ON ORTHOGONAL PAIRS IN CATEGORIES AND LOCALISATION

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1 ON ORTHOGONL PIRS IN CTEGORIES ND LOCLISTION Carle Caacuberta, Georg Pechke and Marku Pfenniger In memory of Frank dam 0 Introduction Special form of the following ituation are often encountered in the literature: Given a cla of Morphim M in a category C, conider the full ubcategory D of object X C uch that, for each diagram g X f with f M, there i a unique morphim h: X with hf = g. The orthogonal ubcategory problem [13] ak whether D i reflective in C, i.e., under which condition the incluion functor D C admit a left adjoint E : C D; ee [17]. Many author have given condition on the category C and the cla of morphim M enuring the reflectivity of D, ometime even providing an explicit contruction of the left adjoint E : C D; ee for example dam[1], oufield [3],[4], Deleanu-Frei-Hilton [9][10], Heller [15], Yoimura [22], Dror-Farjoun [11], Kelly [12]. The functor E i often referred to a a localiation functor of C at the ubcategory D. Mot of the known exitence reult of left adjoint work well when the category C i cocomplete [12] or complete [19]. Unfortunately, thee method cannot be directly applied to the homotopy category of CW-complexe, a it London Mathematical Society Lecture Note Serie. 175, dam Memorial Sympoium on lgebraic Topology: 1, Mancheter 1990, Edited by N. Ray and G. Walker Thi i a copy of the original article. For private ue only 1

2 i neither complete nor cocomplete. Thi difficulty i often circumvented by reorting to emi-implicial technique. In thi paper we offer a contruction of localiation functor depending only on the availability of certain weak colimit in the category C. From a technical point of view, the exitence of uch weak colimit reduce our argument eentially to the ituation in cocomplete categorie. From a practical point of view, however, our reult i a imple recipe for the explicit contruction of localiation functor. It unifie a number of contruction created for pecific purpoe; cf. [4][18],[20]. In fact, it cope goe beyond thee application: For example, it can be ued to how that there i a whole family of functor extending P -localiation of nilpotent homotopy type to the homotopy category of all CW-complexe. We deal with thi iue in [7], where we dicu the geometric ignificance of thee functor a well a their interdependence. Section 1 of the preent paper contain background followed by the tatement and proof of our main reult: the affirmative olution of the orthogonal ubcategory problem in a wide range of cae. In Section 2 we dicu extenion of a localiation functor in a category C to localiation functor in upercategorie of C. Our reult allow u to give, in Section 3, a uniform exitence proof for variou localiation functor and alo to explain their interrelation. The baic feature of our project have been outlined in [8]. cknowledgement. We are indebted to Emmanuel Dror-Farjoun, dicuion with whom ignificantly helped the preent development. We are alo grateful to the CRM of arcelona for the hopitality extended to the author. 1 Orthogonal pair and localiation functor We begin by explaining the baic categorical notion we hall ue. Our main ource are [1],[3],[4],[13]. morphim f : and an object X in a category C are aid to be orthogonal if the function f : C(, X ) C(, X ) i bijective, where C(, ) denote the et of morphim between two given object of C. For a cla of morphim M, we denote by M the cla of object orthogonal to each f M. Similarly, for a cla of object O, we denote by O the cla of morphim orthogonal to each X O. 1.1 Definition n orthogonal pair in C i a pair (S, D) coniting of a cla of morphim S and a cla of object D uch that D = S and S = D. 2

3 If (E, η) i an idempotent monad [1] in C, then the clae S = {f : Ef : E = E} D = {X η X :S = EX} form an orthogonal pair (note that thee could eaily be proper clae). The morphim in S are then called E-equivalence and the object in D are aid to be E-local. Not every orthogonal pair (S, D) arie from an idempotent monad in thi way; cf. [19]. If o, we call E the localiation functor aociated with (S, D). Then the full ubcategory of object in D i reflective and E i left adjoint to the incluion D C. The following propoition enable u to detect localiation functor. 1.2 Propoition Let C be a category and (S, D) an orthogonal pair in C. If for each object X there exit a morphim η X : X EX in S with EX in D, then (i) η X i terminal among the morphim in S with domain X; (ii) η X i initial among the morphim of C from X to an object of D; (iii) The aignment X EX define a localiation functor on C aociated with S, D). For each cla of morphim M, the pair (M, M ) i orthogonal. We ay that thi pair i generated by M and call M the aturation of M. If M = M, then M i aid to be aturated. Thi terminology applie to object a well. Note that if S, D) i an orthogonal pair then both S and D are aturated. The next propertie of aturated clae are eaily checked and well-known in a lightly more general context [3][13]. 1.3 Lemma If a cla of morphim S i aturated, then (i) S contain all iomorphim of C. (ii) If the compoition gf of two morphim i defined and any two of f, g, gf are in S, then the third i alo in S. (iii) Whenever the coproduct of a family of morphim of S exit, it i in the cla S. (iv) If the diagram i a puh-out in which S, then t S. C t D 3

4 (v) If α i an ordinal and F : α C i a directed ytem with direct limit T, uch that for each i < α the morphim i : F (0) F (i) i in S, then α : F (0) T i in S. We call a cla of morphim S cloed in C if it atifie (i), (ii) and (iii) in Lemma 1.3 above. We retrict attention to cloed clae from now on. We proceed with the tatement of our main reult. Recall that a weak colimit of a diagram i defined by requiring only exitence, without initing on uniquene, in the defining univeral property [17]. 1.4 Theorem Let C be a category with coproduct and let S be a cloed cla of morphim in C. Suppoe that: (C1) There i a et S 0 S with S 0 = S. (C2) For every diagram C f with S there exit a weak puh-out with t S. f C t Z (C3) There i an ordinal α uch that, for every β α, every directed ytem f : β C in which the morphim i : F (0) F (i) are in S for i < β admit a weak direct limit T atifying (a) the morphim β : F (0) T i in S; (b) for each : in S 0, every morphim f : T factor through f : F (i) for ome i < α; (c) if two morphim g 1, g 2 : T atify g 1 = g 2 with : in S 0, then they factor through g 1, g 2 : F (i) for ome i < α, in uch a way that g 1 = g 2. Then the cla S i aturated and the orthogonal pair (S, S ) admit a localiation functor E. Furthermore, for each object X, the localizing morphim η X : X EX can be contructed by mean of a weak direct limit indexed by α. Proof For each morphim : in S 0 fix a weak puh-out t1 Z in which t 1 S. Then alo t 2 S becaue S i cloed. 4 t 2

5 1.5 Remark With application in mind, it i worth oberving that part (c) of hypothei (C3) in Theorem 1.4 i atified if each map f : Z T factor through f : Z F (i) for ome i < α. Chooe next a morphim u : Z rendering commutative the diagram t 2 t 1 Z u Id Id and note that u S. Write D for S. We hall contruct, for each object X C, a morphim η X : X EX with EX D and η X S. Set X 0 = X. Given i < α, aume that X i ha been contructed, together with a morphim X X i belonging to S. Define a morphim σ i : X i X i+1 a follow: For each : in the et S, conider all morphim ϕ: X i and ψ : Z X i for which no factoriation through :, rep. u : Z, exit (if there are no uch morphim, then X i D and we may et EX = X i ). Chooe a weak puh-out ( S 0 ( ϕ ) ( ) ψ Z ) f φ S 0 (( ϕ ) ( ψ ) ) X i σ i X i+1 with σ i S, in which f i the coproduct morphim and φ i the correponding coproduct of copie of : and u : Z (which i therefore a morphim in S). Iterate thi procedure until reaching the ordinal α. If β α i a limit ordinal, define X β by chooing a weak direct limit of the ytem {X i, i < β}, according to (C3). Set EX = X α. The contruction guarantee that the compoite morphim η X : X EX i in S. We claim that EX D. Since D = S0, it uffice to check that EX i orthogonal to each morphim in S 0. Take a diagram f EX with S 0. Then f factor through f : X i for ome i < α and hence, either f 5

6 factor through :, or there i a commutative diagram f g X i σ i X i+1 which provide a morphim g : EX uch that g = f. Now uppoe that there are two map g 1, g 2 : EX with g 1 = g 2 = f. Then we can chooe an object X i with i < α, and morphim g 1, g 2 : X i uch that g 1 = g 2. Uing the weak puh-out property of Z, we obtain a morphim h: Z X i rendering commutative the diagram g 1 Z t 2 g 2 u h X i Then, either h factor through diagram u : Z and g 1 = g 2, or there i a commutative u Z h k X i σ i X i+1 which yield σ i g 1 = σ i ht 1 = ku t 1 = k = ku t 2 = σ i ht 2 = σ i g 2 and hence g 1 = g 2. Thi how that EX D. To complete the proof it remain to how that S = S. The incluion S S i trivial. For the convere, let f : be orthogonal to all object in D. Since η : E i in S and E D, there i a unique morphim Ef rendering commutative the diagram η f η E Ef E. ut η f i orthogonal to E and thi provide a morphim g : E E which i twoided invere to Ef. Hence Ef i an iomorphim and f S becaue S i cloed. Given an orthogonal pair (S, D), the cla S i aturated and, a fortiori, cloed. Therefore 6

7 1.6 Corollary Let C be a category with coproduct and (S, D) an orthogonal pair in C. Suppoe that ome et S 0 S generate the pair (S, D) and that the cla S atifie condition (C2) and (C3) in Theorem 1.4. Then the pair (S, D) admit a localiation functor E. Moreover, if the category C i cocomplete, then it follow from Lemma 1.3 that for each orthogonal pair (S, D) condition (C2) and part (a) of condition (C3) are automatically atified. Thi lead to Corollary 1.7 below. n object X ha been called preentable [14] or -definite [3] if, for ome ufficiently large ordinal α, the functor C(X, ) preerve direct limit of directed ytem F : α C. For example, all group are preentable [3]. For finitely preented group it uffice to take α to be the firt infinite ordinal. 1.7 Corollary [3] Let C be a cocomplete category. Let (S, D) be the orthogonal pair generated by an arbitrary et S of morphim of C. Suppoe that the domain and codomain of morphim in S 0 are preentable. Then (S, D) admit a localiation functor. Since any colimit of preentable object i again preentable, the following definition together with the reult of [19] imply Corollary 1.9 below. 1.8 Definition et {E α } of object of a category C i a cogenerator et of C if any morphim f : X Y of C inducing bijection f : C(E α, X) = C(E α, Y ) for each α, i an iomorphim. 1.9 Corollary Let C be a cocomplete category. Suppoe that C ha a cogenerator et whoe element are preentable. Then any orthogonal pair generated by an arbitrary et of morphim of C admit a localiation functor. 2 Extending localiation functor Let E be a localiation functor on the ubcategory C of C. We wih to dicu extenion of E over C. Familiar example include the extenion of P -localiation of abelian group to nilpotent group and further to all group. Two problem arie here: exitence for which we often refer to Theorem 1.4 and uniquene. n appropriate etting for dicuing the latter i obtained by partially ordering the collection of all orthogonal pair in C a follow: For two given orthogonal pair (S 1, D 1 ), (S 2, D 2 ) in C we write (S 1, D 1 ) (S 2, D 2 ) if D 1 D 2 (or, equivalently, if S 1 S 2 ). 7

8 2.1 Remark If E 1, E 2 are localiation functor aociated to (S 1, D 1 ) and (S 2, D 2 ) repectively, and if (S 1, D 1 ) (S 2, D 2 ), then there i a natural tranformation of functor E 1 E 2. In fact, the retriction of E 2 to D 1 i left adjoint to the incluion D 2 D 1. n orthogonal pair (S, D) of C i aid to extend the orthogonal pair (S, D ) of the ubcategory C if both S S and D D. The collection of all extenion of (S, D ) i partially ordered. Moreover we have 2.2 Propoition Let C be a ubcategory of C and (S, D ) an orthogonal pair in C. If (S, D) i an extenion of (S, D ) to C, then where orthogonality i meant in C. ((S ), (S ) ) (S, D) ((D ), (D ) ), In thi ituation, we call the orthogonal pair in C generated by the cla S the maximal extenion of (S, D ), and the one generated by D the minimal extenion. convenient tool for recogniing uch extremal extenion i given in the next propoition. 2.3 Propoition Let C be a ubcategory of C, (S, D ) an extenion of (S, D ) to C. Then (i) (S, D) i the maximal extenion of (S, D ) if and only if there i a ubcla S 0 S uch that S 0 D. (ii) (S, D) i the minimal extenion of (S, D ) if and only if there i a ubcla D 0 D uch that D 0 S. Of coure (S, D ) admit a unique extenion to C if and only if the minimal and the maximal extenion coincide. 2.4 Example Let C be the category of finite group and C the ubcategory of finite nilpotent group. Fix a prime p and conider the orthogonal pair (S, D ) in C aociated to p-localiation [16]. The cla D conit of all p-group, and the orthogonal pair (S, D) = ((D ), D ) in C i both the maximal and the minimal extenion of (S, D ) to C. The pair (S, D) admit a localiation functor namely, mapping each finite group G onto it maximal pquotient, which i therefore the unique extenion to all finite group of the p-localiation of finite nilpotent group. 8

9 3 pplication of the baic exitence reult Example 3.1, 3.2 and 3.3 below dicu well-known functor, each of whoe contruction may be viewed a particular cae of Theorem 1.4. Example 3.4 and 3.7 are new. 3.1 Example Let H be the pointed homotopy category of imply-connected CWcomplexe, and P a et of prime. The P -localiation functor decribed by Sullivan [21] i aociated to the orthogonal pair (S, D) generated by the et S 0 = {ρ k n : S k S k degρ k n = n, k 2, n P }, where P denote the et of prime not in P. Object in D are imply connected CWcomplexe whoe homotopy group are Z P -module. Morphim in S are H ( ; Z P )- equivalence. The hypothee of Corollary 1.6 are fulfilled by taking α to be the firt infinite ordinal and uing homotopy colimit. 3.2 Example Let H denote the pointed homotopy category of connected CW-complexe and h an additive homology theory. Take S to be the cla of morphim f : X Y inducing an iomorphim f : h (X) = h (Y ). We know from [4] that S atifie the hypothee of Theorem 1.4: Chooe α to be the mallet infinite ordinal whoe cardinality i bigger than the cardinality of h (pt); the collection of all CW-incluion ϕ: with h (ϕ) = 0 and card() < card(α) repreent a et S 0 with S0 = S. In the cae h = H ( ; Z P ), the correponding orthogonal pair (S, D) extend the pair (S, D ) aociated with P -localiation of nilpotent pace (ee [4]). It i indeed the minimal extenion of (S, D ), becaue he pace K(Z P, n), n 1, belong to D (cf. Propoition 2.2). 3.3 Example Let G be the category of group and P a et of prime. The P - localiation functor decribed by Ribenboim [20] i aociated to the orthogonal pair (S, D) generated by the et S 0 = {ρ n : Z Z ρ n (1) = n, n P }. Group in D are thoe in which P -root exit and are unique. Such group have been tudied for everal decade (ee [2][20] and the reference there). The hypothee of Theorem 1.4 are readily checked (ue Corollary 1.7). We may chooe α to be firt infinite ordinal. We denote by l : G G P the P -localiation homomorphim. If (S, D ) i the orthogonal pair correponding to P -localiation of nilpotent group, then, ince S 0 S, Propoition 2.2 implie that (S, D) i the maximal extenion of (S, D ). In particular, for each group G there i a natural homomorphim from G P to the oufield HZ P -localiation of G (cf. [5]. 9

10 3.4 Example Example 3.3 can be generalied to the category C of π-group for a fixed group π; that i, object are group with a π-action and morphim are action-preerving group homomorphim. Let F (ξ) be the free π-group on one generator (it can be explicitly decribed a the free group on the ymbol ξ x, x π, with the obviou left π-action; cf. [18]. Define a π-homomorphim ρ n,x : F (ξ) F (ξ) for each x π, n Z, by the rule and conider the et morphim ρ n,x (ξ) = ξ(x ξ)(x 2 ξ)... (x n 1 ξ) S 0 = {ρ n,x :F (ξ) F (ξ) x π, n P }. y Corollary 1.7, the orthogonal pair (S, D) generated by S 0 admit a localiation functor. It again uffice to take the firt infinite ordinal a α in the contruction. Example 3.3 i the pecial cae π = {1}. We extend the term P -local to the π-group in D and term P -equivalence to the morphim in S. They are particularly relevant to the next example. 3.5 Example Thi example i extracted from [7]. Let H be the pointed homotopy category of connected CW-complexe and P a et of prime. We conider the cla D of thoe pace X in H for which the power map ρ n : ΩX ΩX, ρ n (ω) = ω n i a homotopy equivalence for all n P. Then there exit a et of morphim S 0 uch that S 0 = D, namely S 0 = {ρ k n : S 1 (S k pt) S 1 (S k pt) K 0, n P }, where ρ k n = ρ n Id, ρ n : S 1 S 1 denote the tandard map of degree n, and pt denote a dijoint baepoint. Morphim in S = D turn out to be thoe f : X Y for which f : π 1 (X) π 1 (Y ) i a P -equivalence of group and f : H (X; ) H (Y ; ) i an iomorphim for each abelian π 1 (Y ) P -group which i P -local in the ene of Example 3.4. The condition of Corollary 1.6 are atified. One can take α to be the firt infinite ordinal. Space in D will be called P -local and map in S will be called P -equivalence. We denote the P -localiation map by l : X X P. The pair (S, D) extend the pair (S, D ) correponding to P -localiation of nilpotent pace. Since the orthogonal pair correponding to H ( ; Z P )-localiation i minimal among thoe pair extending P -localiation of nilpotent pace (ee Example 3.2), for each pace X there i a natural map from X P to the H ( ; Z P )-localiation of X. 3.6 Example Let H denote the pointed homotopy category of connected CW-complexe and P a et of prime. Conider the orthogonal pair (S, D) generated by the et S 0 = {ρ k n : S k S k degρ k n = n, k 1, n P }. 10

11 The cla D conit of pace whoe homotopy group are P -local, and one find, with the ame method a in [7],[9], that S conit of morphim f : X Y uch that f : π 1 (X) π 1 (Y ) i a P -equivalence of group and f : H k (Y ; ) H k (X; ) i an iomorphim for k 2 and every Z P [π 1 (Y ) P ]-module. Thi cla S i not cloed under homotopy colimit, becaue the natural map from S 1 to K(Z P, 1), which i the homotopy colimit of a certain direct ytem of map ρ 1 n, n P, fail to induce an iomorphim in H 2 with coefficient in the group ring Z P [Z P ], and hence doe not belong to S. Thu, Corollary 1.6 doe not apply in thi cae. In fact, the orthogonal pair (S, D) doe not admit a localiation functor [7]. On the other hand, if we delete from S 0 the map ρ 1 n, n P, then the reulting cla D conit of pace whoe higher homotopy group are P -local, and S conit of morphim f : X Y inducing an iomorphim of fundamental group and uch that f : H k (Y ; ) H k (X; ) i an iomorphim for all k and every Z P [π 1 (Y )]-module. Thi orthogonal pair (S, D) i the maximal extenion to H of the pair decribed in Example 3.1. Now Corollary 1.6 provide a localiation functor aociated to (S, D). Thi functor induce an iomorphim of fundamental group and P -localie the higher homotopy group, i.e., correpond to fibrewie localiation with repect to the univeral covering fibration X X K(π 1 (X), 1). 3.7 Example Fix a group G and let H(G) be the category whoe object are map X K(G, 1) in H and whoe morphim are homotopy commutative triangle. Given an abelian G-group, let S() be the cla of morphim f uch that f : H (X; ) H (H; ) i an iomorphim. Then S() atifie the condition of Theorem 1.4. Example 3.2 correpond to the particular cae G = {1}. In [7] we how that everal idempotent functor on H extending P -localiation of nilpotent pace can be obtained by plicing localiation functor with repect to twited homology in a uitable way. In fact, Example 3.5 can be alternatively obtained a a pecial cae of thi procedure. Reference [1] J.F. dam. Localiation and Completion. Univ. of Chicago Pre , 2, 3 [2] G. aumlag. Some pect of Group with Unique Root. cta Math. 104 (1960) [3].K. oufield. Contruction of factorization ytem in categorie. J. Pure ppl. lgebra 9 (1977) , 2, 3, 7 [4].K. oufield. The localization of pace with repect to homology. Topology 14 no. 2 (1975) , 2, 9 11

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