OPERADS, ALGEBRAS, MODULES, AND MOTIVES IGORKRIZANDJ.P.MAY Abstract. Withmotivationfromalgebraictopology,algebraicgeometry,and stringtheory,westudyvarioustopicsindifferentialhomologicalalgebra. The workisdividedintofivelargelyindependentparts: I Definitionsandexamplesofoperadsandtheiractions II Partialalgebraicstructuresandconversiontheorems III Derivedcategoriesfromatopologicalpointofview IV RationalderivedcategoriesandmixedTatemotives V DerivedcategoriesofmodulesoverE∞ algebras In differential algebra, operads are systems of parameter chain complexes formultiplicationonvarioustypesofdifferentialgradedalgebras“uptohomo- topy”,forexamplecommutativealgebras,n-Liealgebras,n-braidalgebras,etc. Our primary focus is the development of the concomitant theory of modules uptohomotopyandthestudyofbothclassicalderivedcategoriesofmodules over DGA’s and derived categories of modules up to homotopy over DGA’s uptohomotopy. Examplesofsuchderivedcategoriesprovidetheappropriate setting for one approach to mixed Tate motives in algebraic geometry, both rationalandintegral. Contents Introduction 3 Part I. Definitions and examples of operads and operad actions 8 1. Operads 9 2. Algebras over operads 11 3. Monadic reinterpretation of algebras 13 4. Modules over C-algebras 16 5. Algebraic operads associated to topological operads 20 6. Operads, loop spaces, n-Lie algebras, and n-braid algebras 22 7. Homology operations in characteristic p 25 Part II. Partial algebraic structures and conversion theorems 26 1. Statements of the conversion theorems 27 2. Partial algebras and modules 28 3. Monadic reinterpretation of partial algebras and modules 33 4. The two-sided bar construction and the conversion theorems 35 5. Totalization and diagonal functors; proofs 38 6. Higher Chow complexes 42 Part III. Derived categories from a topological point of view 44 1. Cell A-modules 45 1 2 IGORKRIZANDJ.P.MAY 2. Whitehead’s theorem and the derived category 48 3. Brown’s representability theorem 51 4. Derived tensor product and Hom functors: Tor and Ext 53 5. Commutative DGA’s and duality 56 6. Relative and unital cell A-modules 58 Part IV. Rational derived categories and mixed Tate motives 58 1. Statements of results 59 2. Minimal algebras, 1-minimal models, and co-Lie algebras 62 3. Minimal A-modules 64 4. The t-structure on D 66 A 5. Twisting matrices and representations of co-Lie algebras 68 6. The bar construction and the Hopf algebra χ 71 A 7. The derived category of the heart and the 1-minimal model 73 Part V. Derived categories of modules over E algebras 78 ∞ 1. The category of C-modules and the product (cid:163) 80 2. Unital C-modules and the products (cid:67), (cid:66), and (cid:161) 84 3. A new description of A and E algebras and modules 86 ∞ ∞ 4. Cell A-modules and the derived category of A-modules 90 5. The tensor product of A-modules 93 6. The Hom functor on A-modules; unital A-modules 95 7. Generalized Eilenberg-Moore spectral sequences 98 8. E algebras and duality 102 ∞ 9. The linear isometries operad; change of operads 104 References 107 OPERADS, ALGEBRAS, MODULES, AND MOTIVES 3 Part . Introduction Therearemanydifferenttypesofalgebra: associative,associativeandcommuta- tive,Lie,Poisson,etc.,etc. Eachcomeswithanappropriatenotionofamoduleand thus with an associated theory of representations. Moreover, as is becoming more and more important in a variety of fields, including algebraic topology, algebraic geometry, differentialgeometry, andstringtheory, itisveryoftennecessarytodeal with “algebras up to homotopy” and with “partial algebras”. The associated theo- ries of modules have not yet been developed in the published literature, but these notions too are becoming increasingly important. We shall study various aspects of the theory of such generalized algebras and modules in this paper. We shall also developsomerelatedalgebraintheclassicalcontextofmodulesoverDGA’s. While much of our motivation comes from the theory of mixed Tate motives in algebraic geometry, there are pre-existing and potential applications in all of the other fields mentioned above. The development of abstract frameworks in which to study such algebras has a long history. It now seems to be widely accepted that, for most purposes, the most convenient setting is that given by operads and their actions [46]. While the notion was first written up in a purely topological framework, due in large part to the resistance of topologists to abstract nonsense at that period, it was already understood by 1971 that the basic definitions apply equally well in any underlyingsymmetricmonoidal(=tensor)category[35]. Infact,certainchainlevel concepts, the PROP’s and PACT’s of Adams and MacLane [42], were important precursorsofoperads. Fromatopologicalpointofview,theswitchfromalgebraicto topologicalPROP’s,whichwasmadebyBoardmanandVogt[11],wasamajorstep forwards. Perhapsforthisreason,achainlevelalgebraicversionofthedefinitionof an operad did not appear in print until the 1987 paper of Hinich and Schechtman [31]. Applications of such algebraic operads and their actions have appeared in a variety of contexts in other recent papers, for example [27, 28, 29, 32, 34, 33, 56]. Inthealgebraicsetting,anoperadC consistsofsuitablyrelatedchaincomplexes C(j)withactionsbythesymmetricgroupsΣ . AnactionofC onachaincomplex j A is specified by suitably related Σ -equivariant chain maps j C(j)⊗Aj →A, where Aj is the j-fold tensor power of A. The C(j) are thought of as parameter complexes for j-ary operations. When the differentials on the C(j) are zero, we think of C as purely algebraic, and it then determines an appropriate class of (dif- ferential) algebras. When the differentials on the C(j) are non-zero, C determines a class of (differential) algebras “up to homotopy”, where the homotopies are de- termined by the homological properties of the C(j). For example, we saythat C is anE operadifeachC(j)isΣ -freeandacyclic,andwethensaythatAisanE ∞ j ∞ algebra. An E algebra A has a product for each degree zero cycle of C(2). Each ∞ such product is unital, associative, and commutative up to all possible coherence homotopies, and all such products are homotopic. There is a long history in topol- ogy and category theory that makes precise what these “coherence homotopies” are. However, since the homotopies are all encoded in the operad action, there is no need to be explicit. There is a class of operads that is related to Lie algebras as E operadsarerelatedtocommutativealgebras,andthereisaconcomitantnotion ∞ of a “strong homotopy Lie algebra”. In fact, any type of algebra that is defined in 4 IGORKRIZANDJ.P.MAY terms of suitable identities admits an analogous “strong homotopy” generalization expressed in terms of actions by appropriate operads. We shall give an exposition of the basic theory of operads and their algebras and modules in Part I. While we shall give many examples, the deeper parts of the theorythataregearedtowardsparticularapplicationswillbelefttolaterparts. In view of its importance to string theory and other areas of current interest, we shall illustrate ideas by describing the relationship between the little n-cubes operads of iteratedloopspacetheoryontheonehandandn-Liealgebrasandn-braidalgebras on the other. An operad S of topological spaces gives rise to an operad C (S) # of chain complexes by passage to singular chains. On passage to homology with field coefficients, there results a purely algebraic operad H (S). There is a partic- ∗ ular operad of topological spaces, denoted C , that acts naturally on n-fold loop n spaces. For n≥2, the algebras defined by H (C ;Q) are exactly the (n−1)-braid ∗ n algebras. Even before doing any calculation, one sees from a purely homotopical theorem of [46] that, for any path connected space X, H (ΩnΣnX;Q) is the free ∗ H (C ;bQ)-algebra generated by H (X;Q). This allows a topological proof, based ∗ n ∗ on the Serre spectral sequence, of the algebraic fact that the free n-braid algebra generatedbyagradedvectorspaceV isthefreecommutativealgebrageneratedby the free n-Lie algebra generated by V. Actually, the results just summarized are the easy characteristic zero case of Cohen’s much deeper calculations in arbitrary characteristic [15, 16], now over twenty years old. Operads and their actions are specified in terms of maps that are defined on tensor products of chain complexes. In practice, one often encounters structures that behave much like algebras and modules, except that the relevant maps are only defined on suitable submodules of tensor products. For geometric intuition, think of intersection products that are only defined between elements that are in generalposition. Suchpartialalgebrashavebeenusedintopologysincethe1970’s, forexamplein[48]andinunpublishedworkofBoardmanandSegal. InPartII,we shall generalize the notions of algebras over operads and of modules over algebras over operads to the context of partially defined structures. Such partially defined structures are awkward to study algebraically, and it is important to know when they can be replaced by suitably equivalent globally defined structures. We shall show in favorable cases that partial algebras can be replaced by quasi-isomorphic genuine algebras over operads, and similarly for modules. When k is a field of characteristic zero, we shall show further that E algebras and modules can be ∞ replaced by quasi-isomorphic commutative algebras and modules and, similarly, thatstronghomotopyLiealgebrasandmodulescanbereplacedbyquasi-isomorphic genuineLiealgebrasandmodules. Theargumentsworkequallywellforotherkinds of algebras. One of the main features of the definition of an operad is that an operad deter- mines an associated monad that has precisely the same algebras. This interpreta- tion is vital to the use of operads in topology. The proofs of the results of Part II are based on this feature. The key tool is the categorical “two-sided monadic bar construction” that was introduced in the same paper that first introduced operads [46]. This construction has also been used to prove topological analogs of many of the present algebraic results, along with various other results that are suggestive of further algebraic analogs [47, 49, 26, 52]. In particular, the proofs in Part II are OPERADS, ALGEBRAS, MODULES, AND MOTIVES 5 exactly analogous to a topological comparison between Segal’s Γ-spaces [55] and spaces with operad actions that is given in [26]. While these results can be expected to have other applications, the motivation camefromalgebraicgeometry. ForavarietyX,Bloch[7]definedtheChowcomplex Z(X). This is a simplicial abelian group whose homology groups are the Chow groups of X. It has a partially defined intersection product, and we show in Part II that it gives rise to a quasi-isomorphic E algebra, denoted N (X). After ∞ tensoring with the rationals, we obtain a commutative differential graded algebra (DGA) N (X) that is quasi-isomorphic to N (X)⊗Q. The construction of these Q algebrasanswersquestionsofDeligne[20]thatwerethestartingpointofthepresent work. His motivation was the intuition that, when X = Spec(F) for a field F, the associated derived categories of modules ought to be the appropriate homes for categories of integral and rational mixed Tate motives over F. This raises several immediate problems. On the rational level, it is necessary to connect this approach to mixed Tate motives with others. On the integral level, in order to take the intuition seriously, one must first construct the derived category ofmodulesoveranE algebra. Asapreliminarytothesolutionoftheseproblems, ∞ in Part III we shall give a new, topologically motivated, treatment of the classical derived category of modules over a DGA. We shall give a theory of “cell modules” that is just like the theory of “CW spectra” in stable homotopy theory, and we shall prove direct algebraic analogs of such standard and elementary topological results as the homotopy extension and lifting property, the Whitehead theorem, and Brown’s representability theorem. One point is that there is not the slightest difficulty in handling unbounded algebras and modules: except that the details are far simpler, our substitute for the usual approximation of differential modules by projective resolutions works in exactly the same way as the approximation of arbitraryspectraby(infinite)CWspectrawithcellsofarbitrarilysmalldimension, which has long been understood. Similarly, derived tensor products of modules work in the same way as smash products of spectra. In Part IV, we shall specialize this theory to study the derived category D of A cohomologicallyboundedbelowA-modules,whereAisacohomologicallyconnected commutative DGA over a field of characteristic zero. In the language of [3], we shall give the triangulated category D a t-structure. Its heart H will be the A A Abelian subcategory of modules whose indecomposable elements have homology concentrated in degree zero. In the language of [21], we shall show that the full subcategory FH of finite dimensional modules in H is a neutral Tannakian A A category. It is therefore the category of representations of an affine group scheme or, equivalently, of finite dimensional comodules over a Hopf algebra. In fact, without using Tannakian theory, we shall prove directly that H is A equivalenttothecategoryofcomodulesovertheexplicitcommutativeHopfalgebra χ = H0B¯(A). The “cobracket” associated to the coproduct on χ induces a A A structure of “co-Lie algebra” on its vector space γ of indecomposable elements, A andweshallseethatH isalsoequivalenttothecategoryofgeneralizednilpotent A representations of the co-Lie algebra γ . A Part IV is really a chapter in rational homotopy theory, and it may well have applications to that subject. As was observed by Sullivan [58], a co-Lie algebra γ determines a structure of DGA on the exterior algebra ∧(γ[−1]), where γ[−1] is a copy of γ concentrated in degree one. For a cohomologically connected DGA A, 6 IGORKRIZANDJ.P.MAY ∧(γ [−1]) is the 1-minimal model of A. Weshall prove the rather surprising result A that the derived category of modules over the DGA ∧(γ [−1]) is equivalent to the A derived category of the Abelian category H . Curiously, although the theory of A minimal rational DGA’s has been widely studied since Sullivan’s work, the analo- goustheoryofminimalmodulesdoesnotappearintheliterature. Thattheorywill be central to our work in Part IV. In view of the relationship between Chow groups and K-groups, the Beilinson- Soul´e conjecture for the field F is equivalent to the assertion that the DGA N = Q N (Spec(F))iscohomologicallyconnected. Whentheconjectureholds,theresults Q just summarized apply to A=N . Assuming the Beilinson-Soul´e conjecture (and Q assuming our construction of the DGA A), Deligne [20], [17], proposed FH as A a candidate for the Abelian category MTM(F) of mixed Tate motives over F. He (in [18]) and Bloch also proposed the category of finite dimensional comodules overχ asacandidateforMTM(F), and [6]provesrealizationtheoremsin´etale A and Hodge theory starting from this definition. Our work shows that these two categories are equivalent, and it gives a fairly concrete and explicit description of them. WhenAisaK(π,1),inthesensethatAisquasi-isomorphictoits1-minimal model, we shall have the relation Extp (Q,Q(r))∼=grrK (F)⊗Q MTM(F) γ 2r−p between the Abelian category MTM(F) and the algebraic K-theory of F. (Un- defined notations are explained in the introduction to Part IV.) Finally, in Part V, we shall construct the derived category of modules over an A or E k-algebra A, where k is a commutative ground ring. Here A algebras ∞ ∞ ∞ are DGA’s up to homotopy (without commutativity). There are a number of sub- tleties. From Part I, we know that A-modules are equivalent to modules over an associative, but not commutative, universal enveloping DGA U(A). In particular, U(k)=C(1). Inearlierparts,allE operadswereonthesamefooting. InPartV, ∞ weworkwithaparticularE operadC thatenjoysspecialproperties,butweshow ∞ thatrestrictiontothischoiceresultsinnolossofgenerality. Remarkably, withthis choice, the category of E k-modules, alias the category of C(1)-modules, admits ∞ a commutative and associative “tensor product” (cid:163). This product is not unital on the module level, although there is a natural unit map k(cid:163)M →M that becomes an isomorphism in the derived category. This fact leads us to introduce certain modified versions of the product M (cid:163)N that are applicable when one or both of M and N is unital, in the sense that it has a given map k →M. The product “(cid:161)” thatapplieswhenbothM andN areunitaliscommutative,associative,andunital up to coherent natural isomorphism; that is, the category of unital E k-modules ∞ is symmetric monoidal under (cid:161). Conceptually, we now change ground categories from the category of k-modules to the category of E k-modules. It turns out that A and E algebras can be ∞ ∞ ∞ described very simply in terms of products A(cid:163)A → A. In fact, an A k-algebra ∞ is exactly a monoid in the symmetric monoidal category of unital E k-modules, ∞ and an E k-algebra is a commutative monoid. There is a similar conceptual ∞ description of modules over A and E algebras. From here, the development of ∞ ∞ the triangulated derived category D of modules over an A algebra A proceeds A ∞ exactly as in the case of an actual DGA in Part III. When A is an E algebra, the ∞ category of A-modules admits a commutative and associative tensor product (cid:163) A OPERADS, ALGEBRAS, MODULES, AND MOTIVES 7 andaconcomitantinternalHomfunctorHom(cid:2). Again,thereisanaturalunitmap A A(cid:163) M →M thatbecomesanisomorphismonpassagetoderivedcategories. There A are Eilenberg-Moore, or hyperhomology, spectral sequences for the computation of the homology of M (cid:163) N and Hom(cid:2)(M,N) in terms of the classical Tor and Ext A A groups TorH∗(A)(H∗(M),H∗(N)) and Ext∗ (H∗(M),H∗(N)). ∗ H∗(A) Thus our new derived categories of modules over A and E algebras enjoy all of ∞ ∞ the basic properties of the derived categories of modules over DGA’s and commu- tative DGA’s. In view of the unfamiliarity of the constructions in Part V, we should perhaps say something about our philosophy. In algebraic topology, it has long been stan- dard practice to work in the stable homotopy category. This category is hard to construct rigorously, and its objects are hard to think about on the point-set level. (Althoughthedefinitionalframeworkinalgebraicgeometryisnotoriouslyabstract, the objects that algebraic geometers usually deal with are much more concrete than the spectra of algebraic topology.) However, once the machinery is in place, the stable homotopy category gives an enormously powerful framework in which to perform explicit calculations. It may be hoped that our new algebraic derived categories will eventually serve something of the same purpose. Actually, the analogy with topology is more far-reaching. There are analogs of E algebras in stable homotopy theory, namely the E ring spectra that were ∞ ∞ introduced in [47]. With Elmendorf [25], we have worked out a theory of module spectraoverA andE ringspectrathatispreciselyparalleltothealgebraicthe- ∞ ∞ oryofPartV.Althoughitismuchmoredifficult,itsconstructiveandcalculational powerarealreadyevident. Basicspectrathatpreviouslycouldonlybeconstructed bytheBaas-Sullivantheoryofmanifoldswithsingularitiesareeasilyobtainedfrom the theory of modules over the E ring spectrum MU that represents complex ∞ cobordism. Spectral sequences that are the precise analogs of the Eilenberg-Moore (or hyperhomology) spectral sequences in Part V include Ku¨nneth and universal coefficient spectral sequences that are of clear utility in the study of generalized homology and cohomology. Some other applications were announced in [24], and many more are now in place. An exposition of the analogy between the algebraic and topological theories is given in [51]. PartsIIandVconstitutearevisionandexpansionofmaterialinthepreprint[37], whichhadaratherdifferentperspective. Thatdraftwasintendedtolayfoundations for work in both algebra and topology, but it has since become apparent that, despite the remarkably close analogy between the two theories and the resulting expositoryduplication,thetechnicaldifferencesdictateseparateandself-contained treatments. Some of the present results were announced in [38]. Eachparthasitsownintroduction,andwehavetriedtomakethepartsreadable independently of one another. Part III has nothing whatever to do with operads andiswhollyindependentofPartsIandII.Althoughtheexamplesthatmotivated Part IV are constructed by use of Part II, the theory in Part IV also has nothing to do with operads and is independent of Parts I and II. Part V is independent of Part IV and nearly independent of Part II. A reference of the form “II.m.n” is to statement m.n in Part II; within Part II, the reference would be to “m.n”. We shall work over a fixed commutative ground 8 IGORKRIZANDJ.P.MAY ring k. There are no restrictions on k in Parts I, III, and V; k is assumed to be a Dedekind ring in Part II and to be a field of characteristic zero in Part IV. We wish to thank many people who have taken an interest in this work. Part I can serve as an introduction not only to this paper, but also to the closely related papers of Ginzburg and Kapranov [29], Getzler and Jones [27, 28], and Hinich and Schechtman [31, 32]. Some of the more interesting insights in Part I are due to these authors, and we are grateful to them for sharing their ideas with us. The second author wishes to take this opportunity to offer his belated thanks to Max Kelly and Saunders MacLane for conversations in 1970-71. Discussions then about operads in symmetric monoidal categories are paying off now. We are also very grateful to Jim Stasheff, who alerted us to how seriously operads are being used in mathematical string theory, urged us to give the general exposition of Parts I and II, and offered helpful criticism of preliminary versions. We also thank our colleague Spencer Bloch for detecting an error in the first version of Part II and for ongoing spirited discussions about motives. We are especially grateful to our collaborator Tony Elmendorf; the original version of the theory in Part V was far more complicated, and this material has been reshaped by the insights developed in our parallel topological work with him. It is a pleasure to thank Deligne for his letters that led to this paper and for his suggestions for improving its exposition. Part I. Definitions and examples of operads and operad actions We define operads in Section 1, algebras over operads in Section 2, and modules over algebras over operads in Section 4, giving a number of variants and examples. The term “operad” is meant to bring to mind suitably compatible collections of j-ary product operations. It was coined in order to go well with the older term “monad” (= triple), which specifies a closely related mathematical structure that has a single product. As we explain in Section 3, operads determine associated monads in such a way that an algebra over an operad is the same thing as an algebra over the associated monad. While not at all difficult, this equivalence of definitionsiscentraltothetheoryanditsapplications. Section4includesaprecisely analogous description of modules as algebras over a suitable monad, together with aquitedifferent, andmorefamiliar, descriptionasordinarymodulesoveruniversal enveloping algebras. Both points of view are essential. In Section 5, we discuss the passage from topological operads and monads to algebraic operads and monads via chain complexes and homology. We speculate that similar ideas will have applications to other situations, for example in alge- braicgeometry, whereonemayencounteroperadsinacategorythathasasuitable homology theory defined on it. In Section 6, we specialize to the little n-cubes operads C . These arose in iterated loop space theory and are now understood to n be relevant to the mathematics of string theory. We show that H (C ) contains ∗ n a suboperad which, when translated to degree zero, is isomorphic to the operad that defines Lie algebras, and we observe that work in Cohen’s 1972 thesis [15, 16] implies that the full operad H (C ) defines n-braid algebras. While current inter- ∗ n est focuses on characteristic zero information, we shall give some indications of the deeper mod p theory. In particular, in Section 7, we shall describe the Dyer-Lashof operations that are present on the mod p homologies of E algebras. Such opera- ∞ tions are central to infinite loop space theory, and our later work will indicate that they are also relevant to the mod p higher Chow groups in algebraic geometry. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 9 1. Operads WeworkinthetensorcategoryofdifferentialZ-gradedmodulesoverourground ring k, with differential decreasing degree by 1. Thus ⊗ will always mean ⊗ . k Readers who prefer the opposite grading convention may reindex chain complexes C by setting Cn = C . While homological grading is most convenient in Parts ∗ −n I and II, we shall find it convenient to switch to cohomological grading in later parts. We agree to refer to chain complexes over k simply as “k-modules”. As usual, we consider graded k-modules without differential to be differential graded k-modules with differential zero, and we view ungraded k-modules as graded k- modulesconcentratedindegree0. Theseconventionsallowustoviewthetheoryof generalizedalgebrasasaspecialcaseofthetheoryofdifferentialgradedgeneralized algebras. The differentials play little role in the theory of the first four sections. As will become relevant in Part II, everything in these sections works just as well in the still more general context of simplicial k-modules. Webeginwiththedefinitionofanoperadofk-modules. Whilethereareperhaps more elegant equivalent ways of writing the definition, the original explicit version of [46] still seems to be the most convenient, especially for concrete calculational purposes. Wheneverwedealwithpermutationsofk-modules,weimplicitlyusethe standard convention that a sign (−1)pq is to be inserted whenever an element of degree p is permuted past an element of degree q. Definition 1.1. An operad C consists of k-modules C(j), j ≥ 0, together with a unit map η :k →C(1), a right action by the symmetric group Σ on C(j) for each j j, and maps γ :C(k)⊗C(j )⊗···⊗C(j )→C(j) 1 k (cid:80) for k ≥1 and j ≥0, where j =j. The γ are required to be associative, unital, s s and equivariant in the following senses. (cid:80) (cid:80) (a) The following associativity diagrams commute, where j = j and i = i; s t we set g =j +···+j , and h =i +···+i for 1≤s≤k: s 1 s s gs−1+1 gs (cid:79)k (cid:79)j (cid:79)j C(k)⊗( C(j ))⊗( C(i )) γ⊗Id (cid:47)(cid:47)C(j)⊗( C(i )) s r r s=1 r=1 r=1 γ (cid:178)(cid:178) shuffle C((cid:79)(cid:79)i) γ (cid:178)(cid:178) (cid:79)k (cid:79)js (cid:79)k C(k)⊗( (C(j )⊗( C(i ))) (cid:47)(cid:47)C(k)⊗( C(h )). s gs−1+q Id⊗(⊗sγ) s s=1 q=1 s=1 (b) The following unit diagrams commute: 10 IGORKRIZANDJ.P.MAY C(k)⊗(k)k ∼= (cid:47)(cid:47)C(k) k⊗C(j) ∼= (cid:47)(cid:47)C(j) Id⊗ηk (cid:178)(cid:178) (cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)γ(cid:114)(cid:114)(cid:114)(cid:114)(cid:56)(cid:56) η⊗Id(cid:178)(cid:178) (cid:114)(cid:114)(cid:114)(cid:114)(cid:114)(cid:114)γ(cid:114)(cid:114)(cid:114)(cid:114)(cid:57)(cid:57) C(k)⊗C(1)k C(1)⊗C(j) (c) The following equivariance diagrams commute, where σ ∈ Σ ,τ ∈ Σ , k s js σ(j ,...,j ) ∈ Σ permutes k blocks of letter as σ permutes k letters, and τ ⊕ 1 k k 1 ···⊕τ ∈Σ is the block sum: k k C(k)⊗C(j )⊗···⊗C(j )σ⊗σ−1(cid:47)(cid:47)C(k)⊗C(j )⊗···⊗C(j ) 1 k σ(1) σ(k) γ γ (cid:178)(cid:178) (cid:178)(cid:178) C(j) σ(jσ(1),...,jσ(k)) (cid:47)(cid:47)C(j) and C(k)⊗C(j )⊗···⊗C(j ) Id⊗τ1⊗···⊗τk (cid:47)(cid:47)C(k)⊗C(j )⊗···⊗C(j ) 1 k 1 k γ γ (cid:178)(cid:178) (cid:178)(cid:178) C(j) τ1⊗···⊗τk (cid:47)(cid:47)C(j) The C(j) are to be thought of as modules of parameters for “j-ary operations” that accept j inputs and produce one output. Thinking of elements as operations, we think of γ(c⊗d ⊗···⊗d ) as the composite of the operation c with the tensor 1 k product of the operations d . We emphasize that the definition makes sense in any s symmetric monoidal ground category, with product ⊗ and unit object k. In the present algebraic context, the unit map η is specified by a degree zero cycle 1 ∈ C(1). Thedefinitionadmitsseveralminorvariantsandparticulartypes. Recallthat a map of k-modules is said to be a quasi-isomorphism if it induces an isomorphism of homology groups. Variants 1.2. (i) Non-Σ operads. When modelling non-commutative algebras, it is often useful to omit the permutations from the definition, giving the notion of a non-Σ operad. However, one may also keep the permutations in such contexts, using them to record the order in which products are taken. An operad is a non-Σ operad by neglect of structure. (ii) Unital operads. By convention, the 0th tensor power of a k-module A is inter- preted to be k (concentrated in degree 0). The module C(0) parametrizes “0-ary operations” k → A. In practice, one is most often concerned with unital algebras, and one thinks of the unit element 1 ∈ A as specifying a map k → A. In such contexts, it is sensible to insist that C(0) = k, and we then say that C is a unital operad. For types of algebras without units, such as Lie algebras, it is natural to set C(0)=0. (iii) Augmentations. If C is unital, the C(j) have the augmentations (cid:178)=γ :C(j)∼=C(j)⊗C(0)j →C(0)=k.