TOPOLOGICAL ANDRE-QUILLEN COHOMOLOGY AND E1 ANDRE-QUILLEN COHOMOLOGY MICHAEL A. MANDELL Abstract.We compare Andre-Quillen cohomology in various categories of E1 rings. Introduction M. Andre [1] and D. Quillen [16] introduced a cohomology theory for commu- tative rings now called Andre-Quillen cohomology. For a commutative ring k, a commutative k-algebra A, and an A-module M, the Andre-Quillen cohomology of A relative to k with coefficients in M can be defined as a sort of derived func* *tor of derivations. This functor satisfies the appropriate axioms for a cohomology the* *ory and is closely related to "square zero" commutative k-algebra extensions of A. Andre-Quillen cohomology computations play an important role in the construc- tion of the ring spectrum EO2 by Hopkins and Miller. The existence of EO2 and the closely related ring spectrum eo2 have already had a big impact on our un- derstanding of the stable category and of the homotopy groups of spheres. Recent work of Goerss and Hopkins has shown that EO2 is actually a "commutative S- algebra" in the sense of [4], the analogue in the category of spectra of a comm* *utative (differential graded) algebra. Their proof develops an obstruction theory for * *the existence of commutative S-algebra structures. The obstruction groups are alge- braically defined in terms of the homology of the underlying ring spectrum, and are a generalization of Andre-Quillen cohomology to E1 simplicial algebras in a category of comodules. In practice, these groups often reduce to the Andre-Quil* *len cohomology of an E1 simplicial algebra. The category of commutative S-algebras has enough of the formal properties of the category of commutative algebras that many of the different definitions * *of Andre-Quillen cohomology also make sense in this category. Basterra [2] com- pares these definitions, and shows that they are equivalent. We therefore obtai* *n a "Topological Andre-Quillen" cohomology theory for the category of commutative S-algebras. This version of Andre-Quillen cohomology is also closely related to square zero extensions, and it leads to a different obstruction theory for the * *exis- tence of commutative S-algebra structures, and in addition, an obstruction theo* *ry for the existence of commutative S-algebra maps. This is because the Postnikov tower of a connective commutative S-algebra provides a canonical example of a s* *e- quence of square zero extensions. As explained in [2] (following [9]), the k-in* *variants of a commutative S-algebra lift to unique elements of Topological Andre-Quillen cohomology in the following sense. Given an S-module (or spectrum) A, a com- mutative S-algebra structure on the n-th stage of the Postnikov tower A[n] can * *be lifted to the (n + 1)-st stage A[n + 1] if and only if the k-invariant can be l* *ifted to the Topological Andre-Quillen cohomology Dn+2(A[n]; Hssn+1A), and the set 1 2 MICHAEL A. MANDELL of homotopy classes of lifts of the commutative S-algebra structure is in one to one correspondence with the set of lifts of the k-invariant. This is the obstru* *ction theory for the construction of commutative S-algebra structures. From here, ob- struction theory for the construction of commutative S-algebra maps works just as the analogous theory works for spaces or spectra: a map of commutative S- algebras B -! A[n] lifts to A[n + 1] if and only if the k-invariant pulls back * *to zero in Dn+2(B; Hssn+1A) and the set of homotopy classes of lifts has a free transit* *ive action of Dn+1(B; Hssn+1A). In this way, the Topological Andre-Quillen cohomol- ogy of commutative S-algebras provides a natural and complete obstruction theory for connective commutative S-algebras. These generalizations of Andre-Quillen cohomology to very different looking c* *at- egories have similar interesting applications. The purpose of this paper is to * *com- pare both of these theories to a third theory that is somewhat easier to calcul* *ate. This last theory is the Andre-Quillen cohomology in the category of E1 differ- ential graded algebras. We prove that the homotopy theory and Andre-Quillen cohomology of E1 simplicial algebras are equivalent to the homotopy theory and Andre-Quillen cohomology of E1 differential graded algebras. In the case of co* *m- mutative S-algebras, we prove that Topological Andre-Quillen cohomology with coefficients in an Eilenberg-Mac Lane module can be calculated in terms of the Andre-Quillen cohomology of E1 differential graded algebras. We give precise statements in Sections 1 and 5. In a future paper, the Maria Basterra and the author will describe a number of general tools for calculating the Andre-Quillen cohomology of E1 differenti* *al graded algebras, and apply them to calculate Topological Andre-Quillen coho- mology and E1 simplicial algebra Andre-Quillen cohomology in some interesting examples. 1.E1 Simplicial Algebras and E1 Differential Graded Algebras In this section, we state the main results for the comparison of E1 simplici* *al algebras and E1 differential graded algebras. Throughout the paper, we work relative to a base commutative ring k. In algebra we work exclusively in catego* *ries of k-modules and the tensor product should be understood to be the tensor produ* *ct over k. In the context of [6], k is the coefficient ring of a cohomology theory* * and so we must allow k to be a graded ring. In this case, k-modules are graded and differential graded k-modules are bigraded. The internal degree plays no role i* *n the theory below and for the most part can be ignored; in arguments below "degree" always refers to the differential or simplicial degree. Before stating in detail our comparison theorems, we begin with a review of the definition of Andre-Quillen cohomology. The same basic ingredients go into the definition for E1 algebras as go into the definition for commutative algeb* *ras. Recall that for a commutative k-algebra A and a simplicial A-module M, we can define the "square-zero extension" ZM = A M to be the simpicial commutative A-algebra obtained by taking the product of any two elements in M to be zero. The projection ZM -! A is a map of simplicial commutative algebras, and we can look at the set of maps from A to ZM in the "homotopy category" of simplicial commutative k-algebras lying over A. Since ZM is an abelian group object in this category, this set is actually an abelian group. When we take M = nN for an ANDRE-QUILLEN COHOMOLOGY 3 A-module N, the abelian groups we obtain are by definition the Andre-Quillen cohomology of A (as a commutative algebra) with coefficients in M. We give an entirely parallel treatment of the Andre-Quillen cohomology of E1 simplicial and differential graded algebras. For an E1 simplicial or differen* *tial graded k-algebra A, we have a notion of A-module, explained for example in [6, * *10] (but see also Section 2 below). These categories of A-modules have suspension functors that refine the suspension in the category of simplicial k-modules an* *d in the category of differential graded k-modules (where suspension is the shift fu* *nctor). In addition, Definition 2.6 below describes a "square zero multiplication" func* *tor Z from the category of A-modules to the category of E1 algebras lying over A; for an A-module M, the underlying simplicial or differential graded module of ZM is A M. (We write ZA for Z when A is unclear). In both the simplicial and differential graded contexts, we can define Andre-Quillen cohomology as follows. Definition 1.1.Let E denote either the category of E1 simplicial k-algebras or the category of E1 differential graded k-algebras (for a chosen E1 operad). L* *et A be an object of E and let M be an A-module. We define the Andre-Quillen cohomology of A with coefficients in M as Dn(A; M) = h(E=A)(A; ZnM); where h(E=A) denotes the homotopy category of the category of E1 algebras lying over A (the homotopy category of the category of maps in E with codomain A). In this definition n ranges over all values that make sense: all integers in * *the dif- ferential graded context and the non-negative integers in the simplicial contex* *t. The functor Z preserves categorical products, and so naturality in M gives D*(A; M) the structure of a graded k-module. Andre-Quillen cohomology is a functor of the E1 algebra A in the following wa* *y. Let f :A -! B be a map in E. Just as for modules over a ring, we can pull back a B-module structure along f to an A-modules structure. In other words, we obtain a functor f* from the category of B-modules to the category of A-modules that is the identity functor on the underlying (simplicial or differential graded) k-mo* *dules. Likewise we have a functor A xB (-) from E=B to E=A that performs the fiber product on the underlying k-modules; it is the pullback in the category E, and * *so in particular for any C in E=A, the canonical map C -! A xB C is in E=A. We have a natural isomorphism ZA M ~=A xB ZB M in E=A refining the isomorphism of k-modules A xB (B M) ~=A M. The composite h(E=B)(B; ZB M) -! h(E=B)(A; ZB M) ~=h(E=A)(A; ZA M) defines a map D*(B; M) -! D*(A; M) that makes D* appropriately contravari- antly functorial in the algebra variable. In order to compare the Andre-Quillen cohomology of E1 simplicial and differ- ential graded algebras, we first need to compare the categories of E1 simplicia* *l and differential graded algebras. The homotopy category of the over-category h(E=A) up to equivalence does not depend on the particular E1 operad we choose, and so we are free to use the E1 operad of differential graded k-modules C from [10] a* *nd to take advantage of its special properties. In the simplicial context, we use a s* *imilar operad of simplicial k-modules that we denote as E and that we describe precise* *ly in Definition 2.1 below. These operads are both closely related to the linear isom* *etries operad L exploited in [4]. For the whole of this paper, we understand the categ* *ory 4 MICHAEL A. MANDELL of E1 simplicial k-algebras to mean the category EE of E-algebras and the categ* *ory of E1 differential graded k-algebras to mean the category EC of C-algebras. We say an E1 differential graded k-algebra is "non-negative" if its underly- ing differential graded k-module is non-negative, that is, concentrated in non- negative degrees. It turns out that the category of non-negative E1 different* *ial graded k-algebras E+Cis a closed model category with weak equivalences the quas* *i- isomorphisms. The normalization functor N from simplicial k-modules to non- negative differential graded k-modules refines to a functor from E1 simplicial* * k- algebras to E1 differential graded k-algebras. In Section 4, we prove the foll* *owing theorem. Theorem 1.2. The normalization functor from E1 simplicial k-algebras to non- negative E1 differential graded k-algebras is the right adjoint of a Quillen e* *quiva- lence. Quillen equivalence is the natural notion of weak equivalence of model catego* *ries and preserves all homotopy theoretic constructions; see for example [7]. Quill* *en equivalence implies in particular an equivalence of homotopy categories. In add* *i- tion, a Quillen equivalence of model categories induces Quillen equivalences of* * the over-categories of fibrant objects. In any of our model categories of E1 algebr* *as, all objects are fibrant. We therefore obtain as a consequence of the previous theor* *em an equivalence of h(EE=A) with h(E+C=N(A)) for any E1 simplicial k-algebra A. We can consider the categories of non-negative modules over non-negative E1 differential graded k-algebras; these again form a closed model category with w* *eak equivalences the quasi-isomorphisms. For an E1 simplicial k-algebra A, the nor- malization functor N then also refines to a functor from the category of A-modu* *les to the category of non-negative N(A)-modules. In Section 4, we prove the follow* *ing theorem. Theorem 1.3. Let A be an E1 simplicial k-algebra. The normalization functor from A-modules to non-negative N(A)-modules is the right adjoint of a Quillen equivalence. In order to understand the relationship between h(EE=A) and h(EC=N(A)) and to compare the Andre-Quillen cohomology theories, we now just need to understand the relationship between the model category of E1 differential graded k-algebr* *as and the model category of non-negative E1 differential graded k-algebras. We cannot expect a Quillen equivalence, since a general E1 differential graded k- algebra can have homology in negative degrees, but we do have a Quillen adjunct* *ion. The inclusion of the subcategory of non-negative differential graded k-modules * *into the category of all differential graded k-modules has a right adjoint (-)+ sati* *sfying ( Hq(M+ ) = HqM forq 0 0 forq < 0: for any differential graded k-module M. We explain the following theorem in Sec- tion 3. Theorem 1.4. The functor (-)+ refines to a functor EC -! E+Cthat is right ad- joint to the inclusion functor, and which preserves fibrations and weak equival* *ences. The counit of the adjunction A+ -! A is a weak equivalence when HqA = 0 for all q < 0; the unit of the adjunction B -! B+ is always an isomorphism. ANDRE-QUILLEN COHOMOLOGY 5 It follows that for any non-negative E1 differential graded k-algebra B, the functor h(E+ =B) -! h(E=B) is a full and faithful embedding. As a consequence, we obtain the following corollary to Theorem 1.4. Corollary 1.5.For a non-negative E1 differential graded k-algebra B and a non- negative B-module M, we have a natural isomorphism Dn(B; M) ~=h(E+C=B)(B; ZnM) for n 0. When HqM = 0 for all q > 0, we have in addition that Dn(B; M) = 0 for n < 0. The equivalence of virtually all properties of Andre-Quillen cohomology can be deduced from the following theorem together with the usual properties of the normalization functor N. Theorem 1.6. Let A be an E1 simplicial k-algebra, and let M be an A-module. The canonical isomorphism of differential graded k-modules N(A) N(M) -! N(A M) is an isomorphism ZN(A)N(M) -! N(ZA M) in EC=N(A). For A, M, as above, the natural quasi-isomorphism of differential graded k- modules N(M) -! N(M) is a map of N(A)-modules. We therefore get the following corollary of Theorems 1.2, 1.4, and 1.6. Corollary 1.7.Let A be an E1 simplicial k-algebra and let M be an A-module. There is a natural isomorphism Dn(A; M) ~=Dn(N(A); N(M)) for n 0. If ssqM = 0 for q > 0, then Dn(N(A); N(M)) = 0 for all n < 0. 2.Algebras, Modules, and the Operads E, C In this section, we give a short review of the theory of E1 simplicial algeb* *ras, E1 differential graded algebras, and their modules in terms of the operads E a* *nd C. As shown in [10] in the differential graded context, there is a category of * *E1 k-modules with a symmetric weak-monoidal product such that E1 k-algebras over this operad are exactly the commutative monoids for this product. In addition, a module over an E1 k-algebra is exactly a module for this product. We begin with the definition of the operads. Definition 2.1.Let L be the linear isometries operad, the E1 operad of spaces with L(n) the space of linear isometries from (R1 )n to R1 [4, I.3.1]. Let SoL denote the singular simplicial set of L, an E1 operad of simplicial sets. Let * *E be the free k-module on SoL, an E1 operad of simplicial k-modules. Let C be the normalization of E, an E1 operad of differential graded k-modules. The collection of simplicial sets SoL inherits its operad structure from L si* *nce the singular functor commutes with cartesian products. Likewise, the collection E inherits its operad structure from SoL since the free k-module functor conver* *ts cartesian products to tensor products. The collection C inherits its operadic s* *truc- ture from E since the normalization functor is lax symmetric monoidal: the shuf* *fle map gives an associative and symmetric natural transformation C(n) (C(j1) . . .C(jn)) = NE(n) (NE(j1) . . .NE(jn)) -! N(E(n) (E(j1) . . .E(jn))); 6 MICHAEL A. MANDELL see [13, p. 132] or [10, xI.5] for details. As mentioned in the previous section, we work exclusively with the operad E in simplicial k-modules and C in differential graded k-modules, and we denote t* *he categories of E1 simplicial k-algebras, E1 differential graded k-algebras, and * *non- negative E1 differential graded k-algebras as EE, EC, and E+Crespectively. We use the following version of E1 simplicial k-modules and E1 differential grad* *ed k-modules. Definition 2.2.The category ME of E1 simplicial k-modules is the category of modules over the simplicial algebra E(1). The category MC of E1 differential graded k-modules is the category of modules over the differential graded algebra C(1). The category M+Cof non-negative E1 differential graded k-modules is the category of non-negative modules over the differential graded algebra C(1). It is clear from this definition that the tensor product M X of an E1 simplic* *ial k-module M and a simplicial k-module X is naturally an E1 simplicial k-module. In particular, we have a suspension functor on the category of E1 simplicial k-modules; this functor shifts homotopy groups ssnM ~= ssn+1M. Likewise, the tensor product of an E1 differential graded k-module and a differential graded* * k- module is naturally an E1 differential graded k-module, and we have a suspension functor on the category of E1 differential graded k-modules that shifts homolo* *gy groups HnN ~=Hn+1N. The main results of Part V of [10] concern a symmetric weak-monoidal product for E1 differential graded k-modules analogous to the tensor product in the category of differential graded k-modules. Although [10] considered only Z-grad* *ed modules, it is easy to see that restricts to a product on non-negative modules* * with the same formal properties. Essentially the same construction defines an analog* *ous product for E1 simplicial k-modules: We define the bifunctor M N for E1 simplicial k-modules M and N by M N = E(2) E(1)E(1)(M N): The left action of E(1) on E(2) makes M N naturally an E1 simplicial k-module. We have a natural symmetry isomorphism M N ~=N M induced by the symme- try isomorphism M N ~=N M and the transposition isomorphism E(2) ~=E(2) induced by the right 2-action on E(2). We also have unit a map : k N -! N induced by the operad multiplication E(2) (k E(1)) = E(2) (E(0) E(1)) -! E(1): It is important to note that is generally not an isomorphism. Just as in the differential graded case [10, xV.1], the product is coherently associative, an* *d the natural transformations satisfy all the properties of a symmetric monoidal prod* *uct except that the unit map is not an isomorphism. We summarize this in the follow* *ing proposition. Proposition 2.3.The products on ME, MC, and M+Care symmetric weak- monoidal. The proof is the same in all important details as the one in [10]. The main p* *oint is that the operad E has the special property that the maps E(2) E(1)E(1)(E(i) E(j)) -! E(i + j) ANDRE-QUILLEN COHOMOLOGY 7 induced by the operadic multiplication are isomorphisms for all i; j > 0 ("Hopk* *ins' Lemma"). A further consequence of this is the following proposition. Proposition 2.4.The category EE of E1 simplicial k-algebras is exactly the cat- egory of commutative monoids for in ME. The category EC of E1 differential graded k-algebras is exactly the category of commutative monoids for in MC. A commutative monoid in a symmetric weak-monoidal category is defined just as in a symmetric monoidal category: it is an object A together with a maps : k -!* * A and : A A -! A such that the following associativity, commutativity, and unit diagrams commute. id o A A A ______//A A A A> ________//A A k A=________//A A >> === id || || >>> == fflffl| fflffl| >OEOE =OEOE A A __________//A A A The reader unfamiliar with the theory of E1 algebras can take the previous prop* *o- sition as a definition. Likewise, the reader unfamiliar with the theory of modu* *les over E1 algebras can take the following proposition as a definition. Proposition 2.5.For an E1 simplicial k-algebra A, the category of A-modules MA is exactly the category of left A-objects for in ME. For an E1 differential graded k-algebra B, the category of B-modules MB is exactly the category of left B-objects for in MC. A left A-object is an object M in ME together with a map :A M -! M such that the following associativity and unit diagrams commute. id id A A M ______//A M k M@ ________//A M @@ """ id || || @@@ """ fflffl| fflffl| @OO"""" A M __________//M M Finally we can describe the functor Z. Let A be an E1 simplicial or different* *ial graded k-algebra and let M be an A-module. We define a square zero multiplica- tion : ZM ZM -! ZM on ZM = A M as follows. We have a canonical isomorphism ZM ZM ~=(A A) (A M) (M A) (M M): We define to be the map induced by the multiplication of A on the summand A A, the action of A on M on the summands A M and M A, and the zero map on the summand M M. This makes ZM an E1 simplicial or differential graded k-algebra, and the projection map ZM -! A is a map of E1 simplicial or differential graded k-algebras. Definition 2.6.For an E1 simplicial or differential graded k-algebra A, define the functor Z :MA -! E=A by ZM = A M, an object of E=A via the square zero multiplication and the projection map A M -! A. 8 MICHAEL A. MANDELL 3. The Normalization Functor The purpose of this section is to review the background needed to prove Theo- rems 1.2, 1.3, 1.4, and 1.6. The arguments for the last two theorems are relati* *vely straightforward and are given in this section; the proofs of Theorems 1.2 and 1* *.3 require the construction of left adjoint functors and are given in the next sec* *tion. The basis for these theorems is the refinement of the normalization functor to a functor of E1 algebras and E1 modules. We begin with the following observation. Theorem 3.1. The normalization functor refines to a functor from the category ME of E1 simplicial k-modules to the category MC of E1 differential graded k- modules and is lax symmetric monoidal for the product. Proof.Since ME is the category of simplicial E(1)-modules, MC is the category of differential graded C(1)-modules, and C(1) = N(E(1)), the first observation * *is classical: if M is a simplicial E(1)-module, the shuffle map C(1) N(M) = N(E(1)) N(M) -! N(E(1) M) composed with the normalization of the E(1)-action map E(1) M -! M defines the C(1)-action map on N(M). For E1 simplicial k-modules M1, M2, the shuffle map C(2) N(M1) N(M2) -! N(E(2) M1 M2) induces the natural trans- formation N(M1) N(M2) -! N(M1 M2). It is straightforward to check that * * __ this natural transformation is suitably associative, commutative, and unital. * * |__| If A is an E1 simplicial k-algebra, the natural map N(A)N(A) -! N(AA) induces an associative, commutative, and unital multiplication on N(A). Likewis* *e, if M is an A-module, the natural map N(A) N(M) -! N(A M) induces an associative and unital N(A) action on N(M). We obtain the following immediate corollary of the previous theorem. Corollary 3.2.The normalization functor refines to a functor from the category EE of E1 simplicial k-algebras to the category EC of E1 differential graded k- algebras. For an E1 simplicial k-algebra A, the normalization functor refines * *to a functor from the category MA of A-modules to the category MN(A) of N(A)- modules. As another consequence, we get the proof of Theorem 1.6. Proof of Theorem 1.6.The inclusions of A and M in AM induce an isomorphism N(A) N(M) -! N(A M); i.e. an isomorphism of E1 differential graded k-modules ZN(M) -! N(ZM). The composite of the isomorphism (N(A) N(A)) (N(A) N(M)) (N(A) N(M)) (N(M) N(M)) ~= N(ZM) N(ZM) with the multiplication N(ZM) N(ZM) -! N(ZM) is the multiplication of N(A) on the first factor, the action maps on the middle two factors, and the ze* *ro map on the last factor. This therefore coincides with the multiplication_on_ZN(* *M) under the isomorphism above. |__| ANDRE-QUILLEN COHOMOLOGY 9 Before moving on to Theorems 1.2 and 1.3, we need to review the closed model structures. In general, categories of simplicial algebras over simplicial opera* *ds of k-modules admit a right proper simplicial closed model structure [14, 3.3.6]. T* *he following proposition is a special case. Left properness is an immediate conseq* *uence of [12, 7.1-2]. Proposition 3.3.The category EE of E1 simplicial k-algebras admits a proper simplicial closed model structure with the weak equivalences and fibrations the* * weak equivalences and fibrations of the underlying simplicial sets. Since E1 simplicial k-algebras are in particular simplicial abelian groups, a* * map between them is a weak equivalence if and only if it is a quasi-isomorphism aft* *er normalization. Likewise, a map between them is a fibration if and only if it is* * a surjection in positive degrees after normalization. The model structure describ* *ed in the previous proposition therefore appears very similar to the model structu* *re described in the following proposition. The first part is the main result of [1* *2]; the second part is an easy consequence of the same arguments. Proposition 3.4.The category EE of E1 differential graded k-algebras admits a proper closed model structure with the weak equivalences the quasi-isomorphisms and the fibrations the surjections. The category E+Eof non-negative E1 differe* *ntial graded k-algebras admits a proper closed model structure with the weak equivale* *nces the quasi-isomorphisms and the fibrations the maps that are surjections in posi* *tive degrees. Comparing the two closed model structures in the previous proposition, we ob- tain the following portion of Theorem 1.2. As mentioned above, the remainder of the proof is in the next section. Proposition 3.5.The normalization functor EE -! E+Cpreserves fibrations and weak equivalences. Proposition 3.4 also gives us what we need to prove Theorem 1.4. Proof of Theorem 1.4.Recall that the functor (-)+ takes a differential graded k- module M to the differential graded k-module M+ that is the same as M in positi* *ve degrees, the zero cycles of M in degree zero, and zero in negative degrees. For* * any differential graded k-module M, we therefore obtain a natural map C(2) M+ M+ -! C(2) M M; and hence a natural map M+ M+ - ! M M. For an E1 differential graded k-algebra A, composing the map A+ A+ -! A A with the multiplication, we obtain a map A+ A+ -! A that must factor (uniquely) through A+ since A+ A+ is non-negative. This multiplication makes A+ an E1 differential graded k-algebra and the map A+ -! A a map of E1 differential graded k-algebras. Clearly, (-)+ is right adjoint * *to the inclusion E+C -! EC, and it preserves weak equivalences and fibrations_by Proposition 3.4. |__| We now turn to the module categories. For an E1 differential graded k-algebra B, [10, V.4.3] describes a free functor FB from differential graded k-modules to 10 MICHAEL A. MANDELL B-modules. For an E1 simplicial k-algebra A, we can give an entirely similar description of the free functor FA from simplicial k-modules to A-modules: For a simplicial k-module X, the following diagram is a pushout k (E(1) X) _____//A (E(1) X) || || fflffl| fflffl| E(1) X ___________//_FA X: Of course, E(1) X is the free E1 k-module on X. If the product were unital, A (E(1) X) would be the free A-module on X; the pushout along the unit map corrects for the fact that is not an isomorphism. It is convenient to record * *here the following observation. It is proved in [10, V.4.6], [12, 6.1] in the diffe* *rential graded case; the simplicial case is an immediate consequence of [12, 7.1-2]. Proposition 3.6. (i)Let A be an E1 simplicial k-algebra. There is a natural homotopy equivalen* *ce of simplicial k-modules FA X -! A X. (ii)Let B be an E1 differential graded k-algebra. There is a natural homotopy equivalence of differential graded k-modules FB X -! B X. The closed model structures on the module categories follow easily from Quill* *en's small objects argument. Proposition 3.7.Let A be an E1 simplicial k-algebra. The category MA of A- modules is a proper simplicial closed model category with weak equivalences and fibrations the weak equivalences and fibrations of the underlying simplicial se* *ts. Proposition 3.8.Let B be an E1 differential graded k-algebra. The category MB of B-modules is a proper closed model category with weak equivalences the qu* *asi- isomorphisms and fibrations the surjections. If B is non-negative, then the ca* *te- gory M+Bof non-negative B-modules is a proper closed model category with weak equivalences the quasi-isomorphisms and fibrations the maps that are surjection* *s in positive degrees. We get the following portion of Theorem 1.3. The remainder is proved in the next section. Proposition 3.9.Let A be an E1 simplicial k-algebra. The normalization functor MA -! M+N(A)preserves fibrations and weak equivalences. 4. The Proofs of Theorems 1.2 and 1.3 The proofs of Theorems 1.2 and 1.3 begin with the construction of the left adjoint functors. Actually, an easy application of Freyd's adjoint functor theo* *rem [11, V.6.2] shows that the left adjoints must exist, but it is easy to describe* * them relatively explicitly. Recall that the normalization functor from simplicial k-modules to non-negati* *ve differential graded k-modules is an equivalence of categories; we denote its in* *verse as . We denote the free functor from simplicial k-modules to E1 simplicial k-algeb* *ras as E and the free functor from differential graded k-modules to E1 differential graded k-algebras as C. If X is a non-negative differential graded k-module and* * A is an E1 simplicial k-algebra, then the free/forgetful and /N adjunctions indu* *ce natural bijections between the following sets of maps ANDRE-QUILLEN COHOMOLOGY 11 o The set of E1 differential graded k-algebra maps from CX to N(A), o The set of differential graded k-module maps from X to N(A), o The set of simplicial k-module maps from X to A, and o The set of E1 simplicial k-algebra maps from EX to A. Thus, by the Yoneda Lemma, the left adjoint to normalization applied to CX must be isomorphic to EX, naturally in X. Looking at NEX, the inclusion of X = NX induces a map of E1 differential graded k-algebras CX -! NEX. Applying , we obtain a natural transformation of simplicial k-modules C -! E. Now, since left adjoints preserve coequalizers, for an arbitrary E1 differential graded k-algebra B, the left adjoint functor a* *pplied to B must be the E1 simplicial k-algebra B that makes the following diagram a coequalizer in EE. (4.1) E(CB) _____////_EB___//B Here one map on the left is induced by the monad action map CB -! B, and the other is induced by the natural transformation C -! E and the monad multiplication EE -! E. We required that this diagram be a coequalizer of E1 simplicial k-algebras, but since it is a reflexive coequalizer (the reflection * *induced by the unit map B -! CB), it is also a coequalizer of the underlying simplicial k-modules; see for example [4, II.6.6]. As an alternative to the adjoint functor theorem, the left adjoint can be de* *fined by the coequalizer diagram (4.1). In the module case, similar observations apply with FA and FN(A) playing the roles of E and C. We have already shown in both the algebra and module contexts that the nor- malization functor N preserves fibrations and weak equivalences. To show that we have Quillen equivalences, we only have to show that the unit of each adjunctio* *n is a weak equivalence on cofibrant objects. From here the proof in the module case* * is easy using Proposition 3.6, the usual long exact homology sequence of a cofibra* *tion, and passage to colimits. We concentrate on the algebra case. It suffices to consider the case when B is a "cell algebra" [12, 3.1], that i* *s, B = Colim Bn with B0 = k and Bn+1 = Bn qCXn CCXn where X0, X1, : : : are degreewise free differential graded k-modules with zero differential. Here * *CXn denotes the cone on Xn, the contractible differential graded k-module that comes with a conical inclusion Xn -! CXn such that the quotient CXn=Xn is canonically isomorphic to Xn. Since both and N commute with filtered colimits, it suffices to show that the unit is a weak equivalence for each Bn. We argue by induction. As a base case, B1 = CX, and the unit of the adjunction CX -! N(EX) is the map L (n) L (n) C(n) n X -! N(E(n) n (X) ): A basic homological argument shows that this map is a weak equivalence. Now assume by induction that the unit is an equivalence for any cell E1 diffe* *ren- tial graded k-algebra that can be built in n or fewer stages for n 1, and cons* *ider Bn+1 = Bn qCX CCX. We then have that Bn+1 = Bn qEX E(CX). We could proceed by comparing the effect on homology and homotopy groups of the pushouts in the two categories; however, it is easier to compare both construct* *ions 12 MICHAEL A. MANDELL to a third, milder construction, a bar construction version of the pushout. Let fiq = Bn q CX_q_._.q.CX_-z_____"qCCX: q factors Then fio assembles to a simplicial object in E+Cwith face and degeneracy maps j* *ust as in the bar construction: The face maps are induced by the maps CX -! Bn, CX -! CCX, and the codiagonal CXqCX ~=C(XX) -! CX. The degeneracy maps are induced by the inclusions in all but one of the CX factors. The map fi0 = Bn q CCX - ! Bn+1 induces a simplicial map from fio to the constant simplicial object on Bn+1. We can normalize fio to a double complex and we obtain a map of differential graded k-modules from the total complex to Bn+1. It follows for example from [12, 4.1,4.8] that this map is a quasi-isomorphism. The advantage of working with fio is that we can apply to it degreewise, fiq = Bn q (EX)_q_._.q.(EX)_-z________"q(ECX) q factors Again, the map Bn q ECX -! Bn+1 induces a bisimplicial map from fio to Bn+1 considered constant in the "second" simplicial direction. Applying the normalization functor in the "first" simplicial direction, we obtain a commutat* *ive diagram of simplicial differential graded k-modules fio_______//Nfio | | fflffl| fflffl| Bn+1 _____//NBn+1: We know that the left vertical arrow becomes a quasi-isomorphism after passing * *to the total complex of the normalized double complex. The top horizontal arrow is a degreewise quasi-isomorphism since each fiq is a cell algebra that can be bui* *lt in n or fewer stages when n > 2 or by the observation on free algebras above when n = 2. Thus, to show that the bottom horizontal arrow is a quasi-isomorphism, it suffices to show that the right horizontal map is a quasi-isomorphism after pas* *sing to the total complex of the normalized double complex. We prove the slightly stronger statement that the map from the diagonal of fio to Bn+1 is a homotopy equivalence of simplicial k-modules. If we write o[1] for the standard 1-simplex simplicial k-module, then we have an identification fiq ~=Bn qEX E(X q[1]) qEX ECX; and so the diagonal simplicial k-module of fio is canonically isomorphic to Bn qEX E(X o[1]) qEX ECX: The map fio -! Bn+1 is the map that collapses X o[1] to X. Let Y be the simplicial k-module Y = (X o[1]) [X CX: Then we can write diagfio more concisely as Bn+1 qEX EY , and we see that the map diagfio -! Bn+1 is induced by a map of simplicial k-modules under X from Y to CX. Next we observe that the map Y -! CX is a homotopy equivalence of simpli- cial k-modules under X. To see this we only need to check that the normalization ANDRE-QUILLEN COHOMOLOGY 13 of this map NY -! CX is a chain homotopy equivalence under X. If we write I for N(o[1]), then we have that CX = (X I)=X while NY = N(X o[1]) [X (X I)=X: We have a canonical chain homotopy equivalence under X X of N(X o[1]) with X I, given by the shuffle and Alexander-Whitney maps and the classical homotopy from their composite to the identity. The map NY -! CX factors through the map (X I) [X (X I)=X -! (X I)=X that collapses the first X I down to X, and this map is a chain homotopy equiv- alence. We conclude that Y -! CX is a homotopy equivalence of simplicial k-modules under X. Applying the free functor E to the homotopies above and pushing out along the map EX -! Bn, we get that the map diagfio -! Bn+1 is a homotopy equivalence of E1 simplicial k-algebras. Since the forgetful functor from E1 * *sim- plicial k-algebras to simplicial k-modules is simplicial, the map is also a hom* *otopy equivalence of simplicial k-modules. 5. Commutative R-Algebras and Andre-Quillen Cohomology Now we turn to topology and begin our comparison of the Topological Andre- Quillen cohomology of commutative S-algebras with the Andre-Quillen cohomology of differential graded k-algebras. Here we work relative to a commutative S-alg* *ebra R. For our comparison of Andre-Quillen cohomology, we need to assume that R is connective and that R is cofibrant as a commutative S-algebra. In topology, * *we work in the category of R-modules, and ^ denotes the smash product over R. In algebra, we fix k = ss0R. Since we are assuming that R is connective, R admits a map of commutative S- algebras to the Eilenberg-Mac Lane S-algebra Hk. The category of R-modules has an "ordinary homology theory" H*M = ss*(Hk ^ M). As explained in [4, xIV.2], ordinary homology can also be calculated by CW approximation and application of a cellular chain functor. When R = Hk, this cellular chain functor induces an equivalence of the homotopy category of Hk-modules with the homotopy category of differential graded k-modules. We would like to extend this theory in some w* *ay to the category of commutative R-algebras. In attempting to construct some commutative R-algebra version of the cellular chain functor, a difficulty immediately presents itself: There is no obvious c* *ell structure on the free commutative R-algebra on a CW R-module X, W (j) PX = X =j: This problem is intrinsic. For example, (SnR)(p)=p is the quotient of a single * *cell and yet it has homology in infinitely many degrees when p is not invertible in * *k [4, III.5.1]. If we worked with R = S, one solution would be to use a version of the theory of [4] using the geometric realization of the simplicial operad SoL in p* *lace of the linear isometries operad L, and to work with filtrations of the underlyi* *ng spectra. This becomes slightly awkward for the more general choice of R we use. Instead, we replace the category of commutative R-algebras with a category of E1 R-algebras. 14 MICHAEL A. MANDELL Definition 5.1.Let G be the operad of spaces obtained from the operad of simpli- cial sets SoL by geometric realization. We define the category EG of E1 R-algeb* *ras to be the category of G-algebras of R-modules. The trivial maps G(n) -! * induce a map of operads from G to the commutative operad, and we obtain a forgetful functor from the category of commutative R- algebras to the category of E1 R-algebras. We prove the following theorem in t* *he next section. Theorem 5.2. The category EG of E1 R-algebras is a proper closed model category with weak equivalences and fibrations the weak equivalences and fibrations of t* *he underlying R-modules. The forgetful functor from the category of commutative R- algebras to the category of E1 R-algebras is a Quillen equivalence. Recall that the functor G defined by W (j) GX = G(j)+ ^j X is a monad on the category of R-modules with monadic multiplication induced by the operadic multiplication on G. A G-algebra of R-modules is by definition a G-algebra, and G becomes the free functor from R-modules to E1 R-algebras. Now when X is a CW R-module, the free E1 R-algebra GX does have a filtration suitable for calculating its homology; it is an example of a CW E1 R-algebra, which we define in Section 8. In Sections 7-8, we prove the following theorem. Theorem 5.3. The cellular chain functor C* extends to a functor from the cate- gory of CW R-algebras to the category of E1 differential graded k-algebras. In order for this to provide a useful comparison functor for the category of * *E1 R-algebras, we need to understand the relationship between the category of E1 R-algebras and CW E1 R-algebras; it is analogous to the relationship between the category of spaces and the category of CW spaces (and cellular maps). In Sectio* *n 8, we describe a CW approximation functor and a natural weak equivalence fl : -! Id. We prove the following version of the Whitehead theorem for E1 R-algebras. Here CW G denotes the category of CW E1 R-algebras and hCW G denotes the category obtained from CW G by formally inverting the weak equivalences. Theorem 5.4. The forgetful functor and the CW approximation functor induce inverse equivalences between hCW G and hEG. For any E1 R-algebra A, the for- getful functor and CW approximation functor induce inverse equivalences between h(CW G =A) and h(EG=A). We therefore obtain a composite functor = C* O from E1 R-algebras (or commutative R-algebras) to E1 differential graded k-algebras. In order to compa* *re Andre-Quillen cohomology, we need to discuss module categories. In Section 6, we use the special properties of the operad G to give a definition of modules over* * an E1 R-algebra analogous to the description of modules over an E1 differential grad* *ed algebra in Section 2 (from [10, xV]). We denote the category of modules over an* * E1 R-algebra A as MA ; it is a proper closed model category with weak equivalences and fibrations the weak equivalences and fibrations of the underlying R-modules. Note that when A is an E1 R-algebra by virtue of being a commutative R-algebra, MA differs from the category of modules described in [4]; we denote this latter category as MCom;A to distinguish it. We prove the following comparison theorem in Section 6. ANDRE-QUILLEN COHOMOLOGY 15 Theorem 5.5. Let A be a commutative R-algebra. There is a forgetful functor MCom;A -! MA that is the right adjoint of a Quillen equivalence of closed model categories. In both the commutative R-algebra and E1 R-algebra contexts, we have a square zero multiplication functor ZCom;A or ZE;A from the category of A-modules to the category of commutative or E1 R-algebras lying over A; see [2, x3] or Definitio* *n 6.5. The underlying R-module of ZCom;AM or ZE;AM is A _ M and A _ M -! A is the map induced by the trivial map M -! *. We define the Topological Andre-Quillen cohomology of E1 R-algebras as follows. Definition 5.6.Let A be an E1 R-algebra, and let M be an A-module. Define Dn(A; M) = h(EG=A)(A; nZE;AM) for n 2 Z. This is clearly a functor of M, and it becomes a functor of A, analogously as* * in algebra or as described in the commutative case in [2]: For a map of E1 R-algeb* *ras A -! B, the functor A xB (-): EG=B - ! EG=A has a right derived functor AxRB(-): h(EG=B) -! h(EG=A). We have a natural isomorphism AxRBZE;BM ~= ZE;AM, and hence a map h(EG=B)(B; ZE;BM) -! h(EG=B)(A; ZE;BM) ~=h(EG=A)(A; ZE;AM) that induces a map D*(B; M) -! D*(A; M). In [2], Definition 4.1 gives an analogous description of the Topological Andr* *e- Quillen cohomology of a commutative R-algebra. When A is a commutative R- algebra and M is an A-module in the commutative sense, we can form both ZCom;AM and ZE;AM. These are then two E1 R-algebras lying over A whose underlying R-modules over A are the same. The identity map is a map of E1 R- algebras ZE;AM -! ZCom;AM, and it follows that the Topological Andre-Quillen cohomology D*Com(A; M) of A as a commutative R-algebra is the same as Topolog- ical Andre-Quillen cohomology D*(A; M) of A as an E1 R-algebra. To compare with Andre-Quillen cohomology in algebra, we need a comparison functor on the module level. We describe an appropriate version of the CW appro* *x- imation functor and cellular chain functor for modules, and we can use it to de* *fine a functor from the homotopy category of A-modules to the homotopy category of A-modules (in algebra) but this functor goes in the wrong direction. We need its right adjoint. In Section 8, we prove the following theorem. Theorem 5.7. Let A be an E1 R-algebra. There is a functor R: hMA -! hMA with the following properties. (i)There is a natural isomorphism R ~=R. (ii)For M 2 MA , there is an isomorphism of graded k-modules H*M ~=ss*RM, natural in M. (iii)For B 2 E=A, M 2 MA , there is an isomorphism h(E=A)(B; ZA RM) ~=h(E=A)(B; ZA M); natural in B and M. We obtain the following immediate consequence. Corollary 5.8.Let A be an E1 R-algebra, and let M be an A-module. There is a natural isomorphism D*(A; RM) ~=D*(A; M). 16 MICHAEL A. MANDELL Without being more explicit about the construction of the functor R or about the E1 differential graded k-algebra A, it might at first seem that Corollary * *5.8 does not give any real information. However, from the point of view of obstruct* *ion theory, D*(A; N) is most interesting when the coefficients N have homotopy grou* *ps concentrated in degree zero. In this case, when A is connective, the ss0A-module structure on ss0N then uniquely identifies N up to isomorphism in hMA ; usually N is denoted Hss, where ss = ss0N. On the other hand, by the Hurewicz theorem [4, IV.3.6], we have that H0A ~=ss0A, and there exists an M in hMA , unique up to isomorphism, with H*M = ss*N, that is HqM = 0 for q 6= 0 and H0M = ss. We therefore have that N ~=RM. We summarize this as follows. Corollary 5.9.Let A be a connective E1 R-algebra, and let ss be a ss0A-module. Then D*(A; Hss) ~=D*(A; M) for any M with H0M ~=ss and HqM = 0 for q 6= 0. Since the comparison functor is not part of a Quillen equivalence, we do not immediately obtain a comparison of the formal properties of Topological Andre- Quillen cohomology of E1 R-algebras with those of the Andre-Quillen cohomology of E1 differential graded k-algebras. We discuss some further relations in the* * last two sections. In particular, in Section 10, we show that the isomorphism of And* *re- Quillen cohomology preserves the fundamental cohomological exact sequence (the transitivity sequence) and in Section 11 we compare the operations in the Topo- logical Andre-Quillen cohomology of E1 R-algebras with the operations in the Andre-Quillen cohomology of E1 differential graded k-algebras. Finally, we have the following analogue of [4, IV.2.4]. Theorem 5.10. Let R = Hk. The functor : hEG -! hEC is an equivalence of categories. For any E1 Hk-algebra A, the functors : h(EG=A) -! h(EC=A) and R: hMA -! hMA are equivalences of categories. In general, up to natural isomorphism, the functors and R for R factor throu* *gh the equivalences and R for Hk; see Theorem 9.4 for a precise statement. 6.E1 R-Modules and the Products , C, and For an E1 R-algebra A, although it is possible to define the category of A- modules directly in terms of the operad G and its action on A, the study of the category of A-modules becomes easier if we define it in terms of a product. This product is defined on the category of E1 R-modules. To define the category of E1 R-modules, recall that the space G(1) is a topological monoid, and G(1)+ ^ R is an associative R-algebra, weakly equivalent to R. A left module over G(1)+ ^* * R is exactly an R-module M together with an associative and unital map :G(1)+ ^ M -! M. Definition 6.1.The category MG of E1 R-modules is the category of left modules over G(1)+ ^ R. We define the bifunctor : MG x MG -! MG by M N = G(2)+ ^G(1)xG(1)(M ^ N): with G(1)+ ^ R action induced by the left G(1) action on the space G(2), Just as in [10] and Section 2, the product is coherently associative, commut* *a- tive, and weakly unital. We have the following propositions. Proposition 6.2.The product on MG is symmetric weak-monoidal. ANDRE-QUILLEN COHOMOLOGY 17 Proposition 6.3.The category EG of E1 R-algebras is exactly the category of commutative monoids for in MG. Using the previous proposition, we can define the category of modules over an E1 R-algebra as follows. Definition 6.4.Let A be an E1 R-algebra. The category MA of E1 A-modules is the category of left A-objects for in MG. Likewise, we can now define the square zero multiplication functor. Given an E1 R-algebra A, and an E1 A-module M, let ZM = A _ M, and let : ZM ZM ~=(A A) _ (A M) _ (M A) _ (M M) -! A _ M = ZM be the map that is the multiplication A A -! A on the first wedge summand, is the action map A M -! M and M A ~=A M -! M on the middle wedge summands, and is the trivial map M M -! * on the last wedge summand. Definition 6.5.For an E1 R-algebra A, define the functor ZE;A:MA -! EG=A by ZE;AM = ZM = A_M with the square zero multiplication and the projection map A _ M -! A. We now move on to the comparison theorems 5.2 and 5.5. The proofs of these theorems are complicated slightly by the fact that the product is only weakly unital and so does not share all the formal properties of a symmetric monoidal product. For example, A B is not the coproduct of E1 R-algebras A and B, and A M is not the free A-module on an E1 R-module M. For the correct constructions we need "unital" variants given by the and C products introduced in [10] and [4, XIII]. Definition 6.6.The category MuGof unital E1 R-modules is the category of E1 R-modules under R. For unital E1 R-modules K and L and an E1 R-module M, we define the products K C M and K L as the following pushouts. R M ____//_K M K R [RR R A L _____//K L || || [ || || fflffl| fflffl| fflffl| fflffl| M ______//_K C M K [R L __________//_K L We define M B L symmetrically. The following associativity relations can be proved just as their analogues i* *n [10]. Proposition 6.7.The bifunctor is a symmetric monoidal product on the cate- gory of unital E1 R-modules. There are canonical isomorphisms K C (M N) ~=(K C M) N; K C (L C M) ~=(K L) C M natural in the unital E1 R-modules K, L and the E1 R-modules M, N. These associativity relations lead immediately to the following observations. Proposition 6.8.Let A and B be E1 R-algebras. Then AB is an E1 R-algebra and is the coproduct of A and B in the category of E1 R-algebras. Proposition 6.9.Let A be an E1 R-algebra. Then A C (-) is the free functor from E1 R-modules to E1 A-modules. 18 MICHAEL A. MANDELL The comparison theorems are based on a comparison of the product with the smash product of R-modules. We have a forgetful functor from E1 R-modules to R-modules, which amounts to pulling back the action of G(1)+ ^ R along the unit map R -! G(1)+ ^ R; this functor has as left adjoint the free functor G(1)+ ^ (* *-). We could compare the smash product of R-modules with the product of E1 R-modules using these functors on the homotopy category, but on the point-set category another pair of adjoint functors works a little better. The trivial ma* *p of monoids G(1) -! * induces a map of associative R-algebras G(1)+ ^ R -! R, and so we obtain a "trivial action" functor from R-modules to E1 R-modules. The trivial action functor is a right adjoint; its left adjoint is the "indecomposi* *bles" functor R ^G(1)+^R(-). The category of E1 R-modules is a proper closed model category with weak equivalences and fibrations the weak equivalences and fibrat* *ions of the underlying R-modules by [4, VII.4.8]. The following comparison result is elementary. Proposition 6.10.The free, forgetful adjunction and the trivial action, indecom- posibles adjunction are both Quillen equivalence pairs. The indecomposibles fun* *c- tor is strong symmetric monoidal and the trivial action functor is lax symmetric monoidal. The derived functor of each of these functors is a symmetric monoidal equivalence of homotopy categories. Since the trivial action functor from R-modules to E1 R-modules is lax symmet- ric monoidal, it takes a left A-object for ^ to a left A-object for , that is, * *it refines to a functor MCom;A- ! MA . This is the "forgetful functor" of Theorem 5.5. Proof of Theorem 5.5.We have that the category of E1 A-modules is the category of algebras for a continuous monad on the category of R-modules, namely the mon* *ad A C (G(1)+ ^ (-)). We obtain the model structure from [4, VII.4.10]. Now suppose that A is a commutative R-algebra. The trivial action functor preserves all weak equivalences, and so to see that we have a Quillen equivalence, we just need to* * see that the unit map M -! R ^G(1)+^RM is a weak equivalence for all cell A-modules R. Passing to sequential colimits * *and using the usual long exact sequences of homotopy groups, it suffices to treat t* *he case when M is the free E1 A-module on a sphere R-module, M = A C (G(1)+ ^ SnR). Unwinding the definition of C, we see that A C (G(1)+ ^ SnR) is the pushout (G(2)=(G(1) x *))+ ^ (R ^ SnR)_//(G(2)=(G(1) x *))+ ^ (A ^ SnR) | | | | fflffl| fflffl| G(1)+ ^ SnR_________________//A C (G(1)+ ^ SnR); and the unit map is the quotient of the left action of G(1) on G(2). Since G(2) is isomorphic to G(1) as a left G(1)-space (cf. [4, I.6.1]), it follows that t* *he in- decomposibles functor takes A C (G(1)+ ^ SnR) to A ^ SnR. In addition, since the singular simplicial set functor and geometric realization preserve homotopies, * *the homotopies in [12, 7.1-2] (in the case m = 1) show that the right vertical arro* *w in the diagram above is a homotopy equivalence of R-modules. The map (G(2)=(G(1) x *))+ ^ (A ^ SnR) -! A ^ SnR ANDRE-QUILLEN COHOMOLOGY 19 is a homotopy equivalence of R-modules since G(2)=(G(1) x *) is contractible (by [12, 7.1], for example). It follows that the unit map A C (G(1)+ ^ SnR) -! A ^ SnR is a homotopy equivalence of R-modules and therefore a weak equivalence_of E1 A-modules. |__| As a consequence of the last observation of the proof, that for any E1 R-alge* *bra A, the natural map A C (G(1)+ ^ SnR) -! A ^ SnRis a homotopy equivalence of R-modules, we also obtain the following corollary. Corollary 6.11.Let f :A -! B be a map of E1 R-algebras. The induced functor f* :MB -! MA is the right adjoint of a Quillen adjoint pair. If f is a weak equivalence, then the Quillen adjoint pair is a Quillen equivalence. Finally, we close the section with the proof of Theorem 5.2. Proof of Theorem 5.2.The closed model structure is again an easy application of [4, VII.4.9]. Left properness follows from [12, 7.1-2]. For the Quillen equival* *ence, define the functor : EG -! Com by the following coequalizer PGA _____////_PA_//_A: Here, one of the lefthand arrows is the G action map on A and the other is the composite of the natural map PG -! PP (induced by the map G -! P) and the monadic multiplication PP -! P. A check of universal properties reveals that is left adjoint to the forgetful functor Com -! EG. Since the weak equivalences and fibrations in Com are also the weak equivalences and fibrations of the underlyi* *ng R- modules, a map in Com is a weak equivalence or fibration if and only if the for* *getful functor takes it to a weak equivalence or fibration in EG. It follows that and* * the forgetful functor are a Quillen adjunction. To see that this is a Quillen equiv* *alence, we just have to see that the unit map is a weak equivalence for every cell G-al* *gebra, i.e. for every cell G-algebra [4, VII.4.11]. The unit B -! B is a weak equivale* *nce for B = GX for any CW R-module X by [4, III.5.1] and [4, VII.6.7] (here is where we use the assumption that R is cofibrant). The remainder of the argument is entirely similar to the proof of Theorem 1.2, with geometric realization replac* *ing normalization and diagonalization. The maps |fio| -! Bn+1 and |fio| -! Bn+1 are weak equivalences in this case by virtue of being homotopic to isomorphisms_ [4, VII.3.8]. |__| 7.The Cellular Chain Functor Before we can even define CW E1 R-algebras for our comparison of topology to algebra, we need a number of preliminary observations on the cellular chain fun* *ctor. In fact, this section contains only these preliminary observations, and we post* *pone the definition of CW E1 R-algebras until the next section. We do introduce in t* *his section a category of E1 R-algebras, "quasi-CW" E1 R-algebras, on which we can define the cellular chain functor, but this category is not closed under coprod* *ucts or pushouts. Using this category as a stepping stone, we introduce "E-CW" algebras, whose underlying R-modules are built out of extended power "cells". This catego* *ry of algebras is closed under algebra cell attachment and provides the foundation* * for the definition of CW E1 R-algebras in the next section. 20 MICHAEL A. MANDELL We begin with a quick review of CW R-modules. Recall that a cell R-module is an R-module M = ColimMn such that M0 = *, and each Mn+1 is formed from Mn as the cofiber of a map Fn -! Mn, where Fn is a wedge of sphere R-modules SqR (of varying dimensions). The filtration Mn is called the sequential filtration.* * The only role of the sequential filtration is to allow inductive arguments; it is u* *nrelated to any of the dimensions of the sphere R-modules the cells are attached along. A CW R-module is a cell R-module M = ColimMn that has a "skeletal filtration" Mmn, m 2 Z, such that the attaching maps Fn -! Mn are filtered, and the skeletal filtration on Mn+1 is the obvious one induced on the cofiber; for example, for * *an attaching map f :SqR-! Mn, we understand that f must factor through Mqnand the filtration on the cofiber Cf is Cfm = Mmn for m < q + 1, and Cfm = Cf for m q + 1. The skeletal filtration on M, Mm = ColimMmn, is then closely tied to the formal dimensions of the cells out of which that M is built. A cellular map* * of CW R-modules is a map of R-modules that preserves the skeletal filtration. An E1 R-algebra A has a unit map R -! A. Technically, R is not a cell R-module and this makes a cell R-module structure on A inconvenient as well as unlikely; nevertheless, we can understand R as a cell of dimension zero and def* *ine a unital variant of CW R-modules. We define a CW unital R-module in the same way as a CW R-module except that we start with M0 = R concentrated in skeletal filtration level zero (i.e. Mm0 = * for m < 0, Mm0 = R for m 0). A cellular map of CW unital R-modules is a map of R-modules under R that preserves the skeletal filtration. For defining the cellular chain functor, only two properties of the skeletal * *filtra- tion are needed: Mm =Mm-1 is weakly equivalent to a wedge of SmR's and H*Mm is zero above degree m. We define the cellular chains by C*M = ssm (Mm =Mm-1 ) with differential induced by the connecting homomorphism of the cofiber sequence Mm-1 =Mm-2 - ! Mm =Mm-2 - ! Mm =Mm-1 with an appropriate sign (namely the sign oem , where oe = 1 is the sign of com- muting the suspension isomorphism with the connecting homomorphism). These two properties then imply that H*M ~= H*(C*M). This leads to the following definition. Definition 7.1.A skeletally filtered R-module is an R-module M together with a filtration (by cofibrations) Mm for m 2 Z such that Mm =Mm-1 is weakly equivale* *nt to a wedge of SmR's and H*Mm is zero above degree m. A cellular map of skeletal* *ly filtered R-modules is a map that preserves the skeletal filtration. Proposition 7.2.The cellular chain functor C* extends to a functor from the cat- egory of skeletally filtered R-modules and cellular maps to the category of dif* *ferential graded k-modules. There is a natural isomorphism H*M ~=H*C*M. If X is a CW space and M is a CW unital R-module, then X+ ^ M is not a CW unital R-module, but it is a skeletally filtered R-module. We have introduced t* *he definition of skeletally filtered R-modules just for this reason, to give us a * *language to talk about such objects and their cellular chains. Skeletally filtered R-mod* *ules are not closed under enough of the usual constructions in homotopy theory to be useful for much else. In the E1 R-algebra context, it is convenient to have a category of objects for which we can make sense of the cellular chain functor, * *but which is more general than the category of CW E1 R-algebras. ANDRE-QUILLEN COHOMOLOGY 21 Definition 7.3.A quasi-CW (unital) E1 R-module is a (unital) E1 R-module M that is also a CW (unital) R-module for which the action map G+ ^M -! M is cellular. A quasi-CW E1 R-algebra is an E1 R-algebra A that is also a quasi-CW unital E1 R-module for which the multiplication G(2)+ ^ (A ^ A) -! A is cellula* *r. We understand a map between quasi-CW (unital) E1 R-modules to be cellular if it is a map of E1 R-modules (under R) and a cellular map of CW (unital) R- modules. Likewise a map between quasi-CW E1 R-algebras is cellular if it is a map of E1 R-algebras and a cellular map of CW unital R-modules. When we refer to the category of quasi-CW (unital) E1 R-modules or quasi-CW R-algebras, we understand the maps to be the cellular maps. Theorem 7.4. The cellular chain functor extends to functors: (i)From the category of quasi-CW E1 R-modules to the category of E1 differ- ential graded k-modules. (ii)From the category of quasi-CW unital E1 R-modules to the category of unital E1 differential graded k-modules. (iii)From the category of quasi-CW E1 R-algebras to the category of E1 diffe* *r- ential graded k-algebras. Proof.The first two statements are clear. The map G(3)+ ^ (A ^ A ^ A) -! A is cellular since the map G(2)+ ^ ((G(2)+ ^ A ^ A) ^ A) -! A is, and so the multiplication C(2) C*(A) C*(A) -! C*(A) is associative. It is easy to see that it is also commutative and unital, and so C*A is an E1 differential_graded k-algebra. |__| If N is a quasi-CW (unital) E1 R-module, we say that M is a subcomplex if it is a subcomplex of the underlying CW (unital) R-module and the inclusion is a map of E1 R-modules (under R). We have the following closure properties for quasi-CW unital E1 R-modules. They are immediate consequences of analogous closure properties for CW unital R-modules Proposition 7.5.Let N be a quasi-CW unital E1 R-module and let M be a sub- complex of N. If P is a quasi-CW unital E1 R-module and M -! P is a cellular map, then P [M N is a quasi-CW unital E1 R-module with P as a subcomplex. In addition, it is the pushout in the category of quasi-CW E1 R-modules, and t* *he universal map C*P [C*M C*N -! C*(P [M N) is an isomorphism. Proposition 7.6.Let M0 -! M1 -! . .b.e a sequence of subcomplexes of quasi- CW unital E1 R-modules. Then ColimMn is a quasi-CW unital E1 R-module with each Mn as a subcomplex. In addition, it is the colimit in the category of* * quasi- CW unital R-modules, and the universal map ColimC*(Mn) -! C*(Colim Mn) is an isomorphism. Similar statements hold for quasi-CW E1 R-modules. The analogue of Propo- sition 7.6 also holds for quasi-CW E1 R-algebras, but we need to include more structure before we can have a category of skeletally filtered E1 R-algebras t* *hat satisfies the analogue of Proposition 7.5. The extra structure we add is a cell* * struc- ture based on extended powers G(n)+ ^n M(n)for CW R-modules M, and with subcomplexes based on the map induced by an inclusion of a subcomplex M N. Definition 7.7.We define an E-cell to be a quasi-CW R-module of the form B(n; X) = G(n)+ ^n X 22 MICHAEL A. MANDELL for some n > 0, where X is a finite CW R-module with a cellular n-action. We say that B(n; X) is a subcell of B(n; Y ) when X is a subcomplex of Y and the inclusion X -! Y is n-equivariant. Definition 7.8.An E-CW (unital) module is a quasi-CW (unital) E1 R-module M = ColimMn such that M0 = * (M0 = R), and Mn+1 = Mn [Bn Cn where Bn, Cn are wedges of E-cells, Bn -! Cn is a wedge of subcell inclusions, and Bn -! Mn is a cellular map. An E-CW (unital) module is then by neglect of structure a quasi-CW (unital) E1 R-module by the previous two propositions and their variants. We have the notion of a subcomplex M of an E-CW module N, where the wedges of cells Bn Cn out of which M is constructed are wedge summands of the wedges of cells out of which N is constructed. The analogues of Propositions 7.5 and 7.6 clearly hold for E-* *CW modules. In addition, we have the following result. Theorem 7.9. Let M and N be E-CW unital modules. Then M N is an E-CW unital module with M [R N as a subcomplex. There is a canonical isomorphism C*(M) C*(N) ~=C*(M N). The commutativity isomorphism M N ~=N M is cellular, and for an E-CW unital module P , the associativity isomorphism M (N P ) ~=(M N) P is cellular. If L is a subcomplex of M, then L N is a subcomplex of M N. Proof.We have an isomorphism E1 R-modules B(n; X) B(q; Y ) ~=B(p + q; p+q+ ^pxq (X ^ Y )): The construction of the E-CW unital module structure and the verification that * * __ the maps are cellular are obtained by induction up the sequential filtrations. * * |__| Analogous theorems hold for the product of E-CW modules and the C product of an E-CW unital module and an E-CW module. We can now define an E-CW algebra to be an E1 R-algebra A that is an E-CW unital module for which the multiplication A A -! A is cellular; it is easy to see that this is just the s* *ame thing as a quasi-CW E1 R-algebra that is an E-CW unital module. The following theorem is the motivation for introducing E-CW modules. Theorem 7.10. Let Y be a wedge of finite CW R-modules and let X be a subcom- plex. Let A be an E-CW algebra, and let X -! A be a cellular map. Then the pushout of E1 R-algebras AGX GY is an E-CW algebra that contains A as a sub- complex. In addition, it is the pushout in the category of quasi-CW E1 R-algeb* *ras, and the universal map C*A CC*X CC*Y -! C*(A GX GY ) is an isomorphism. Proof.We can write A GX GY as a sequential colimit of pushouts of modules, A0 = A, A1 = A0 [AC(G(1)+^X)A C (G(1)+ ^ Y ), and in general An = An-1 [AC(G(n)+^n Zn)A C (G(n)+ ^n Y (n)); where Zn is the subcomplex of Y (n)that is the union of the subcomplexes X ^_ Y (n-1), Y ^ X ^ Y (n-2), : :,:Y (n-1)^ X. |__| ANDRE-QUILLEN COHOMOLOGY 23 8. CW E1 R-Algebras We can now define a CW E1 R-algebra exactly analogously to a CW R-module. We start with A0 = R the initial E1 R-algebra, and we form An+1 from An as an E1 R-algebra cofiber of a wedge of spheres along cellular attaching maps: An+1 = An GXn+1 GCXn+1: Inductively each An is an E-CW algebra, and An+1 becomes an E-CW algebra by Theorem 7.10. The colimit A = ColimAn is both a cell E1 R-algebra and an E-CW algebra. Definition 8.1.A CW E1 R-algebra is a cell E1 R-algebra for which the at- taching maps are cellular. Define the category CW G of CW E1 R-algebras to have objects the CW E1 R-algebras and maps the cellular maps. The usual process constructs a CW approximation functor. Construction 8.2. We construct the functor : EG -! CW G and the natural transformation fl : -! Idin EG as follows. For an E1 R-algebra A, let 0A = R and let fl0: 0A -! A be the initial map. Inductively, having constructed the CW E1 R-algebra nA and the map fln :nA -! A, let Dn+1 be the set of commutative diagrams SmR______//CSmR f|| |g| |fflffl fflffl| nA __fln_//_A where the map f is cellular, and m ranges over all integers. Let W m Xn+1 = SRff; ff2Dn+1 and let n+1A = nA GXn+1 GCXn+1, attached by the maps f in the diagrams in Dn+1. We define fln+1: n+1A -! A using fln and the maps g in the diagrams in Dn+1. Let A = ColimnA, and let fl = Colimfln. As an easy consequence of [4, III.2.9], we have the following proposition. Proposition 8.3.For any E1 R-algebra A, the natural map fl is a weak equiva- lence. When A is a quasi-CW E1 R-algebra, there is no reason to expect the map fl to be a cellular map. In this context we can define a functor L by attaching on* *ly those cells from diagrams (f; g) where g is also cellular. The CW E1 R-algebra LA is a subcomplex of A, and the composite LA -! A -! A is cellular. Using [4, III.2.9] again, we obtain the following proposition. Proposition 8.4.Let A be a quasi-CW E1 R-algebra. Then the natural trans- formation LA -! A is a weak equivalence. Theorem 5.4 is now an easy consequence. When A is a CW E1 R-algebra, we define a CW A-module to be a cell A-module where the attaching maps are cellular. We define the category CWM A to be the category of CW A-modules and cellular maps. We can define a CW approximation functor from A-modules to CW A-modules and prove the following theorem. 24 MICHAEL A. MANDELL Theorem 8.5. Let A be a CW E1 R-algebra. The forgetful functor and CW approximation functor induce inverse equivalences between hMA and hCWM A . Proposition 7.5 implies that the cellular chain functor preserves pushouts al* *ong inclusions of subcomplexes of CW A-modules. Similarly, Theorem 7.10 applied in- ductively implies that the cellular chain functor preserves pushouts along incl* *usions of subcomplexes of CW E1 R-algebras. The cellular chain functor also preserves sequential colimits of inclusions of subcomplexes. These two properties are ess* *en- tially all that are needed in this context to apply Brown's Representability Th* *eorem [3] to the functors h(EC=C*A)(C*(-); B) and hMC*A(C*(-); M) for a CW E1 R-algebra A and objects B and M. Choosing representing objects (B) and (M) defines functors : h(EC=C*A) -! h(CW G =A) and : hMC*A -! hCWM A: The defining property of these functors give natural bijections h(CW G =A)(-; (B)) ~=h(EC=C*A)(C*(-); B) and hCWM A(-; (M)) ~=hMC*A(C*(-); M): In other words, and give right adjoint functors to the cellular chain functor. We can now prove Theorem 5.7 Proof of Theorem 5.7.Let R be the composite of : hMA - ! hCWM A and the equivalences hCWM A ' hMA ' hMA : Since C* commutes with suspension, it follows that R commutes with the right adjoint of suspension, and since suspension is an equivalence on the homotopy categories, it follows that R commutes with suspension. Applying the adjunction to the free CW A-module on SnR, we get a bijection HnM ~=ssnM ~=ssnRM, and by naturality, it is an isomorphism of k-modules. Finally, if N is a CW A-modul* *e, then by inspection we have an isomorphism C*ZA N ~=ZA C*N in EC=A. Applying this to M and using the C*; adjunction, we obtain a map (8.6) ZA M -! (ZA C*M) -! (ZA M): We can make GSnRan object of CW G=A via the trivial map SnR-! A. Looking at the set of maps in h(CW G =A) out of GSnR, we identify the map ssn(M) ~=h(CW G =A)(GSnR; ZA M) -! h(EC=A)(CSnR; ZA M) ~=Hn(M) as the isomorphism ssnM - ! HnM. It follows that the map (8.6) is a weak equivalence. Via the equivalence h(CW G =A) ' h(EG=A), the map (8.6) induces a natural transformation h(E=A)(-; ZA RM) -! h(EC=A)((-); ZA M); which is therefore a natural isomorphism. Plugging in A, we get a natural_isomo* *r- phism D*(A; RM) ~=D*(A; M). |__| ANDRE-QUILLEN COHOMOLOGY 25 9.Eilenberg Mac Lane Commutative S-Algebras In this section we discuss the stronger results of Theorem 5.10 for the base commutative S-algebra Hk and the relationship of these equivalences with the isomorphisms we have for a general R as the base commutative S-algebra. In this section, to avoid confusion, we subscript with R or Hk to denote the category in which each construction is made, that is, whether we take R or Hk as the base commutative S-algebra. The first basic fact is the following proposition, which* * is clear from [4, III.4.1] and the constructions. Proposition 9.1.The change of base rings functor Hk ^ (-) and the forgetful functor (MG)Hk -! (MG)R are the left and right adjoint of a Quillen pair. The functor Hk ^ (-) is strong symmetric monoidal and the forgetful functor is lax symmetric monoidal. As a consequence, we also get the following proposition. Proposition 9.2.The change of base rings functor Hk ^ (-) and the forgetful functor (EG)Hk -! (EG)R are the left and right adjoint of a Quillen pair. The previous proposition in particular gives us a natural map of E1 R-algebras A -! Hk ^ A that allows us to regard (Hk ^ A)-modules (of E1 Hk-modules or of E1 R-modules) as A-modules. Since the forgetful functor takes the E1 Hk- algebra (ZHk^A M)Hk to the E1 R-algebra (ZHk^A M)R , we obtain the following proposition. Proposition 9.3.Let A be an E1 R-algebra, and let M be an (Hk ^ A)-module of E1 Hk-modules. There is a natural isomorphism DnR(A; M) ~=DnHk(Hk ^ A; M): The change of rings functor Hk ^ (-) takes sphere R-modules to sphere Hk- modules and so it converts CW (unital) R-modules to CW (unital) Hk-modules. Moreover, it induces an isomorphism of the cellular chain functors. It follows * *that in the homotopy category, R factors through Hk up to natural isomorphism. Likewise, RR factors through RHk up to natural isomorphism. In fact, we can prove the following more precise theorem. Theorem 9.4. There is a natural map of CW E1 Hk-algebras Hk ^ R (-) -! Hk (Hk ^ (-)) such that the diagram Hk ^ R (-)L______________//Hk (Hk ^ (-)) LLL~LLL ~qqqqq id^flLL%%L xxqqflqqq Hk ^ (-) commutes. The induced map R (-) -! Hk (Hk ^ (-)) on cellular chains induces a map on Andre-Quillen cohomology that makes the following diagram commute 26 MICHAEL A. MANDELL for any Hk (Hk ^ A)-module M. ~= DnHk(Hk ^ (-); RM)_____//Dn(Hk (Hk ^ (-)); M) ~=|| || fflffl| fflffl| DnR(-; RM) ______~=_____//Dn(R (-); M) Proof.We have Hk^(R )0(-) = Hk = (Hk )0(-) as a base case, and by induction, we construct a natural cellular map Hk ^ (R )n(-) -! (Hk )n(Hk ^ (-)) making the obvious diagram with id^fln and fln commute. Then smashing with Hk takes a diagram in (Dn+1)R to a diagram in (Dn+1)Hk , and tells us how to construct t* *he map Hk^(R )n+1(-) -! (Hk )n+1(Hk^(-)). In the colimit we obtain the desired_ map. The second diagram above commutes by an easy adjunction argument. |__| We close with the proof of Theorem 5.10. Proof of Theorem 5.10.Since the functor is the composite of the CW approxi- mation functor and the cellular chain functor C* on CW algebras, and since R is the composite of an equivalence with the right adjoint to the cellular chain fu* *nctor C* on CW modules, we just need to show that the functors C*: h(CW G =A) -! h(EC=C*A) and C*: hMA -! hMC*A are equivalences for a CW E1 Hk-algebra A. Using the adjunctions and , the equivalence is easily deduced from the fact that the ordinary homology groups H* are naturally isomorphic to the homotopy__ groups ss* for Hk-modules, cf. [4, IV.2.4]. |__| 10. The Transitivity Sequence Corollary 5.8 provides a natural isomorphism of functors from the Topological Andre-Quillen cohomology of an E1 R-algebra to the Andre-Quillen cohomology of an E1 differential graded k-algebra, but we would like to have an isomorphi* *sm of cohomology theories. Since the functor R is the right adjoint to the functor* * C*, which is a triangulated functor between triangulated categories, we see that R * *is a triangulated functor, and so the isomorphism of Corollary 5.8 is homological in* * the coefficient variable. In this section, we show that this isomorphism is cohomol* *ogical in the algebra variable by showing that it commutes with the fundamental long exact sequence, the transitivity sequence [16, 5.1], [2, 4.3]. Because we are working in the category of E1 algebras instead of the category of commutative algebras, we have an artificial distinction between a commutative algebra like R or k that we can work relative to and a general E1 algebra. For this reason, we need to define the following relative version of Topological An* *dre- Quillen cohomology; in the commutative context, it is equivalent to working ove* *r a different base commutative algebra. Definition 10.1.Let A be an E1 R-algebra. The category EA of E1 A-algebras is the category of E1 R-algebras lying under A. Let B be an E1 A-algebra, and let M be a B-module. We define the Topological Andre-Quillen cohomology of B relative to A with coefficients in M as Dn(B\A; M) = h(EA =B)(B; ZB nM); where ZB nM is regarded as an E1 A-algebra via the composite of the map A -! B and the inclusion of B in B _ nM. We make the analogous definitions in the category of E1 differential graded k-algebras. ANDRE-QUILLEN COHOMOLOGY 27 A map of E1 A-algebras over B is in particular a map of E1 R-algebras over B, and we get a natural transformation D*(B\A; M) -! D*(B; M). Theorem 10.2. (The Transitivity Sequence) Let A be an E1 R-algebra, let B be an E1 A-algebra, and let M be a B-module. There is a long exact sequence . .-.! Dn-1(A; M) -! Dn(B\A; M) -! Dn(B; M) -! Dn(A; M) -! . .:. We have an analogous sequence in the differential graded context. The followi* *ng theorem compares them. Theorem 10.3. Let A be an E1 R-algebra, B an E1 A-algebra, and M a B- module. There is a natural isomorphism D*(B\A; RM) ~=D*(B\A; M) and the following diagram commutes. Dn-1(A; RM) ____//_Dn(B\A; RM)____//_Dn(B; RM)___//_Dn(A; RM) | | | | | | | | fflffl| fflffl| fflffl| fflffl| Dn-1(A; M) ____//Dn(B\A; M) ____//Dn(B; M) _____//Dn(A; M) Theorem 10.2 can be proved just as the commutative case is proved in [2]; we summarize the argument below to facilitate the proof of Theorem 10.3. The long exact sequence arises from a cofiber sequence of E1 R-algebras. To describe thi* *s, it is convenient to work in the category EB =B, where the initial object is isomor* *phic to the final object. Proposition 10.4.Let A be a cofibrant E1 R-algebra, let B be a cofibrant E1 A-algebra, and let M be a B-module. There are natural isomorphisms Dn(B\A; M) ~=h(EB =B)(B A B; ZB nM) Dn(B; M) ~=h(EB =B)(B B; ZB nM) Dn(A; M) ~=h(EB =B)(B A; ZB nM): Proof.The functor BA (-) is left adjoint to the forgetful functor EB =B -! EA =* *B, the functor B (-) is left adjoint to the forgetful functor EB =B -! EG=B, and these adjoint pairs are Quillen pairs. These give the first two isomorphisms. T* *he last isomorphism is the composite h(EG=A)(A; ZA nM) ~=h(EG=B)(A; ZB nM) ~=h(EB =B)(B A; ZB nM): |___| We also need the following observation and its analogue in the differential g* *raded context. As in any pointed model category, the category EB =B has a (Quillen) suspension functor on its homotopy category; we denote it as B . Theorem 10.5. Let B be an E1 R-algebra or E1 differential graded k-algebra. There is an isomorphism h(EB =B)(B C; ZB M) ~=h(EB =B)(C; ZB M); natural in M 2 hMB and C 2 h(EB =B). Proof.Let B denote the Quillen loop functor in EB =B. Although ZB does not preserve fibrations, we still clearly have ZB M ~= B ZB M in h(EB =B). The displayed map is the composite of the map induced by this isomorphism and_the B , B adjunction. |__| 28 MICHAEL A. MANDELL Proof of Theorem 10.2.We can assume without loss of generality that A is cofi- brant and A -! B is a cofibration. Then the sequence B A -! B B -! B A B is a cofiber sequence in EB =B. We therefore get a long exact sequence (10.6) . .-.! h(EB =B)(B (B A); ZB nM) -! h(EB =B)(B A B; ZB nM) -! h(EB =B)(B B; ZB nM) -! h(EB =B)(B A; ZB nM): Applying Proposition 10.4, we can identify the last three terms as Topological Andre-Quillen cohomology. Theorem 10.5 allows us to identify the first term as h(EB =B)(B (B A); ZB nM) ~=h(EB =B)(B A; ZB nM) ~= h(EG=A)(A; ZA n-1M) = Dn-1(A; M) and also allows us to extend the sequence to the right. |_* *__| We see from the proof of Theorem 10.2 that to prove Theorem 10.3, we just need to see that takes cofiber sequences in h(EB =B) to cofiber sequences in h(EB * *=B) and converts the isomorphism in Theorem 10.5 to the analogous isomorphism in algebra. The following two lemmas give precise statements. Lemma 10.7. Let f :X -! Y be a cofibration in EB =B and let Cf be the cofiber. Then f is a cofibration in EB =B, and the universal map Cf -! Cf is a quasi-isomorphism. Proof.An easy induction argument shows that converts categorical monomor- phisms into cofibrations. Since EG is left proper, the map of pushouts Cf = Y X B -! Y X B = Cf is a weak equivalence, and so the universal map Cf -! Cf (induced by the map Y -! Cf) is a weak equivalence. Inductive use of Theorem 7.10 shows that this latter map is also cellular and identifies the induced map on cellular_chains a* *s the universal map Cf -! Cf. |__| Passing to the homotopy category, the previous lemma in particular gives us a natural isomorphism B ~=B in h(EB =B). By adjunction, we obtain a natural map B -! B in h(EB =B). In the proof of Theorem 5.7, we con- structed the natural transformation (8.6), and this induces a natural transform* *ation ZB R -! ZB . The following lemma relates these natural transformations with the natural isomorphisms B ZB ~=ZB and B ZB ~=ZB . Lemma 10.8. Let M be a B-module. The following diagram commutes. B ZB RM ______//ZB RM _____//ZB RM | | | | fflffl| fflffl| B ZB RM ____//_B ZB M______//ZB M ANDRE-QUILLEN COHOMOLOGY 29 Proof.Note that the horizontal arrows in the top row of the diagram are quasi- isomorphisms and so we obtain a map ZB RM -! B ZB M, and we only need to see that the diagram ZB RM _____//ZB RM | | (10.9) | | fflffl| fflffl| B ZB M ______//ZB M commutes. Let E be a CW approximation functor in the category of B-modules, and let N be a B-module representing RM. Then by the definition of R, we have that EN represents M. Factoring the map (10.10) ZB C*(E(NI)) -! (ZB C*EN) xB (ZB C*EN) as an acyclic cofibration followed by a fibration in EB =B, we get a model of * *the Quillen loop object B C*ZB EN together with a map C*ZB E(N) -! B ZB C*EN: The map (10.10) is ZB of a map C*(E(NI)) -! C*EN x C*EN: Factoring as a quasi-isomorphism followed by a fibration in MB , we get a map C*ZB E(N) -! ZB C*EN: where denotes the Quillen loop functor. Since the fibrations in EB =B and in MB are just the surjections, applying the functor ZB to the fibration in t* *he previous factorization gives us a fibration in EB =B; the left lifting propert* *y of acyclic cofibrations with respect to fibrations then constructs the map B ZB C*EN -! ZB C*EN on our Quillen loop models in such a way that the diagram C*ZB E(N) __=__//_C*ZB E(N) | | | | fflffl| fflffl| B ZB C*EN _____//_ZB C*EN commutes. Composing with the natural map C*EN -! M, we get a diagram __ that represents (10.9) in h(EB =B). |__| Proof of Theorem 10.3.Lemma 10.7 gives an isomorphism between the long exact sequence (10.6) with the analogous sequence in algebra. This gives the map Dn(B\A; RM) -! Dn(B\A; M) in the first statement and shows that the last two squares in the diagram commu* *te. Lemma 10.8 shows that the composite map Dn-1(A; RM) ~=h(EB =B)(B (B A); ZB RM) ~= h(EB =B)(B (B A); ZB M) ~=Dn-1(A; M) is the usual isomorphism, and it follows that the first square also commutes. * * |___| 30 MICHAEL A. MANDELL 11. Operations In this section we compare the cohomology operations in the Topological Andre- Quillen cohomology of E1 R-algebras with those in the Andre-Quillen cohomology of E1 k-algebras. As we explain, the local nature of the coefficients lead to * *char- acteristic classes or "constants" in Andre-Quillen and Topological Andre-Quillen cohomology, which lead to "constant" operations. We see below that every opera- tion can be written uniquely as a constant operation plus a "based" operation. * *The constants in Topological Andre-Quillen cohomology generally differ from the con- stants in Andre-Quillen cohomology, but we show that the isomorphism in Corol- lary 5.8 induces an isomorphism of the based operations. The based operations include all the additive, linear, and multi-linear operations. Although we cou* *ld argue in the full generality of Corollary 5.8, the motivation for the definitio* *n and the usefulness of operations is better described when we restrict to the exampl* *e of main interest: where the coefficient module has only a single non-trivial homot* *opy group (Corollary 5.9). We begin by explaining what we mean by operations in Topological Andre- Quillen cohomology. For the entirety of this section, let ff denote a fixed co* *m- mutative k-algebra (with no grading) and ss a fixed ff-module (with no grading). We denote by Hff a fixed E1 R-algebra with ssqHff = 0 for q 6= 0 and with a fixed isomorphism ss0Hff = ff. (Any other choice would be isomorphic in hEG by a unique isomorphism.) For any ff-module ss (with no grading), we choose and fix an Hff-module Hss with ssqHss = 0 for q 6= 0 and with a fixed isomorphism ss0Hss = ss. (Again, any other choice would be isomorphic in hMHffby a unique isomorphism.) A map of E1 R-algebras A -! Hff allows us to regard Hss as an A-module, and therefore allows us to define D*(A; Hss). Thus, D*(-; Hss) is naturally regarded as a functor on h(EG=Hff). In fact, this is the correct gene* *rality, since for connective E1 R-algebras A, the set of isomorphism classes of A-modu* *le structures on Hss (isomorphism classes in hMA that map to the isomorphism class of Hss in hMR ) are in one to one correspondence with ss0A-module structures on ss. Moreover, factorizations ss0A -! ff -! End ss are in one to one corresponde* *nce with isomorphism classes in h(EG=Hff) that map to the isomorphism class of A in hEG. We therefore make the following definition. Definition 11.1.An operation Dm1 (A; Hss) x . .x.Dmr (A; Hss) -! Dn(A; Hss0) is a natural transformation of functors h(EG=Hff) -! Set. An operation is based if it is a natural transformation of based sets. The base point is of course the zero element in the k-module structure. We can also define additive, linear, and multi-linear operations, but these play no sp* *ecial role in what follows. To save space, we write that the operation displayed abov* *e is an operation "from (m1; : :;:mr; ss) to (n; ss0)". We define operations and based operations in the Andre-Quillen cohomology of E1 differential graded k-algebras similarly. Since H*Hff is connective and H0Hf* *f = ff, we can choose a map of E1 differential graded k-algebras Hff -! ff, and in hEC, this map the unique one that on homology is inverse to the isomorphism ff = ss0Hff -! H0Hff. For each A, we can identify Hss as Rss. Then any operation from (m1; : :;:mr; ss) to (n; ss0) in the Andre-Quillen cohomology of E1 k-alge* *bras ANDRE-QUILLEN COHOMOLOGY 31 induces an operation from (m1; : :;:mr; ss) to (n; ss0) in the Topological Andr* *e- Quillen cohomology of E1 R-algebras via the isomorphism of Corollary 5.8. Of course, when the operation in Andre-Quillen cohomology is based (or additive, linear, or multi-linear), the corresponding operation in Topological Andre-Quil* *len cohomology is as well. The following is the main result of this section. Theorem 11.2. The based operations in the Andre-Quillen cohomology of E1 differential graded k-algebras are in one-to-one correspondence with the based * *oper- ations in the Topological Andre-Quillen cohomology of E1 R-algebras. To prove this, we need to understand the relationship between based operations and all operations. For this, note that the structure map ffl: A -! Hff induces* * a map of Topological Andre-Quillen cohomology ffl*: D*(Hff; Hss0) -! D*(A; Hss0). Each element x 2 Dn(Hff; Hss0) induces an operation x from (m1; : :;:mr; ss) to (n; ss0) (for any (m1; : :;:mr; ss)) that sends every element of Dm1 (A; Hss) x . .x.Dmr (A; Hss) to ffl*x 2 Dn(A; Hss0). We call elements of the form ffl*x constants and the op* *erations obtained in this way constant operations. The analogous observation and constru* *c- tion gives us constants and constant operations in the Andre-Quillen cohomology of E1 differential graded k-algebras. Proposition 11.3.Every operation in the Topological Andre-Quillen cohomology of E1 R-algebras or the Andre-Quillen cohomology of E1 differential graded k- algebras can be written uniquely as a based operation plus a constant operation. Proof.Let f be an operation from (m1; : :;:mr; ss) to (n; ss0) and let x be the* * image of (0; : :;:0) in Dn(Hff; Hss0) or in D(ff; ss0). Then by naturality, f sends (* *0; :_:;:0)_ to ffl*x in Dn(A; Hss0) or in Dn(A; ss0) for all A. Thus, f - x is based. * * |__| We now turn to the proof of Theorem 11.2, which is essentially a Yoneda Lemma argument. The functor Dm1 (-; Hss) x . .x.Dmr (-; Hss) on h(EG=Hff) is a representable functor, represented by ZHffM where M = m1Hss x . .x.mrHss: Thus, the set of operations from (m1; : :;:mr; ss) to (n; ss0) is in one-to-one* * corre- spondence with the set Dn(ZHffM; Hss0) ~=Dn(ZHffM; ss0): The following lemma picks out the based operations. Lemma 11.4. The map Dn(ZHffM\Hff; Hss0) -! Dn(ZHffM; Hss0) is an injection and its image is the set of based operations. Proof.We use the transitivity sequence (Theorem 10.2). The map Dn(ZHffM; Hss0) -! Dn(Hff; Hss0) in the transitivity sequence associated to Hff -! ZHffRM is easily identified a* *s the one that sends an operation f to f(0; : :;:0) 2 Dn(Hff; Hss0). By Proposition 1* *1.3, __ this map is a surjection for every n, and its kernel is the set of based operat* *ions. |__| 32 MICHAEL A. MANDELL At this point it is convenient to work entirely in the topological context. A* *s in Section 9, we denote with a subscript R or Hk the category in which the constru* *c- tions are formed. The natural isomorphism DnR(-; Hss) ~= DnHk(Hk ^ (-); Hss) allows us to associate an operation in the Topological Andre-Quillen cohomology of E1 R-algebras to each operation in the Topological Andre-Quillen cohomology of E1 Hk-algebras. The equivalence of Theorem 5.10 gives us a one-to-one cor- respondence between the operations for E1 differential graded k-algebras and t* *he operations for E1 Hk-algebras. The second diagram in Theorem 9.4 implies that the operation of E1 R-algebras associated to a given operation of E1 differen- tial graded k-algebras agrees with the operation associated to the corresponding operation of E1 Hk-algebras. Thus, Theorem 11.2 is equivalent to the following theorem. Theorem 11.5. The based operations in the Topological Andre-Quillen cohomol- ogy of E1 Hk-algebras are in one-to-one correspondence with the based operatio* *ns in the Topological Andre-Quillen cohomology of E1 R-algebras. Proof.To avoid confusing notation, we assume without loss of generality that the E1 Hk-algebra model Hff is sent by the forgetful functor to the E1 R-algebra model Hff, and likewise for the Hff-modules Hss. The association of an E1 Hk- algebra operation to an E1 R-algebra operation is the map DnHk(ZHffM; Hss0) = h(EG=Hff)Hk (ZHffM; ZHffHss0) -! h(EG=Hff)R (ZHffM; ZHffHss0) = DnR(ZHffM; Hss0) induced by the forgetful functor from E1 Hk-algebras to E1 R-algebras. It follo* *ws that the inclusion of the based operations is the map DnHk(ZHffM\Hff; Hss0) = h(EHff=Hff)Hk (ZHffM; ZHffHss0) -! h(EHff=Hff)R (ZHffM; ZHffHss0) = DnR(ZHffM\Hff; Hss0): induced by the forgetful functor from the category of Hff-algebras of E1 Hk- algebras to the category of Hff-algebras of E1 R-algebras. The following propo* *si-_ tion implies that this map is an isomorphism. |__| Proposition 11.6.The forgetful functor from the category of Hff-algebras of E1 Hk-algebras to the category of Hff-algebras of E1 R-algebras induces an equival* *ence h(EHff=Hff)Hk -! h(EHff=Hff)R Proof.We can assume without loss of generality that Hff is a commutative Hk- algebra and the map Hk -! Hff is a cofibration. Then the forgetful functor from commutative algebras to E1 algebras induces equivalences h(Com Hff=Hff) -! h(EHff=Hff)Hk and h(Com Hff=Hff) -! h(EHff=Hff)R making the following diagram commute. h(Com Hff=Hff) m QQQ 'mmmmm QQQ'Q mmm QQQQ vvmmmm Q(( h(EHff=Hff)Hk _______________________//_h(EHff=Hff)R __ |__| ANDRE-QUILLEN COHOMOLOGY 33 References [1]M. Andre, Methode Simpliciale en Algebre Homologique et Algebre Commutative* *, Lecture Notes in Math., 32, Springer, Berlin, 1967. [2]M. Basterra, "Andre-Quillen Cohomology of Commutative S-Algebras," J. P. A.* * A. 144 (1999), pp. 111-143. [3]E. 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May, Simplicial Objects in Algebraic Topology, Van Nostrand, 1967. [14]C. W. Rezk, "Spaces of Algebra Structures and Cohomology of Operads," thesi* *s, M. I. T. 1996. [15]D. G. Quillen, Homotopical Algebra, Springer Lecture Notes 43, 1967. [16]D. G. Quillen, "On the (Co-)Homology of Commutative Rings," in Applications* * of Cate- gorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), 65-8* *7, Amer. Math. Soc., Providence, R.I., 1970 Department of Mathematics, University of Chicago, Chicago, IL E-mail address: mandell@math.uchicago.edu