M L φ h L 1 ML θ L S h M L We have hθφ = eφ = h. By the uniquene condition of the property of h (Theorem ), we conclude that θφ 1 ML. Thu M L L S.

38 Chapter 3 Homotopy Approximation of Module Behera and Nanda [11] have obtained the Potnikov approximation of a 1-connected baed CW -complex with the help of a uitable et of morphim. They have obtained the aid decompoition by introducing a Serre cla C of abelian group. Thi chapter contain a Potnikov-like decompoition of a module over a ring with unity with repect to a Serre cla. The relative homotopy theory of module, including the (module) homotopy exact equence wa introduced by Peter Hilton ([33], Chapter 13). In fact he ha developed homotopy theory in module theory, parallel to the exiting homotopy theory in topology. Unlike homotopy theory in topology, there are two type of homotopy theory in module theory, the injective theory and projective theory. They are dual but not iomorphic [34 36]. Uing injective theory we have obtained, by conidering a Serre cla C of module, the Potnikov-like factorzation of a module. 3.1 Homotopy of Module The narrative may be recalled from [4, 33]. We briefly decribe ome of the concept toward notational view-point. Definition [33] Let Λ be a ring with unity. Let M be the category of Λ-module and Λ-module homomorphim. Let A and B be right Λ-module and f : A B Λ-homomorphim in the category M. The map f i i-nullhomotopic denoted f i 0, if f can be extended to ome injective module Ā containing A. Alo if g : A B then f i g if f g i 0. The i-homotopy cla of f i denoted by [f] i. Definition [18, 33] Let A and B be right Λ module and f : A B. A mapping cylinder of f i the module Ā B together with map λ : A Ā B, given by λ(a) = i(a)+f(a) where i : A Ā i the incluion and κ : Ā B B i defined by κ(ā+b) = b. Definition We now move toward a definition of upenion. Conider the hort exact equence 0 A Ā Ā/A 0 29

39 Chapter 3 Homotopy Approximation of Module where Ā i injective. We define a upenion of A, S(A) a Ā/A and S2 (A) i defined a S(A)/S(A). Then repeating thi proce we get a equence, namely, SA, S 2 A,, S n A,. Thi enable u to conider unambiguouly the group π(sa, B) or more generally π n (A, B). Notice that thee group have effectively been defined by mean of an injective reolution of A, namely A Ā SA Sn A with ucceive cokernel SA, S 2 A, S n A,. Then π n (A, B) i the nth homology group of the complex obtained by applying the functor Hom(, B) to thi reolution. we may decribe the (injective) procedure for defining Ext n (B, A) in imilar term. Hom(B, A) Hom(B, Ā) Hom(B, SA) Hom(B, Sn A) Then C n, the group of n-cochain, i Hom(B, S n A) and δ n, the coboundary operator, i the map C n C n+1 induced by i n+1 p n S n A pn S n+1 A i n+1 S n+1 A, where p n i the natural projection and i n+1 the injection. Alo, Z n, the group of n-cocycle, may be identified with Hom(B, S n A). For f : B S n A i a cocycle if and only if i n+1 p n f = 0, that i, p n f = 0. Thi mean, by exactne, that f(b) i n (S n A). Thu f may be regarded a a map B S n. Then B n, the group of n-coboundarie, will be identified with p n 1 Hom(B, S n 1 A). For, in order that f : B S n A be a coboundary, f mut equal i n p n 1 g for ome g : B S n 1 A. Thu we ee that Ext n (B, A) = Z n /B n = Hom(B, S n A)/p n 1 Hom(B, S n 1 A) where n 1. Propoition [33] Let A and B be right Λ-module. The following tatement are equivalent, for n 0 (a) S n (A) i injective. (b) Ext n+1 (B, A) = 0 for all B. (c) π n (A, B) = 0 for all B. 3.2 The category M Let U be a fixed Grothendieck univere. Let M denote the category of all Λ-module and Λ-homomorphim and let M be the correponding i-homotopy category, that i, 30

40 Chapter 3 Homotopy Approximation of Module the object of M are all Λ-module and the morphim of M are i-homotopy clae of Λ-homomorphim. We aume that the underlying et of the element of M are element of U. We fix a uitable et of morphim in uch that for any module M, M. Let S n denote the et of all map α : A B α : π m (M, A) π m (M, B) i a C-iomorphim for m n and a C-epimorphim for m = n + 1. We will how that the et of morphim S n of the category fraction. Propoition S n admit a calculu of left fraction. M admit a calculu of left Proof. Clearly S n i cloed under compoition. We hall verify condition (a) and (b) of Theorem Let βα S n and α S n where α : A B and β : B C. Since βα, α S n, for any module M in M, (βα) : π m (M, A) π m (M, C) and α : π m (M, A) π m (M, B) are C-iomorphim for m n and C-epimorphim for m = n + 1. It i to be hown that β : π m (M, B) π m (M, C) i C-iomorphim for m n and C-epimorphim for m = n + 1. Since β α and α are C-iomorphim for m n and C-epimorphim for m = n + 1, it follow that β : π m (M, B) π m (M, C) i a C-epimorphim for m = n + 1. It i enough to how that β i a C-monomorphim for m n. For any [b], [ b] π m (M, B) with β [b] = β [ b] there exit [a], [ b] π m (M, A) uch that α [a] = [b] and α [ã] = [ b], ince α i a C-iomorphim for m n; hence (βα) [a] = β α [a] = β [b] = β [ b] = β α [ã] = (βα) [ã] giving [a] = [ã] a (β α ) i a C-iomorphim for m n. Hence β S n. 31