On the derived category of an algebra over an operad

ON THE DERIVED CA TEGOR Y OF AN ALGEBRA O VER AN OPERAD CLEMENS BERG ER AND IEKE MOERD IJK Abstract. W e presen t a general construction of the derived catego ry of an algebra ov er an operad and establish its inv ariance prop erties. A central role is play ed b y the en v eloping operad of an alge bra ov er an operad. Introduction It is a classica l device in homolo gical algebra to ass o ciate to an as so ciative r ing R t he homotop y category of different ial gra ded R -mo dules , the so-called deriv e d c a te gory D ( R ) o f R . One of the imp orta nt issues is to know when two rings hav e equiv a lent deriv ed categories; positive answers to this question may b e obtained b y means of the theo ry of tilting complexes, which is a kind of deriv ed Mo rita theory , cf. Rick a rd [1 5], Keller [10], Sc h w ede [1 6], T o¨ en [19]. In this pap er, we pr ovide a solution to the problem o f giving a suitable constructio n of the derived ca tegory asso ciated to an a lgebra over an op erad in a no n-additive context. Building on earlier w ork of ours’ (cf. [2, 3] and the Appendix to this pap er) we establish general inv a riance prop erties of this derived categor y under c hange of alg ebra, change of op erad and c hange of ambien t categor y . In the sp ecia l case of the op erad fo r differential g raded algebras , our constructio n agrees with the classical one. A central role in our pro ofs is pla y ed by the envelo ping op er ad P A of A whose algebras are the P -algebra s under A . Indeed, the monoid of unary op er a tions P A (1) may b e ident ified with the enveloping algeb r a Env P ( A ) of A . The latter has the characteristic pr o p erty that Env P ( A )-modules (in the classical sense) corresp o nd to A -mo dules (in the op era dic sense). This indirect construction of the env eloping algebra o cc ur s in sp ecific cases at several places in the liter ature (cf. Getzler -Jones [6], Ginzburg-Kapra nov [7], F ress e [4, 5], Spitzwec k [18], v a n der Laan [20], B a sterra - Mandell [1 ]). The main p oint in the use of the env eloping op era d P A rather than the en v eloping algebr a P A (1) is that the assignment ( P , A ) 7→ P A extends to a left adjoint functor whic h on a dmissible Σ-cofibr a nt opera ds P and cofibr ant P -algebras A be hav es like a left Quillen functor on cofibrant ob jects. It is prec is ely this go o d homotopical behaviour that allows the de finitio n of the der ived categor y D P ( A ) for any P -alg ebra A . A cknow le dgements : W e ar e grateful to B. J ahren, K. Hess and B. Oliver, the o r- ganizing committee of the Algebr aic T op olog y Semester 2 006 of the Mittag-Leffler- Institut in Sto ckholm. Most o f this work has b een car ried out during the author s’ visit at the MLI in Spring 20 06. W e are also grateful to B. F resse for helpful comments o n an earlier version of this paper. Date : 16 Janu ary 2008. 1991 Mathema tics Subje ct Classific ation. Pri mary 18D50; Secondary 18G55, 55U35. 1 2 CLEMENS BERGER AND IEKE MOERDIJK 1. E nveloping o perads and envelo ping a l gebras Let E be a (bicomplete) closed s ymmetric monoida l category . W e write I for the unit, − ⊗ − for the monoidal structure and Hom E ( − , − ) for the internal hom of E . This section is a recollection of k nown results on categories of modules over a P -algebra A , wher e P is any s ymmetric opera d in E . The main ob jective of this section is to fix the notations and definitions we use . The category of A -modules is a mo dule categ ory in the classica l sens e for a suita ble monoid in E , the so-called enveloping algebr a o f A . W e will explain in detail that the en v eloping a lgebra of A is isomor phic to the monoid of unary op er ations P A (1) of the so- called enveloping op er ad of the pair ( P , A ). The env eloping op erad P A is c haracter ised b y a universal prop erty which implies in par ticular that P A -algebra s are the P -alg ebras under A . Definition 1.1. L et A b e a P -algebr a in E . An A -mo dule (u nder P ) c onsists of an obje ct M of E to gether with action ma ps µ n,k : P ( n ) ⊗ A ⊗ k − 1 ⊗ M ⊗ A ⊗ n − k → M , 1 ≤ k ≤ n , subje ct to the fol lowing thr e e axioms: (1) (Un it axiom) Th e op er ad-unit I → P (1) induc es a c ommutative triangle I ⊗ M ∼ = ✲ M P (1) ⊗ M ❄ µ 1 , 1 ✲ (2) (As s o ciativity axiom) F or e ach n = n 1 + · · · + n s ≥ 1 , the fol lowing diagr am c o mmutes: P ( s ) ⊗ P ( n 1 ) ⊗ · · · ⊗ P ( n s ) ⊗ A ⊗ k − 1 ⊗ M ⊗ A ⊗ n − k β ✲ P ( s ) ⊗ A ⊗ l − 1 ⊗ M ⊗ A ⊗ s − l P ( n ) ⊗ A ⊗ k − 1 ⊗ M ⊗ A ⊗ n − k α ❄ µ n,k ✲ M µ s,l ❄ wher e α is induc e d by the op er ad structure of P , and β is induc e d by the P -algebr a structu r e of A and the A -mo dule struct u r e o f M ; in p articular, l is the un ique natur al numb er such that n 1 + · · · + n l − 1 < k ≤ n 1 + · · · + n l . (3) (Equivarianc e axiom) F or e ach n ≥ 1 , the µ n,k induc e a total action µ n : P ( n ) ⊗ Σ n ( k = n a k =1 A ⊗ k − 1 ⊗ M ⊗ A ⊗ n − k ) ✲ M wher e t he symmetric gr oup Σ n acts on the c opr o duct by p ermuting factors. A morphism f : M → N of A -mo dules under P is a morphism in E r enderi ng c ommutative al l diagr ams of t he form P ( n ) ⊗ A ⊗ k − 1 ⊗ M ⊗ A ⊗ n − k id ⊗ id ⊗ f ⊗ i d ✲ P ( n ) ⊗ A ⊗ k − 1 ⊗ N ⊗ A ⊗ n − k M µ M n,k ❄ f ✲ N . µ N n,k ❄ ON THE DERIVED CA TEGOR Y OF AN ALGEBRA OVER AN O P ERAD 3 The categor y of A -modules under P will be deno ted by Mod P ( A ); the forg etful functor ( M , µ ) 7→ M will be denoted by U A : Mo d P ( A ) → E . R emark 1.2 . The pairs ( A, M ) co nsisting of a P -algebra A and an A -module M define a categor y with morphisms the pa irs ( φ, ψ ) : ( A, M ) → ( B , N ) consisting of a ma p of P -algebr as φ : A → B and a map of A -modules ψ : M → φ ∗ ( N ). This category ca n b e iden tified with the full s ubc ategory o f left P -mo dules co ncentrated in degr ees 0 and 1 . In o rder to make this more explicit, recall that a left P -mo dule M consists o f a co llection ( M k ) k ≥ 0 of Σ k -ob jects together with a map of collections P ◦ M → M satisfying the usual a xioms of a left a ction. Suc h a collection ( M k ) k ≥ 0 is concentrated in degrees 0 and 1 precisely w he n all M k for k ≥ 2 are initial ob jects in E . The left P -mo dule structur e r estricted to M 0 endows M 0 with a P - algebra s tructure, while the left P -mo dule structure restricted to ( M 0 , M 1 ) amounts precisely to an M 0 -mo dule structure on M 1 under P . There is yet another wa y to sp ecify such a pair ( A, M ) if E is an addi tive categor y . Recall that a P -alg ebra structure o n the ob ject A of E is equiv alen t to a n o pe r ad map P → End A taking v alues in the endomorphism op er ad of A . The latter is defined by End A ( n ) = Hom E ( A ⊗ n , A ) where the op erad structure ma ps are giv en by substitution and permutation of the factors in th e do main. F or a pa ir of ob jects A a nd M in an a dditive categ ory E , w e define a line ar endomorphism op er ad E nd M | A of M relative to A such that oper a d maps P → End M | A corres p o nd to a P -alg ebra structure on A together with an A -mo dule structure on M . This linea r endomorphism op er ad E nd M | A is defined as a sub op er ad of the en- domorphism op e r ad End M ⊕ A , where − ⊕ − stands for the direct sum in E . Since Hom E ( X ⊕ Y , Z ⊕ W ) ∼ =  Hom E ( X, Z ) Hom E ( Y , Z ) Hom E ( X, W ) Hom E ( Y , W )  with the usual matrix rule for c o mp osition, it makes sense to define End M | A ( n ) as that sub ob ject of End M ⊕ A ( n ) that takes the summand A ⊗ n to A , the summands of the form A ⊗ k − 1 ⊗ M ⊗ A ⊗ n − k to M , and all other s ummands to a n ull (i.e. initial a nd t erminal) ob ject of E . It is then readily verified that this sub collectio n (End M | A ( n )) n ≥ 0 of (End M ⊕ A )( n )) n ≥ 0 defines a s ubo p erad End M | A of End M ⊕ A , and that an ope rad map P → End M | A determines, and is determined by , a P - algebra structure on A together with a n A -module structure on M . It fo llows from the preceding considerations that for each A -mo dule M in an additive category E , the direct sum M ⊕ A ca rries a canonical P -alg ebra structure, induced b y the comp osite oper ad map P → End M | A → End M ⊕ A ; the resulting P -algebra is often denoted by M ⋊ A , cf. [8]. Pro jection on the second factor defines a map of P -alg ebras M ⋊ A → A , hence an ob ject of the category Alg P / A of P -a lgebras ov er A . This as signment extends to a functor ρ : Mod A → Alg P / A . The following lemma is due to Quillen [14]; it is the starting point of the definition of the c otangent c omplex of the P -alg ebra A . Lemma 1.3. L et A b e an algebr a over an op er ad P in an additive, close d symmetric monoidal c ate gory E . Under the a b ov e c onstruction, the c ate gory of A -mo dules is isomorphi c to the c ate gory of ab elia n gr oup obje cts of Alg P / A . 4 CLEMENS BERGER AND IEKE MOERDIJK Pr o of. Since E is additive, the catego r y of A -mo dules is additive and a ny A -mo dule carries a canonical ab elian gro up structure in Mo d A . By ins pe c tion, the functor ρ : Mo d A → Alg P / A preser ves finite pro ducts, thus abelian group ob jects, so that for any A -mo dule M , the im age ρ ( M ) carr ies a canonica l ab elian g roup struc tur e in Alg P / A . The zero element of this ab elia n group structure is given by the section A → M ⊕ A ; in particular , the functor ρ is full a nd faithful, pr ovided ρ is considere d as taking v a lues in catego r y of abelian gr o up ob jects of Alg P / A . It remains to b e shown that an y a b elian group o b ject of Alg P / A arises as ρ ( M ) for a uniquely determined A - mo dule M . Indeed, an abelian gr oup o b ject N → A has a section A → N by the ze r o element so that N splits canonically as N = M ⊕ A . The P - algebra structure on N = M ⊕ A restricts to the given P -algebr a structure on A . The ab elia n gro up structure ( α, id A ) : ( M ⊕ M ) ⊕ A = N × A N − → N = M ⊕ A commutes with the P -algebr a structure of N ; thus, the square P (2) ⊗ ( M ⊕ M ) ⊗ ( M ⊕ M ) id P (2) ⊗ α ⊗ α ✲ P (2) ⊗ M ⊗ M M ⊕ M µ 2 ⊕ µ 2 ❄ α ✲ M µ 2 ❄ is co mm utative which implies that µ 2 is zer o. This sho ws that the op erad a ction P → End M ⊕ A factors through End M | A and we are done.  Lemma 1 . 4. L et P b e an op er ad. The c ate gory of P (0) - mo d ules u nder P is c anon- ic al ly isomorphic to the m o dule c a te gory of the monoid P (1) . Pr o of. F o r a P (0)-mo dule M under P , the action map µ 1 , 1 : P (1 ) ⊗ M → M defines an action on M by the monoid P (1). Co nversely , an ac tion on M by P (1) extends uniquely to action ma ps µ n,k : P ( n ) ⊗ P (0) ⊗ k − 1 ⊗ M ⊗ P (0) ⊗ n − k → M where we use the symmetry of the monoida l structure as w ell as the op era d structure maps P ( n ) ⊗ P (0) ⊗ k − 1 ⊗ I ⊗ P (0) ⊗ n − k → P ( n ) ⊗ P (0 ) ⊗ k − 1 ⊗ P (1) ⊗ P (0) ⊗ n − k → P (1 ) .  Definition 1. 5. L et P b e an o p er ad and A b e a P -algebr a . The env eloping op era d P A of t he P -algebr a A is defin e d by the u niversal pr op erty that op er ad maps P A → Q c orr esp ond pr e cisely to p airs ( φ, ψ ) c onsisting of an op er ad map φ : P → Q and a P -algebr a map ψ : A → φ ∗ Q (0) , and that this c o rr esp ondenc e is natur al in Q . Alternatively , w e can co nsider the catego ry Pairs( E ) o f pairs ( P , A ) consisting of an op er ad P and a P - a lgebra A , with morphisms the pairs ( φ, ψ ) : ( P , A ) → ( Q , B ) consisting of an oper ad map φ : P → Q and a P -alg ebra map ψ : A → φ ∗ ( B ). There is a ca nonical embedding of the categ ory Op er( E ) o f op er a ds in E in to the category Pairs ( E ) giv en by P 7→ ( P , P (0)). The univ ersal pr op erty of the env eloping o p er ad then ex pr esses (provided it ex is ts for all P a nd A ) that Op er( E ) is a r efle ctive sub c ate gory of Pairs ( E ) and that the left adjoint of the embedding is precisely the env eloping oper ad construc tio n Pairs( E ) → Op er( E ) : ( P , A ) 7→ P A . In particular , if this left adjoint exists, it preserves all colimits. Prop ositi on 1.6. The enveloping op era d P A exists for any P -algebr a A . ON THE DERIVED CA TEGOR Y OF AN ALGEBRA OVER AN O P ERAD 5 Pr o of. F o r a free P - algebra A = F P ( X ), wher e X is an ob ject of E , the en veloping op erad of F P ( X ) is given by P F P ( X ) ( n ) = a k ≥ 0 P ( n + k ) ⊗ Σ k X ⊗ k , see for instance Getzler and Jones [6 ]. A general P -alg ebra A is part of a canonica l co equalizer F P F P ( A ) ⇒ F P ( A ) → A, whence the corres p o nding coequa lizer of op erads P F P F P ( A ) ⇒ P F P ( A ) → P A (1) has the required universal prop e rty of the enveloping o p er ad of A .  The iden tit y P A → P A corres p o nds by the univ ersal pr op erty to an op erad map η A : P → P A together with a map of P -a lgebras ¯ η A : A → η ∗ A P A (0). W e will now show that the la tter map is an isomo rphism. Lemma 1.7. F or any P -algebr a A , the c ate gory of P A -algebr as is c anonic al ly iso- morphic t o the c a te gory of P -algebr as under A , and P A (0) is isomorphic to A . Pr o of. The pa ir ( η A , ¯ η A ) induces a functor fro m the category of P A -algebra s to the category of P -a lgebras under A whic h is co mpa tible with the forgetful functors. This functor is an isomorphism of ca tegories since a P A -algebra s tr ucture on B is given equiv alen tly by an op erad map P A → E nd B or by an op er ad map P → End B (i.e. a P -algebra structure on B ) together with a ma p of P - algebra s A → B .  Lemma 1.8. L et P b e a n op er ad and α : A → B b e a map of P -algebr as. Write B α for the P A -algebr a define d by α . The enveloping op er a d of the P -algebr a B is isomorphi c to the enveloping op er ad of the P A -algebr a B α . Pr o of. An op erad map ( P A ) B α → Q gives r ise to a pair ( φ, ψ ) consisting of an op erad map φ : P A → Q and a P A -algebra map ψ : B α → φ ∗ Q (0). According to Lemma 1.7, the la tter y ields a P -alg e bra map ψ ′ : B → η ∗ A φ ∗ Q (0) (under A ) for the cano nical o p erad map η A : P → P A . Conversely , the pair ( φη A , ψ ′ ) uniquely determines the op erad map ( P A ) B α → Q we sta rted from. Therefore, the en v eloping op erad ( P A ) B α has the same univ ersal property as the env eloping op erad P B so that b oth op era ds are isomorphic.  The following prop osition is pr eparato ry for the relationship be tw een env eloping op erad and en veloping algebra. The result is implicitly used b y Go er ss and Hopkins, compare [8, Lemma 1.13 ]. Prop ositi on 1.9. L et T b e a monad on a close d symmetric monoidal c ate gory E . The c ate gory of T -algebr as is a m o dule c ate gory for a monoid M T in E if and only if the tens or-c o tensor adjunction of E lifts to the c ate gory of T -algebr as along the for getful functor U T : Alg T → E . If the latter is t he c ase, the monad T is isomorphi c to T ( I ) ⊗ ( − ) , i.e. t he m onoid M T is given by T ( I ) . Pr o of. Assume first that the mona d T is given b y tens oring with a mono id M T . Then the tensor- cotensor adjunction of E lifts as follows. F or a ny M T -mo dules M , N and ob ject X of E , the tensor M ⊗ X inher its an M T -mo dule structure by M T ⊗ M ⊗ X ǫ M ⊗ X ✲ M ⊗ X ; 6 CLEMENS BERGER AND IEKE MOERDIJK the cotensor Hom E ( X, N ) inherits a n M T -mo dule structure by the adjoin t of M T ⊗ Hom E ( X, N ) ⊗ X M T ⊗ ev X ✲ M T ⊗ N ǫ N ✲ N . It f ollows that the a djunction E ( M ⊗ X , N ) ∼ = E ( M , Ho m E ( X, N )) lifts to an ad- junction Alg T ( M ⊗ X , N ) ∼ = Alg T ( M , Ho m E ( X, N )). Assume conv ersely that such a lifted tensor-co tensor adjunction exis ts for a given monad T on E . By adjointness w e g e t for any ob jects X , Y binatura l iso morphims of T -algebr as F T ( X ) ⊗ Y ∼ = F T ( X ⊗ Y ) . Since by assumption U T preserves tensors this implies (setting X = I ) that the monad T = U T F T is isomorphic to T ( I ) ⊗ ( − ); in par ticular, T ( I ) carr ies a ca nonical monoid structure.  Theorem 1.10. F or any algebr a A over an op er ad P in a close d symmetric monoidal c ate gory E , the c ate gory of A -mo dules under P is c anonic al ly isomor- phic t o the mo dule c ate gory of the monoid P A (1) . Pr o of. First o f all, it follows immediately from Definition 1 .1 that the forg etful functor U A : Mo d A → E crea tes co limits (hence p er mits an application of Beck’s tripleability theorem) a nd allows a lifting to Mo d A of the tensor s a nd cotenso rs by ob jects of E . In order to a pply Propo sition 1.9, and to compare the resulting monoid to P A (1), we give an explicit descriptio n of the left adjoint F A : E → Mo d A of U A , again following Goers s and Hopkins, co mpare [8, Prop o sition 1.14]. F or ob jects M and A of E , denote by Ψ( A, M ) the p ositive collection in E given b y Ψ( A, M )( n ) = n a k =1 A ⊗ k − 1 ⊗ M ⊗ A ⊗ n − k , n ≥ 1 , the symmetric group Σ n acting by p ermutation of the facto rs. Moreov er, define the ob ject P ( A, M ) = ` n ≥ 1 P ( n ) ⊗ Σ n Ψ( A, M ) . The axioms of an A -mo dule M amount then to the existence of an action map µ M : P ( A, M ) → M which is unitar y and asso ciative in a natural sense. F or instance, the associa tivity constraint uses a canonical is o morphism ( P ◦ P )( A, M ) ∼ = P ( F P ( A ) , P ( A, M )) where F P ( A ) is the free P - a lgebra on A . It follo ws that for a free P -alg ebra A = F P ( X ) the free A - mo dule o n M is given by P ( X , M ), the A -mo dule structure b eing induced by the isomorphim just cited. A gener al P - a lgebra A is part of a reflexive coequa lizer F P F P ( A ) ⇒ F P ( A ) → A which is pr eserved under the forg etful functor Alg P → E . Therefore, the underlying ob ject of the free A -mo dule F A ( M ) on M is part of a reflexive co e q ualizer in E P ( F P ( A ) , M ) ⇒ P ( A, M ) → U A F A ( M ) . (2) Prop os itio n 1.9 implies that the categor y Mo d A is a mo dule ca tegory for the monoid M A ∼ = U A F A ( I ). Putting M = I in (2) we end up w ith the following reflexive co equalizer diagra m in E P ( F P ( A ) , I ) ⇒ P ( A, I ) → M A . (3) F or the seco nd step of the pro of obser ve first that the co equa lizer (1) in Ope r( E ) is preserved under the forgetful functor from op erads to collections, since operads a re ON THE DERIVED CA TEGOR Y OF AN ALGEBRA OVER AN O P ERAD 7 monoids in co llections with r e sp ect to the circle product, and since the co equa lizer is reflexive. There fo re w e get the following r eflexive coequalize r diagram in E P F P F P ( A ) (1) ⇒ P F P ( A ) (1) → P A (1) . (4) It follows fro m the definitions that (3) and (4) are isomorphic diagrams in E . It remains to be shown that the mo noid structures of M A and of P A (1) coincide under this iso morphism. Lemmas 1.4 and 1.7 imply that the categ ory o f P A (1)-mo dules is isomorphic to the category of A -mo dules under P A ; the canonical operad map η A : P → P A induces thus a functor (over E ) from the categ ory of P A (1)-mo dules to the category of A -modules under P , and therefore (b y P rop ositio n 1.9 ) a map of mono ids from P A (1) to M A ; this map of monoids may b e iden tified with the isomorphism be tw een the co equa lizers of (4) and (3).  Definition 1.11. F or any algebr a A over an op er ad P in a close d symmetric monoidal c ate gory E , t he env eloping algebr a Env P ( A ) is the monoid P A (1) of unary op er ations of the enveloping op er ad of A . The e nveloping a lgebra co nstruction is a functor that takes maps of pairs ( φ, ψ ) : ( P , A ) → ( Q , B ) to maps of monoids En v P ( A ) → E nv Q ( B ) in E . Theorem 1.10 shows that the category of A -mo dules under P is canonica lly iso morphic to the mo dule category of the env eloping algebra Env P ( A ). The purp o se of the remaining part o f this se c tio n is to give a sufficient condition for the e nveloping algebra Env P ( A ) to b e a bialgebr a , i.e. to hav e a c ompatible comonoid structure; this amounts to the existence of a monoidal str ucture on the category of A -mo dules under P . Recall that a Hopf op er ad P in E is b y definition an op erad in the symmet- ric monoidal categor y Comon( E ) o f comono ids in E ; fo r such a Hopf op era d, a P -bialgebr a is defined to b e a P - a lgebra in Co mon( E ). Alternatively , a “Ho pf structure” on an opera d P amoun ts to a mo noidal stru ct ur e on the categor y o f P - algebras suc h tha t the forgetful functor is stro ngly monoida l, cf. [1 2]; P - bialgebra s are then precisely comonoids in this mono ida l category of P -algebra s. F or a ny tw o op erads P and Q , the tensor pro duct P ⊗ Q denotes the op er a d defined by ( P ⊗ Q )( n ) = P ( n ) ⊗ Q ( n ). Prop ositi on 1.12. F or any Hopf op er ad P and P -bialgebr a A , the enveloping op er ad P A is again a Hopf op er ad. I n p articular, the enveloping algebr a of A is a bialgebr a in E . Pr o of. An y Hopf o p erad P has a diagona l P → P ⊗P . Therefore, for any P - algebra s A and B , there is a canonica l opera d map P → P ⊗ P η A ⊗ η B − → P A ⊗ P B , and hence by the universal prop erty of P A ⊗ B , a canonical op erad map P A ⊗ B → P A ⊗ P B . If A is a P - bialgebra, there is a diagona l A → A ⊗ A in the category of P -algebra s and hence a diagonal P A → P A ⊗ A → P A ⊗ P A in th e category of op erads . This shows that P A is a Hopf op era d, and that P A (1) is a bialgebra in E .  2. The derived ca tegor y of an algebra over a n operad In this section, we study the derived categ ory of a n algebra over an op erad. F or any P - algebra A , the derived ca tegory D P ( A ) is defined to b e the homotopy category of the category of cA -mo dules, where cA is a cofibrant resolution o f A in the ca tegory of P -alg ebras. Thanks to T heo rem 1.10, in v ar iance pr o p erties 8 CLEMENS BERGER AND IEKE MOERDIJK of D P ( A ) corr esp ond to inv ar ia nce pr op erties of the env eloping algebra Env P ( A ). Since the env eloping algebra may b e identified with the mono id of unar y op er ations of the enveloping op era d P A , the metho ds of [2] apply (see the Appe ndix for a small correctio n to [2 ]), and we get q uite prec is e information on the inv ariance pro p erties of the derived category D P ( A ). F r o m now on, E de no tes a monoidal m o del c ate gory . Recall (cf. Hov ey [9]) that a monoidal mo del category is simultaneously a closed symmetric mono ida l catego ry and a Quillen mo de l catego ry such that tw o compatibility axioms ho ld: the pushout- pr o duct axiom and the unit axiom . The unit a xiom require s the existence of a cofibrant reso lutio n of the unit cI ∼ − → I such that tensoring with c o fibrant ob jects X induces w eak equiv alences X ⊗ cI ∼ − → X . The latter is of course automatic if the unit of E is already cofibran t. W e assume th rougho ut that E is c o c omplete and c o fibr antly gener ate d as a mo del catego ry . F or any unitar y asso ciative ring R , the category of simplicial R -mo dules is an additive monoida l mo del ca tegory with w eak equiv alences (resp. fibrations) those maps o f simplicial R -mo dules whose underlying map is a w eak equiv alence (r esp. fibration) o f simplicial sets. Simila rly , the category of differ ent ial gr ade d R - mo dules is an additiv e monoidal mo del c ategory with weak equiv alences (resp. fibr ations) the quas i-isomor phisms (resp. epimor phisms) of differential graded R -mo dules. These tw o examples generalise to a ny Gr othendie ck ab eli an c ate gory A equipp ed with a set of generato rs. Recall from [3] that a map of op er ads is called a Σ -c ofibr ation if the underlying map is a co fibr ation o f collectio ns and an op er ad P is called Σ -c ofibr ant if the unique map from the initial op erad to P is a Σ-cofibratio n. In particular, for a Σ-cofibrant o p erad P , the unit I → P (1) is a cofibration in E . This termino logy differs sligh tly from [2] where a n oper ad with the latter proper ty has been called well-pointed. An op era d P is ca lled admissibl e if the model str ucture on E trans fer s to the category Alg P of P -algebras along the free-for getful adjunction F P : E ⇆ Alg P : U P , i.e. if Alg P carries a mo del structure whose w eak equiv alences (resp. fibrations) are those maps f : A → B of P -algebra s for whic h the underlying map U P ( f ) : U P ( A ) → U P ( B ) is w eak equiv alence (resp. fibr ation) in E . See [2, Propo sition 4 .1] for conditions on P which imply admissibility . Lemma 2.1. F or any admissible op er ad P a nd P -algebr a A , the envelop ing op er ad P A is again admissible. Pr o of. This follows immediately from Lemma 1.7.  A (trivial) c el lular extension of P -a lgebras is an y s equential colimit of pushouts A → A [ u ] o f the form F P U P ( A ) ǫ A ✲ A F P ( X ) F P ( u ) ❄ ✲ A [ u ] ❄ ON THE DERIVED CA TEGOR Y OF AN ALGEBRA OVER AN O P ERAD 9 where ǫ A denotes the counit o f the fre e-forgetful adjunction and u : U P ( A ) → X is a gene r ating (tr iv ial) cofibration in E . A c el lular P -algebr a A is a cellular extension of the initial P -algebr a P (0). Lemma 2. 2. L et P b e a Σ -c ofibr a nt op er ad and A b e a P -algebr a. If the unique map of P -algebr as P (0) → A is a (t rivial) c el lular ext ension of P -algebr as, then the induc e d map P → P A is a (trivial) Σ -c ofibr ation of op er ads. Pr o of. The case o f a cellular extension is [2, Prop os itio n 5.4]. E xactly the same pro of applies to a trivial cellular extensio n as w ell.  A monoid M in E will b e ca lled wel l-p ointe d if the unit map I → M is a cofibration in E . In particular, if I is cofibrant then a well-pointed monoid M ha s a cofibrant underly ing ob ject. Prop ositi on 2.3. L et P b e an admissible Σ -c ofibr ant op er ad and A b e a c ofibr ant P -algebr a. Then t he enveloping op er ad P A is an admissible Σ - c o fibr ant op er ad. In p articular, the enveloping algebr a En v P ( A ) is wel l-p ointe d. Pr o of. Admissibility was dealt with in Lemma 2.1. A ny cofibrant P -a lgebra is retract of a cellula r P -alge br a, whence P A is retr act of Σ-cofibra nt o p erad and therefore Σ- cofibrant. The second statement follows from the identification of the env eloping algebra En v P ( A ) with P A (1).  Corollary 2.4. L et P b e an admi ssible Σ -c ofibr ant op er ad and α : A → B b e a we ak e quivalenc e of c ofibr a nt P -algebr as. The induc e d map P α : P A → P B is a we ak e quivalenc e of admissible Σ -c ofibr ant op er ads. In p articular, t he induc e d map of envelo ping algebr as Env P ( A ) → Env P ( B ) is a we ak e quivalenc e of wel l-p ointe d monoids. Pr o of. By K. Br own’s Lemma, it suffices to cons ider the case of a trivial cellular extension α . The statement then f ollows from Lemmas 1 .8, 2.1 and 2 .2, since P B may b e identified with ( P A ) B α , and P A is an admissible Σ-cofibr ant op era d.  R emark 2.5 . The homotopical prop erties of the env eloping op era d constr uction ( P , A ) 7→ P A , as express ed by Lemma 2.2 and Coro llary 2.4, are the main technical ingredients in establishing the homotopy inv ariance o f the derived ca tegory o f an algebra o v er an op era d. Beno ˆ ıt F resse pointed out to us that in re cent work [5], he independently obtained simila r ho motopical prop er ties of the assig nment ( R , A ) 7→ R ◦ P A whe r e R is a Σ-co fibr ant right P - mo dule a nd A is a cofibrant P -alg ebra. Since the env eloping op era d P A may be identified with R ◦ P A for a certain right P -mo dule R , se e [5, Section 1 0], Lemma 2.2 and Corollary 2.4 ma y b e recovered from [5, Lemma 13.1 .B] a nd [5, Theorem 13 .A.2 ]. The mor e general context of right P -mo dules how ever makes the pro ofs of these sta tement s mo re inv olved than t hose of our results which are immediate consequences of [2, Section 5]. Theorem 2.6. L et P b e an a dmissible Σ -c ofibr ant op er ad in a c ofibr antly gener ate d monoidal mo del c ate gory. F or any c ofibr ant P -algebr a A , the c ate gory Mo d P ( A ) of A -mo d ules c arries a tra nsferr e d mo del structur e. Any map of c ofibr ant P - algebr as f : A → B induc es a Qu il len adjunction f ! : Mo d P ( A ) → Mo d P ( B ) : f ∗ . If f is a we ak e quivalenc e, then ( f ! , f ∗ ) is a Quil len e quiva lenc e. 10 CLEMENS BERGER AND IEKE MOERDIJK Pr o of. By Pr o p osition 2.3, the env eloping algebra of a cofibrant P -a lgebra is well- po inted. Theorem 1.10, Coro llary 2.4 and Prop osition 2.7 then yield the conclus ion.  Prop ositi on 2.7. L et E b e a c ofibr antly gener ate d monoidal mo del c ate go ry. (a) F or any wel l-p ointe d m onoid M , the c ate gory Mo d M of M -mo dules c arries a tr ansferr e d mo del structur e; if in addition M has a c omp atible c o c ommu- tative c omonoid structur e, then Mo d M is a monoidal mo de l c ate gory; (b) Each map of wel l-p ointe d mo noids f : M → N induc es a Q u il len adjunction f ! : Mo d M ⇆ Mo d N : f ∗ ; if f is a we ak e quivalenc e, then ( f ! , f ∗ ) is a Quil len e quiva lenc e. Pr o of. The first par t of (a) follows by a transfer argument (cf. [2, Sectio n 2.5]) fro m the fact that tensoring with M preserves colimits, as well as cofibra tions and trivia l cofibrations; the pr eserv ation of cofibrations and trivial cofibrations follows from the pu shout-pro duct axiom and the well-p ointedness of M . If in addition M has a co mpatible co monoid structure, then Mo d M carries a clos ed mono idal s tructure which is strictly preserved by the forgetful functor Mod M → E . Since the forgetful functor pr e serves and reflects limits as well as fibrations a nd w eak equiv alences, this implies that Mo d M satisfies the axioms of a monoida l model ca teg ory . The first statemen t of (b) follows, since f ∗ preserves fibrations and trivial fibr a - tions b y definition of the transferre d model structur es on Mo d M resp. Mo d N . F or the se cond statement of (b), we use that ( f ! , f ∗ ) is a Quillen equiv a le nc e if a nd o nly if the unit η X : X → f ∗ f ! ( X ) is a w eak equiv alence at each cofibr a nt M -mo dule X . Assume that f is a weak equiv alence; since any co fibrant M -mo dule is retra ct of a “cellular extension” o f the initial M -mo dule, Reedy’s patching and teles cop e lemmas (cf. [3, Section 2.3]) imp ly that it is sufficient to consider M -mo dules of the form M ⊗ C , wher e C is a cofibra nt ob ject of E . In this c a se, the unit η M ⊗ C may be identified with f ⊗ id C : M ⊗ C → N ⊗ C ; the latter is a weak equiv alence by an application of the pushout-pro duct axiom and of K. Br own’s Lemma to the functor ( − ) ⊗ C : I / E → C / E .  R emark 2.8 . It follows from Pr op osition 2.7 (b) that for ea ch weak equiv alence f : M → N of well-pointed mo no ids, the unit η X : X → f ∗ f ! ( X ) is a weak equiv a lence a t cofibran t M -mo dules X . F o r later use, we obser ve that this holds also if X is only cofibra nt as an ob ject of E pr ovided N is co fibrant as an M -mo dule. R emark 2.9 . If E satisfies the m onoid axiom of Sch w ede and Shipley [17], the category of M -mo dules carr ies a transferred model structure for any monoid M in E . The monoid axio m holds in many interesting situations, in particula r if either all ob jects of E ar e cofibr ant, or all ob jects of E ar e fibrant. How ev er, even if the monoid ax io m holds, P rop osition 2.7 (b) do es not car r y ov er to a base- change along arbitr a ry monoids; indeed, the unit of the base-ch ange adjunction behav es in general badly at cofibra nt M -mo dules if M is not supp osed to b e well-p ointed. In genera l it is more restrictive for f : A → B to b e a weak equiv alence than to induce a Quillen equiv alence. A complete character is ation of those f whic h induce a Quillen equiv a lence on mo dule categories would requir e a ho motopical Morita theory . W e giv e here, in a par ticular case, a precise criterion for when a Quillen equiv a lence betw een mo dule categor ies comes from a w eak equiv alence. ON THE DERIVED CA TEGOR Y OF AN ALGEBRA OVER AN O P ERAD 11 Prop ositi on 2.10. U nder the hyp otheses of 2. 6, let f : A → B b e a map of c ofi- br a nt P -algebr as. Assume that either A or t he enveloping algebr a of B is c ofibr ant as an A -mo dule. Then f is a we ak e quival enc e if and only if ( f ! , f ∗ ) is a Q uil len e quiva lenc e and the induc e d map of B -mo dules f ! ( A ) → B is a we ak e quival enc e. Pr o of. The g iven f can b e considered as a ma p of A -mo dules A → f ∗ ( B ), and as such it factors thr o ugh the unit o f the adjunction: A → f ∗ f ! ( A ) → f ∗ ( B ). It follows from [2, Coro llary 5.5 ] that A ha s a cofibrant underlying ob ject, and it f ollows fro m Prop os itio n 2.3 tha t the env eloping algebras of A and of B a re well-pointed. Assume firs t that f is a weak equiv alence. Then, b y Theorem 2.6 , ( f ! , f ∗ ) is a Quillen eq uiv a le nc e , and by Remark 2.8, either of the hyp o theses implies that the unit A → f ∗ f ! ( A ) is a w eak equiv a lence, whence f ! ( A ) → B is a w eak equiv alence. Conv ersely , assume th at ( f ! , f ∗ ) is a Quillen e quiv a lence, and that f ! ( A ) → B , and hence f ∗ f ! ( A ) → f ∗ ( B ), is a w eak equiv a lence; then, by Remark 2.8, either of the h ypotheses implies that A → f ∗ f ! ( A ) is a weak equiv alence, whence f is a weak equiv alence.  Definition 2. 11. L et P b e an admissible Σ -c ofibr ant op er ad. The deriv ed category D P ( A ) of a P -algebr a A is the homotopy c ate gory of the c ate gory of cA -mo dules under P , wher e cA is any c ofibr ant r esolution of A in the c ate gory of P -algebr a s. Up to adjoin t equiv alence, this definition do es not depend o n the c hoice o f the cofibrant resolution, by Theorem 2.6. F ur thermore, a ny map f : A → B o f P - algebras induces an adjunction f ! : D P ( A ) ⇆ D P ( B ) : f ∗ since the cofibrant resolutions can be chosen functorially . F or weak equiv alence s f , this adjunction is an adjoint equiv alence. W e shall now dis cuss the functorial b ehaviour of the der ived category under change of op erads φ : P → Q . Recall that φ induces an adjunction φ ! : Alg P ⇆ Alg Q : φ ∗ which is a Quillen adjunction for admissible opera ds P , Q . In particular, for any P -algebra A , φ induces a map of env eloping oper ads P A → Q φ ! ( A ) , and hence a Quillen adjunction Mod P ( A ) ⇆ Mo d Q ( φ ! A ). The derived adjunction o f the Quillen pair ( φ ! , φ ∗ ) is denoted by Lφ ! : Ho (Alg P ) ⇆ Ho(Alg Q ) : Rφ ∗ . Theorem 2.12. L et E b e a lef t pr op er, c ofibr a ntly gener ate d monoidal mo del c at- e go ry and φ : P → Q b e a we ak e quiva lenc e of admissible Σ -c ofibr ant op er ads. Then, (a) for a c ofibr ant P -algebr a A , t he Quil len adjunction Mo d P ( A ) ⇆ Mo d Q ( φ ! A ) is a Quil len e quivalenc e; (b) for an arbitr ary P -algebr a A , the map φ induc es a n e quivalenc e o f derive d c a te gori es D P ( A ) ≃ D Q ( Lφ ! ( A )) ; (c) for an arbitr ary Q - algebr a B , φ induc es an e quivalenc e o f d erive d c ate gories D P ( Rφ ∗ ( B )) ≃ D Q ( B ) . Pr o of. Part (a) is a sp ecial case of [2, Theorem 4.4]. More pr ecisely , since φ is a weak equiv a le nce, the canonical map of o pe rads P A → Q φ ! ( A ) is a w eak eq uiv ale nce of admis s ible Σ- c ofibrant opera ds , cf. the pr o of of [2, Pr op osition 5 .7]. Therefore, Env P ( A ) → Env Q ( φ ! ( A )) is a weak equiv a lence a nd Mo d P ( A ) ⇆ Mod Q ( φ ! ( A )) is a Quillen equiv a le nce by Theorem 2.6. 12 CLEMENS BERGER AND IEKE MOERDIJK Part (b) follows fr om part (a), applied to a co fibrant re s olution cA of A . F or (c), le t B be a Q - algebra , and let B ∼ − → f B be a fibrant resolution of B in the categor y of Q -a lgebras. Then, D Q ( B ) ≃ D Q ( f B ) as asserted ea r lier. By part (a), the canonica l map φ ! cφ ∗ ( f B ) → f B (adjoint to cφ ∗ ( f B ) ∼ − → φ ∗ ( f B )) is a weak equiv a le nce inducing a nother equiv alence of derived ca tegories. Also, part (b) gives a n equiv alence D P ( φ ∗ ( f B )) ≃ D Q ( φ ! cφ ∗ ( f B )). Putting all these equiv alences together, w e get as requir e d D P ( Rφ ∗ ( B )) = D P ( φ ∗ ( f B )) ≃ D Q ( φ ! cφ ∗ ( f B )) ≃ D Q ( f B ) ≃ D Q ( B ) .  Example 2.13 . Let E = T op b e the categ ory of co mpactly generated spaces en- dow ed with Quillen’s model structure, let P = A ∞ be any Σ-cofibra nt A ∞ -op erad, and let X b e an A ∞ -space. Then, D A ∞ ( X ) is equiv a lent to the homo topy cat- egory Ho(T op /B X ) o f s paces ov er the classifying space B X of X . Indeed, b y Theorem 2.12, X may b e rectified to a monoid M = φ ! ( cX ) a lo ng the canon- ical ma p of op erads φ : A ∞ → Ass without c hanging the derived categor y : D A ∞ ( X ) ≃ D Ass ( Lφ ! X ), a nd B X may b e iden tified with the usual classifying s pace B M . The Bor el constr uction then yields an eq uiv ale nce D Ass ( M ) ≃ Ho(T op /B M ). In particular, if X is the loop space o f a p ointed connected s pa ce Y , endo wed with the A ∞ -action by th e little interv a ls opera d, w e get D A ∞ (Ω Y ) ≃ Ho(T op / Y ). F or the la st comparis on theorem, recall that a monoida l Quil len adjunction Φ ! : F ⇆ E : Φ ∗ betw een monoida l mo del ca tegories is a n adjoint pair which is simultaneously a Quillen adjunction (with resp ect to the mo del str uctures) and a monoidal adjunction (with res p ec t to the closed symmetric monoida l structures). The latter mea ns that t he left and r ight adjo int s of the adjunction are lax sy mmet- ric monoidal functors, and that unit and counit o f the adjunction are symmetric monoidal natural transfor mations, see [11, I I I.20 ] for details. Observe that if the left adjoint of an adjunction b etw een closed symmetric monoidal categorie s is stro ng symmetric monoida l, then the rig ht adjoint car ries a natural symmetric m onoidal structure for which t he adjunction is monoidal. Theorem 2.14. L et Φ ! : F ⇆ E : Φ ∗ b e a monoid al Q uil len adjunction b etwe en c o fibr antly gener ate d monoidal mo d el c ate gories such that t he induc e d map on units Φ ! ( I F ) → I E is a c ofibr ation. L et Q b e a Σ -c ofibr ant admissible op er ad in F , and B b e a Q -algebr a, and assume that Φ ! Q is an admissible op era d in E . Then ther e is a c anonic al adjunction of de- rive d c ate gori es D Q ( B ) ⇆ D Φ ! Q ( L Φ ! B ) . This adjunction is a Qu il len e quivalenc e, whenever (Φ ! , Φ ∗ ) is a Q uil len e qu ivalenc e. Pr o of. First notice that the monoida l adjunction induces adjoint functor s Φ ! : Op er( F ) ⇆ Oper( E ) : Φ ∗ , and tha t if Q is a Σ-co fibr ant opera d in F , th en Φ ! ( Q ) is so in E . By assumption, Q and Φ ! ( Q ) are b oth admissible so that w e hav e an induced Quillen adjunction Φ ! : Alg F ( Q ) ⇆ Alg E (Φ ! ( Q )) : Φ ! , where Φ ! ( A ) is Φ ∗ ( A ) with the Q -alge br a structure induced b y η Q : Q → Φ ! Φ ∗ ( Q ). Now let cB ∼ − → B be a co fibrant resolution of the Q -a lgebra B . Then the derived category D Φ ! ( Q ) ( L Φ ! B ) is the der ived c ategory o f mo dules for the monoid Φ ! ( Q ) Φ ! ( cB ) (1), while D Q ( B ) is that for the mo noid Q cB (1). By Lemma 2.1 5 be low ON THE DERIVED CA TEGOR Y OF AN ALGEBRA OVER AN O P ERAD 13 and the assumption on units, we hav e an iso morphism of w ell-p ointed monoids Φ ! ( Q ) Φ ! ( cB ) (1) ∼ = Φ ! ( Q cB )(1) in E w he nc e Lemma 2.16 gives the result.  Lemma 2. 15. Un der the hyp otheses of The or em 2.14, e ach Q -algebr a B induc es a c anonic al iso morphism of op er ads Φ ! ( Q B ) ∼ = Φ ! ( Q ) Φ ! ( B ) . Pr o of. It is sufficien t to chec k that b oth op er ads enjoy the same universal prop erty . This in tur n fo llows from the fact that for each β : Φ ! ( Q ) → P with adjoint γ : Q → Φ ∗ ( P ), the composite map Φ ! ◦ β ∗ : Alg E ( P ) → Alg E (Φ ! ( Q )) → Alg F ( Q ) coincides with the comp osite map γ ∗ ◦ Φ ∗ : Alg E ( P ) → Alg F (Φ ∗ ( P )) → Alg F ( Q ).  Lemma 2.16 . Under the hyp otheses of The or em 2.14, e ach wel l-p o inte d monoid M in F induc es a wel l-p ointe d monoid Φ ! ( M ) in E , and ther e is a Quil len adjunction Φ ! : Mo d F ( M ) ⇆ Mo d E (Φ ! M ) : Φ ! . This adjunction is a Quil len e quivalenc e, pr ov ide d Φ ! : F ⇆ E : Φ ∗ is a Quil len e quiva lenc e. Pr o of. This follows fro m the fact that co fibrant mo dules over well-pointed monoids hav e cofibrant underlying ob jects, and that Φ ! is just Φ ∗ on underlying o b jects.  3. A ppendix Section 5 of [2] is used several times in this pap er (for instance in the pr o ofs of Lemma 2.2 and Theorems 2.12 and 2.14). Ther e is a small mistak e in the proof of [2, Pr op osition 5.1] in that construction 5 .11 in lo c . cit. does not take car e of the unit of the op era d extension P [ u ]. In or der to remedy this, o ne has to include in the definition o f a Σ- c o fibrant op er ad P the co ndition that the unit I → P (1) is a cofibration in E (as done in Section 2 of this pap er and in [3]). Mor eov er, construction 5.11 has to b e slightly adapted in the following w a y: First of all, the inductive construction of F k ( n ) should be ov er the set A k ( n ) of admissible coloured trees with n inputs and k vertices whic h ar e either c olour e d or unary . Next, u − ( T , c ) ֌ u ( T , c ) should b e replaced b y u ∗ ( T , c ) ֌ u ( T , c ) where u ∗ ( T , c ) cons is ts o f those labelled trees which lie in u − ( T , c ) or ha v e a unary vertex lab elled by the identit y . In o rder to s how that u ∗ ( T , c ) ֌ u ( T , c ) is a g ain an Aut( T , c )-c o fibration (as w as do ne for u − ( T , c ) ֌ u ( T , c )), one has the following v ar iation of [2, Lemma 5.9]. Lemma 3.1. L et I b e the c ol le ction given by the unit I c onc entr ate d in de gr e e 1 , and let I ֌ K 1 u ֌ K 2 b e c ofibr ations of c ol le ctions. Then u ∗ ( T , c ) ֌ u ( T , c ) is an Aut( T , c ) -c ofibr atio n. Pr o of. The inductive definition of u ( T , c ) g iven in [2, page 82 8, lines 2-3] comes together with a similar inductive description of u ∗ ( T , c ): u ∗ ( T , c ) =      K 1 ( n ) ⊗ v ∗ ( T , c ) if the ro ot is uncolour e d, not unary , K 1 (1) ⊗ v ∗ ( T , c ) ∪ I ⊗ v ( T , c ) if the ro ot is uncolour e d, unary , K 2 ( n ) ⊗ v ∗ ( T , c ) ∪ K 1 ( n ) ⊗ v ( T , c ) if the ro ot is colour ed , where we have abbre v iated v ∗ ( T , c ) = n [ i =1 u ( T 1 , c 1 ) ⊗ · · · ⊗ u ∗ ( T i , c i ) ⊗ · · · ⊗ u ( T n , c n ) , v ( T , c ) = u ( T 1 , c 1 ) ⊗ · · · ⊗ u ( T n , c n ) . 14 CLEMENS BERGER AND IEKE MOERDIJK One now proves by induction that u ∗ ( T , c ) ֌ u ( T , c ) is an Aut( T , c )-c o fibration, exactly as in the pro of of [2, Lemma 5 .9]; to w ards the end, the ca se where the roo t is uncoloured splits in to t w o s ubca ses: the one where the roo t is not una ry is as in [2], while for the o ne where the ro ot is unar y , the map u ∗ ( T , c ) ֌ u ( T , c ) is of the form I ⊗ B ∪ I ⊗ A K 1 (1) ⊗ A ֌ K 1 (1) ⊗ B , and [2, Lemma 5.1 0] applies again. 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