A ATEGORIAL ONSTRUTION OF MINIMAL MODEL A. Behera, S. B. houdhury M. Routaray Department of Mathematic National Intitute of Technology ROURKELA - 769008 (India) abehera@nitrkl.ac.in 512ma6009@nitrkl.ac.in 512ma302@nitrkl.ac.in Abtract - Under a reaonable aumption, the minimal model of a 1-connected d.g.a. can be expreed a the Adam cocompletion of the d.g.a. with repect to a choen et of d.g.a.-map. MS : 55 P 60, 18 A 40 Keyword - ategory of fraction, calculu of right fraction, Grothendieck univere, Adam cocompletion, differential graded algebra, minimal model. I. INTRODUTION It i to be emphaized that many algebraic geometrical contruction in Algebraic Topology, Differential Topology, Differentiable Manifold, Algebra, Analyi, Topology, etc., can be viewed a Adam completion or cocompletion of object in uitable categorie, with repect to carefully choen et of morphim. 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 generalized homology (or cohomology) theorie. Subequently, thi notion ha been conidered in a more general framework by Deleanu, Frei Hilton [4], where an arbitrary category 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. The central idea of thi note i to invetigate a cae howing how uch an algebraic geometrical contruction i characterized in term of Adam cocompletion. II. ADAMS OOMPLETION We recall the definition of Grothendieck univere, category of fraction, calculu of right fraction, Adam cocompletion ome characterization of Adam cocompletion. 2.1 Definition. ([11], 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 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 Y are element of U, then {X, Y}, the ordered pair (X, Y) X Y are element of U. We fix a univere U that contain N the et of natural number ( o Z, Q, R, ).
2.2 Definition. ([11], p. 267) A category i aid to be a mall U-category, U being a fixed Grothendeick univere, if the following condition hold: S(1) : The object of form a et which i an element of U. S(2) : For each pair (X, Y) of object of, the et Hom(X, Y) i an element of U. 2.3 Definition. ([11], p. 269) Let be any arbitrary category S a et of morphim of. A category of fraction of with repect to S i a category denoted by [S 1 ] together with a functor F S [S 1 ] having the following propertie: F(1) : For each S, F S () i an iomorphim in [ S 1 ]. F(2) : F S i univeral with repect to thi property: If G D i a functor uch that G() i an iomorphim in D, for each S, then there exit a unique functor H [ S 1 ] D uch that G = HF S. Thu we have the following commutative diagram: F S [S 1 ] G H D 2.4 Note. For the explicit contruction of the category [ S 1 ], we refer to [11]. We content ourelve merely with the obervation that the object of [ S 1 ] are ame a thoe of in the cae when S admit a calculu of left (right) fraction, the category [ S 1 ] can be decribed very nicely [7, 11]. 2.5 Definition. ([11], p. 267) A family S of morphim in a category i aid to admit a calculu of right fraction if (a) any diagram Z X Y in with S can be completed to a diagram with t S ft = g, W t X g Z Y (b) given W t X f Y S Z g with S f = g, there i a morphim t W X in S uch that ft = gt. A imple characterization for a family S to admit a calculu of right fraction i the following.
2.6 Theorem. ([4], Theorem1.3, p. 70) Let S be a cloed family of morphim of atifying (a) if vu S v S, then u S, (b) any diagram in with S, can be embedded in a weak pull-back diagram g t with t S. Then S admit a calculu of right fraction. 2.7 Remark. There are ome et-theoretic difficultie in contructing the category [S 1 ]; thee difficultie may be overcome by making ome mild hypothee uing Grothendeick univere. Preciely peaking, the main logical difficulty involved in the contruction of a category of fraction it ue, arie from the fact that if the category belong to a particular univere, the category [S 1 ] would, in general belong to a higher univere ([11], Propoition 19.1.2). 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 [5]. Alo the following theorem how that if S admit a calculu of left (right) fraction, then the category of fraction [S 1 ] remain within the ame univere a to the univere to which the category belong. 2.8 Theorem. ([10], Propoition, p. 425) Let be a mall U-category S a et of morphim of that admit a calculu of left (right) fraction. Then [S 1 ] i a mall U-category. 2.9 Definition. [4] Let be an arbitrary category S a et of morphim of. Let [S 1 ] denote the category of fraction of with repect S F: [S 1 ] be the canonical functor. Let S denote the category of et function. Then for a given object Y of, [S 1 ](Y, ) S define a covariant functor. If thi functor i repreentable by an object Y S of, [S 1 ](Y, ) (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 [4]. We recall ome reult on the exitence of Adam cocompletion. We tate Deleanu theorem [5] that under certain condition, global Adam cocompletion of an object alway exit. 2.10 Theorem. Let be a complete mall U-category (U i a fixed Grothendeick univere) S a et of morphim of that admit a calculu of right fraction. Suppoe that the following compatibility condition with product i atified. () If each i X i Y i, i I, i an element i an element of U, then i I i i I X i i I Y i
i an element of S. Then every object X of ha an Adam cocompletion X S with repect to the et of morphim S. The concept of Adam cocompletion can be characterized in term of a couniveral property. 2.11 Definition. [4] Given a et S of morphim of, we define S, the aturation of S a the et of all morphim u in uch that F(u) i an iomorphim in [S 1 ]. S i aid to be aturated if S = S. 2.12 Propoition. ([4], Propoition 1.1, p. 63) A family S of morphim of i aturated if only if there exit a factor F D uch that S i the collection of morphim f uch that Ff i invertible. Deleanu, Frei Hilton have hown that if the et of morphim S i aturated then the Adam cocompletion of a pace i characterized by a certain couniveral property. 2.13 Theorem. ([4], Theorem 1.4, p. 68) Let S be a aturated family of morphim of admitting a calculu of right fraction. Then an object Y S of i the S-cocompletion of the object Y with repect to S if 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 S Y ha to be in S; thi i the cae when S i aturated the reult i a follow: 2.14 Theorem. ([4], dual of Theorem 2.9, p. 76) Let S be a aturated family of morphim of let every object of admit an S-cocompletion. Then the morphim e Y S Y belong to S i univeral for morphim to S-cocomplete object couniveral for morphim in S. III. THE ATEGORY DGA Let DGA be the category of 1-connected differential graded algebra over Q (in hort d.g.a. ) d.g.a.-homomorphim. Let S denote the et of all d.g.a.-epimorphim in DGA which induce homology iomorphim in all dimenion. The following reult will be required in the equel. 3.1 Propoition. S i aturated. Proof. The proof i evident from Propoition 2.12. 3.2 Propoition. S admit a calculu of right fraction. Proof. learly, S i a cloed family of morphim of the category DGA. We hall verify condition (a) (b) of Theorem 2.6. Let u, v S. We how that if vu S v S, then u S. learly u i an
epimorphim. We have (vu) = v u v are both homology iomorphim implying u i a homology iomorphim. Thu u S. Hence condition (a) of Theorem 2.6 hold. To prove condition (b) of Theorem 2.6 conider the diagram A B in DGA with S. We aert that the above diagram can be completed to a weak pull-back diagram D g t A B in DGA with S. Since A, B are in DGA we write A = Σ A n, B = Σ B n, = Σ n, are d.g.a.-homomorphim. Let A n n. We have to how that i a differential graded algebra. Let f = Σ f n, = Σ n f n A n B n, n n B n, D n = {(a, c) A n n f n (a) = n (c)} D = Σ D n t n D n A n g n D n n be the uual projection. Let t = Σ t n g = Σ g n. learly the above diagram i commutative. It i required to how that D i a d.g.a.. Define a multiplication in D in following way: (a, c) (a, c ) = (aa, cc ), where (a, c) D n, (a, c ) D m. Let Define by the rule Let Since d A = Σ d n A, d n A A n A n+1 d = Σ d n, d n n n+1. d n D D n D n+1 d n D (a, c) = (d n A (a), d n (c)), (a, c) D n. d D = Σ d n D. d D d D (a, c) = (d A d A (a), d d (c)) = (0,0) for all (a, c) D we have that d D i a differential. Next we how that d D i a derivation: For (a 1, c 1 ) D n (a 2, c 2 ) D m,
d D ((a 1, c 1 ) (a 2, c 2 )) = d D (a 1 a 2, c 1 c 2 ) = (d A (a 1 a 2 ), d (c 1 c 2 )) = (d A (a 1 ) (a 2 ) + ( 1) n (a 1 ) d A (a 2 ), d (c 1 ) (c 2 ) + ( 1) n (c 1 ) d (c 2 )) = (d A (a 1 ) a 2, d (c 1 ) c 2 ) +(( 1) n a 1 d A (a 2 ), ( 1) n c 1 d (c 2 )) = (d A (a 1 ), d (c 1 )) (a 2, c 2 ) + (( 1) n a 1, ( 1) n c 1 ) (d A (a 2 ), d (c 2 )) = d D (a 1, c 1 ) (a 2, c 2 ) + ( 1) n (a 1, c 1 ) d D (a 2, c 2 ). Thu D become a d.g.a.. We how that D i 1-connected, H 0 (D) Q H 1 (D) 0. We have H 0 (D) = Z 0 (D) B 0 (D) = Z 0 (D) = {(a, c) Z 0 (A) Z 0 () f 0 (a) = 0 (c)}. Let 1 A A 1. Then d D (1 A, 1 ) = (d A (1 A ), d (1 )) = 0 implie that (1 A, 1 ) Z 0 (D). H 0 (A) = Z 0 (A) Q implie that Z 0 (A) = Q1 A. Similarly, H 0 () = Z 0 () Q implie that Z 0 () = Q1. Thu (a, c) H 0 (D) = Z 0 (D) Z 0 (A) Z 0 () if only if a = r1 A c = r1 for ome r Q. Thu H 0 (D) Q. In order to how H 1 (D) 0, let (a, c) Z 1 (D). Thi implie that a Z 1 (A), c Z 1 () f 1 (a) = 1 (c). Since A i 1-connected we have H 1 (A) 0, Z 1 (A) B 1 (A) = B 1 (A); hence a = d A 0 (a ), a A 0. Similarly ince i 1-connected we have H 1 () 0, Z 1 () B 1 () = B 1 (); hence c = d 0 (c ), c 0. Now f 1 (a) = 1 (c), f 1 (d A 0 (a )) = 1 (d 0 (c )). Thi give d B 0 f 0 (a ) = d B 0 0 (c ), d B 0 (f 0 (a ) 0 (c )) = 0. Thu f 0 (a ) 0 (c ) Z 0 (B). But S. Hence H 0 () H 0 (B) i an iomorphim, 0 Z 0 () Z 0 (B) i an iomorphim. Hence there exit an element c Z 0 () uch that
0 (c ) = f 0 (a ) 0 (c ). Moreover d 0 D (a, c + c ) = (d 0 A (a ), d 0 (c ) + d 0 (c )) = (d 0 A (a ), 0 + d 0 (c )) = (d 0 A (a ), d 0 (c )) = (a, c) howing that (a, c) B 1 (D). Thu H 1 (D) 0. learly t i a d.g.a.-epimorphim. We how that t H (D) H (A) i an iomorphim. Firt we how that t H (D) H (A) i a monomorphim. The hollowing commutative diagram will be ued in the equel. A n 2 f n 2 B n 2 n 2 n 2 A B d n 2 d n 2 d n 2 A n 1 B n 1 f n 1 n 1 n 1 A B d n 1 d n 1 d n 1 A n f n B n n n Since t n D n A n i the uual projection we have t n (a, c) = a for every (a, c) D n. Hence the algebra homomorphim t H n (D) H n (A) i given by t [(a, c)] = [t n (a, c)] = [a] for [(a, c)] H n (D). We note that H n (D) Hence = Z n (D) B n (D) (Z n (A) Z n () ) (B n (A) B n ()). H n (D) = (Z n (A ) Z n ( )) (B n (A ) B n ( )) for ome A n A n n n. For any [(a, c)] H n (D) we have [(a, c)] = (a, c) + B n (D) = (a, c) + (B n (A ) B n ( )), (a, c) Z n (D) D n. Then (a, c) + d D n 1 (a, c ) (a, c) + B n (D), for every d D n 1 (a, c ) B n (D) where (a, c ) D n 1 D n, (a, c) + d D n 1 (a, c ) = (a, c) + (d A ), d n 1 (c )) (a, c) + (B n (A ) B n ( )),
for every d D n 1 (a, c ) = (d A n 1 (a ), d A n 1 (c )) Thu B n (A ) B n ( ). (a + d A ), c + d n 1 (c )) (a, c) + (B n (A ) B n ( )), = [(a, c)] H n (D). We note that Now let aume that thi give Since We have So Therefore, from the above give thi give [(a + d A ), c + d n 1 (c )] Since i an iomorphim we have Hence we have [a] = [a + d A n 1 (a ) ] [c] = [c + d n 1 (c )]. [(a 1, c 1 )], [(a 2, c 2 )] H n (D) t [(a 1, c 1 )] = t [(a 2, c 2 )]; [a 1 ] = [a 2 ], [a 1 + d A n 1 (a )] = [a 2 + d A n 1 (a )]. (a 1, c 1 ), (a 2, c 2 ), (d A n 1 f n (a 1 ) = n (c 1 ), f n (a 2 ) = n (c 2 ) (a ), d n 1 (c )) D n f n d A ) = n d n 1 (c ). f n (a 1 + d A )) = n (c 1 + d n 1 (c )) f n (a 2 + d A )) = n (c 2 + d n 1 (c )). t [(a 1, c 1 )] = t [(a 2, c 2 )] f [a 1 + d A n 1 (a )] = f [a 1 + d A n 1 (a )] [f n (a 1 + d A n 1 (a ))] = [f n (a 2 + d A n 1 (a ))]; [ n (c 1 + d n 1 (c ))] = [ n (c 2 + d n 1 (c ))] [c 1 + d n 1 = ([a 2 + d A )], [c 2 + d n 1 (c )]); (c )] = [c 2 + d n 1 (c )]. [c 1 + d n 1 (c )] = [c 2 + d n 1 (c )]. ([a 1 + d A )], [c 1 + d n 1 (c )]) we apply the iomorphim α (Z n (A ) B n (A )) (Z n ( ) B n ( )) (Z n (A ) Z n ( )) (B n (A ) B n ( )) to the above to get = α ([a 2 + d A )], [c 2 + d n 1 (c )]), α ([a 1 + d A )], [c 1 + d n 1 (c )])
Thu [(a 1 + d A ), c 1 + d n 1 (c ))] = [(a 2 + d A ), c 2 + d n 1 (c ))]. howing that i a monomorphim. [(a 1, c 1 )] = [(a 2, c 2 )], t H (D) H (A) Next we how that t H (D) H (A) i an epimorphim. Let [a] H (A) be arbitrary. Then f n (a) B n. Since i an epimorphim f n (a) = n (c) for ome c n. Hence (a, c) D n. Then t [(a, c)] = [t n (a, c)] = [a] howing t i an epimorphim. Since t i an epimorphim t i an iomorphim we conclude that t S. Next for any d.g.a. E = Σ E n d.g.a.-homomorphim u = {u n E n A n } v = {v n E n n } in DGA let the following diagram E v u A B commute, fu = v. onider the diagram E v h D g u t A B Define h = {h n E n D n } by the rule h(x) = (u(x), v(x)) for x E. learly h i well defined i a d.g.a. homomorhim. Now for any x E, th(x) = t(u(x), v(x)) = u(x) gh(x) = g(u(x), v(x)) = v(x), th = u gh = v. Thi complete the proof of Propoition 3.2. 3.3 Propoition. If each i A i B 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 I i i I A i i I B i Proof. The proof i trivial. The following reult can be obtained from the above dicuion. 3.4 Propoition. The category DGA i complete.
From Propoition 3.3-3.4, it follow that the condition of Theorem 2.10 are fulfilled by the ue of Theorem 2.13, we obtain the following reult. 3.5 Theorem. Every object A of the category DGA ha an Adam cocompletion A S with repect to the et of morphim S there exit a morphim e A S A in S which i couniveral with repect to the morphim in S, that i, given a morphim B A in S there exit a unique morphim t A S B uch that t = e. In other word the following diagram i commutative: A S e A t B We recall the following algebraic preliminarie. IV. MINIMAL MODEL 4.1 Definition. [6, 12] A d.g.a. M i called a minimal algebra if it atifie the following propertie: (a) M i free a a graded algebra. (b) M ha decompoable differential. (c) M 0 = Q, M 1 = 0. (d) M ha homology of finite type, for each n, H n (M) i a finite dimenional vector pace. Let M be the full ubcategory of the category DGA coniting of all minimal algebra all d.g.a.-map between them. 4.2 Definition. [6, 12] Let A be a imply connected d.g.a.. A d.g.a. M = M A i called a minimal model of A if the following condition hold: (i) M M. (ii) There i a d.g.a.-map ρ M A A which induce an iomorphim on homology, ρ H (M) H (A). Henceforth we aume that the d.g.a.-map ρ M A A i a d.g.a.-epimorphim. 4.3 Theorem. (Theorem 2.13, p. 48 [6]; Lemma 2, p. 38 Theorem 2, p. 45 [12]) Let A be a imply connected d.g.a. M A be it minimal model. The map ρ M A A ha couniveral property, for any d.g.a. Z d.g.a.-map φ Z A, there exit a d.g.a.-map θ M A Z uch that ρ φθ; furthermore if the d.g.a.-map φ Z A i an epimorphim then ρ = φθ, the following diagram i commutative:
M A ρ A θ φ Z V. THE RESULT We how that under a reaonable aumption, the minimal model of a 1-connected d.g.a. can be expreed a the Adam cocompletion of the d.g.a. with repect to the choen et of d.g.a.-map. 5.1 Theorem. M A A S. Proof. Let e A S A be the map a in Theorem 3.5 ρ M A A be the d.g.a.-map a in Theorem 4.3. Since the d.g.a.-map ρ M A A i a d.g.a.-epimorphim, by the couniveral property of e there exit a d.g.a.- map θ A S M A uch that e = ρθ. A S e A θ ρ By the couniveral property of ρ there exit a d.g.a.-map φ M A A S uch that eφ = ρ. ρ M A A φ e M A onider the following diagram A S e A θ 1 AS M A e φ A S A S Thu we have eφθ = ρθ = e. By the uniquene condition of the couniveral property of e (Theorem 3.5), we conclude that φθ = 1 AS. Next conider the diagram
M A ρ A φ 1 MA A S ρ θ M A Thu we have ρθφ = eφ = ρ. By the couniveral oroperty of ρ (Theorem 4.3), we conclude that θφ = 1 MA. Thu M A A S. Thi complete the proof of Theorem 5.1. REFERENES [1] J.F. Adam, Idempotent Functor in Homotopy Theory, : Manifold, onf. Univ. of Tokyo Pre, Tokyo, 1973. [2] J.F. Adam, Localization ompletion Lecture Note in Mathematic, Univ. of hicago, 1975. [3] J.F. Adam, Localization ompletion, Lecture Note in Mathematic, Univ. of hicago, 1975. [4] A. Deleanu, A. Frei P.J. Hilton, Generalized Adam completion, ahier de Top. et Geom. Diff., vol. 15, pp. 61-82, 1974. [5] A. Deleanu, Exitence of the Adam completion for object of cocomplete categorie J. Pure Appl. Alg. vol. 6, pp. 31 39, 1975. [6] A.J. Dechner, Sullivan Theory of Minimal Model Thei, Univ. of Britih olumbia, 1976. [7] P. Gabriel M. Ziman, alculu of Fraction Homotopy Theory Springer-Verlag, New York, 1967. [8] S. Halperin, Lecture on Minimal Model, Publ. U. E. R. de Mathe matique, Univ. de Lille I, 1977, 1981. [9] S. Mac Lane, ategorie for the working Mathematician, Springer-Verlag, New York, 1971. [10] S. Na, A note on the univere of a category of fraction, anad. Math. Bull., vol. 23(4), pp. 425 427, 1980. [11] H. Schubert, ategorie, Springer Verlag, New York, 1972. [12] Wu wen-tun, Rational Homotopy Type, LNM 1264, Springer-Verlag, Berlin,1980.