aa r X i v : . [ m a t h . A T ] F e b E n ring spectra and Dyer–Lashof operations Tyler Lawson
Cohomology operations are absolutely essential in making cohomology an ef-fective tool for studying spaces. In particular, the mod- p cohomology groupsof a space X are enhanced with a binary cup product, a Bockstein deriva-tion, and Steenrod’s reduced power operations; these satisfy relations such asgraded-commutativity, the Cartan formula, the Adem relations, and the insta-bility relations [92]. The combined structure of these cohomology operations isvery effective in homotopy theory because of three critical properties. These operations are natural.
We can exclude the possibility of certain mapsbetween spaces because they would not respect these operations.
These operations are constrained.
We can exclude the existence of certainspaces because the cup product and power operations would be incompat-ible with the relations that must hold.
These operations are complete.
Because cohomology is representable , wecan determine all possible natural operations which take an n -tuple ofcohomology elements and produce a new one. All operations are built,via composition, from these basic operations. All relations between theseoperations are similarly built from these basic relations.In particular, this last property makes the theory reversible: there are mech-anisms which take cohomology as input and converge to essentially completeinformation about homotopy theory in many useful cases, with the principalexamples being the stable and unstable Adams spectral sequences. The stableAdams spectral sequence begins with the Ext-groups Ext( H ∗ ( Y ) , H ∗ ( X )) inthe category of modules with Steenrod operations and converges to the stableclasses of maps from X to a p -completion of Y [1]. The unstable Adams spectralsequence is similar, but it begins with nonabelian Ext-groups that are calculatedin the category of graded-commutative rings with Steenrod operations [20, 19].Our goal is to discuss multiplicative homotopy theory: spaces, categories, orspectra with extra multiplicative structure. In this situation, we will see thatthe Dyer–Lashof operations play the role that the Steenrod operations did inordinary homotopy theory.In ordinary algebra, commutativity is an extremely useful property possessedby certain monoids and algebras. This is no longer the case in multiplicative ho-motopy theory or category theory. In category theory, commutativity becomes1 tructure : to give symmetry to a monoidal category C we must make a choiceof a natural twist isomorphism τ : A ⊗ B → B ⊗ A . Moreover, there are moredegrees of symmetry possible than in algebra because we can ask for weaker orstronger identities on τ . By asking for basic identities to hold we obtain thenotion of a braided monoidal category, and by asking for very strong identitiesto hold we obtain the notion of a symmetric monoidal category. In homotopytheory and higher category theory we rarely have the luxury of imposing iden-tities, and these become replaced by extra structure. One consequence is thatthere are many degrees of commutativity, parametrized by operads.The most classical such structures arose geometrically in the study of iteratedloop spaces. For a pointed space X , the n -fold loop space Ω n X has algebraicoperations parametrized by certain configuration spaces E n ( k ), which assembleinto an E n -operad ; moreover, there is a converse theorem due to Boardman–Vogt and May that provides a recognition principle for what structure on Y is needed to express it as an iterated loop space. As n grows, these spacespossess more and more commutativity, reflected algebraically in extra Dyer–Lashof operations on the homology H ∗ Y that are analogous to the Steenrodoperations.In recent years there is an expanding library of examples of ring spectrathat only admit, or only naturally admit, these intermediate levels of structurebetween associativity and commutativity. Our goal in this chapter is to give anoutline of the modern theory of highly structured ring spectra, particularly E n ring spectra, and to give a toolkit for their study. One of the things that wewould like to emphasize is how to usefully work in this setting, and so we willdiscuss useful tools that are imparted by E n ring structures, such as operationson them that unify the study of Steenrod and Dyer–Lashof operations. We willalso introduce the next stage of structure in the form of secondary operations.Throughout, we will make use of these operations to show that structured ringspectra are heavily constrained, and that many examples do not admit thisstructure; we will in particular discuss our proof in [48] that the 2-primaryBrown–Peterson spectrum does not admit the structure of an E ∞ ring spectrum,answering an old question of May [61]. At the close we will discuss some ongoingdirections of study. The author would like to thank Andrew Baker, Tobias Barthel, Clark Barwick,Robert Bruner, David Gepner, Saul Glasman, Gijs Heuts, Nick Kuhn, MichaelMandell, Akhil Mathew, Lennart Meier, and Steffen Sagave for discussions re-lated to this material, and Haynes Miller for a careful reading of an earlier draftof this paper. The author also owes a significant long-term debt to Charles Rezkfor what understanding he possesses.The author was partially supported by NSF grant 1610408 and a grantfrom the Simons Foundation. The author would like to thank the Isaac New-ton Institute for Mathematical Sciences for support and hospitality during the2rogramme HHH when work on this paper was undertaken. This work wassupported by: EPSRC grant numbers EP/K032208/1 and EP/R014604/1.
Throughout this section, we will let C be a fixed symmetric monoidal topologicalcategory. For us, this means that C is enriched in the category S of spaces, thatthere is a functor ⊗ : C × C → C of enriched categories, and that the underlyingfunctor of ordinary categories is extended to a symmetric monoidal structure.We will write Map C ( X, Y ) for the mapping space between two objects, andHom C ( X, Y ) for the underlying set. Associated to C there is the (ordinary)homotopy category h C , with morphisms [ X, Y ] = π Map C ( X, Y ). Associated to any object X ∈ C there is an endomorphism operad End C ( X ).The k ’th term is Map C ( X ⊗ k , X ) , with an operad structure given by composition of functors. For any operad O , this allows us to discuss O -algebra structures on the objects of C , maps of O -algebras, and further structure.If O is the associative operad A ssoc , then O -algebras are monoid objectsin the symmetric monoidal structure on C . If O is the commutative operad C omm , then O -algebras are strictly commutative monoids in C . However, theseoperads are highly rigid and do not take any space-level structure into account.Mapping spaces allow us to encode many different levels of structure. Example 3.1.
There is a sequence of operads A → A → A → . . . built outof the Stasheff associahedra [91]. An A -algebra has a unital binary multipli-cation; an A -algebra has a chosen homotopy expressing associativity, and hasMassey products; an A -algebra has a homotopy expressing a juggling formulafor Massey products; and so on. Moreover, each operad is simply built from theprevious: extension from an A n − -structure to an A n -structure roughly asks toextend a certain map S n − × X n → X to a map D n − × X n → X expressingan n -fold coherence law for the multiplication [3]. This gives A n a perturbative property: if X → Y is a homotopy equivalence, then A n -algebra structures onone space can be transferred to the other. Example 3.2.
The colimit of the A n -operads is called A ∞ , and it is equivalentto the associative operad. It satisfies a rectification property. In a well-behavedcategory like the category S of spaces or the category Sp of spectra, any A ∞ -algebra is equivalent in the homotopy category of A ∞ -algebras to an associativeobject. Example 3.3.
There is a sequence of operads E → E → E → . . . , wherethe space E n ( k ) is homotopy equivalent to the configuration space of ordered3 -tuples of points in R n . These have various models, such as the little cubes or little discs operads. The E -operad is equivalent to the associative operad, andthe E ∞ -operad is equivalent to the commutative operad. We refer to an algebraover any operad equivalent to E n as an E n -algebra . These play an important rolein the recognition principle [64, 17]: given an E n -algebra X we can construct an n -fold classifying space B n X ; and if the binary multiplication makes π ( X ) intoa group then X ≃ Ω n B n X . The relation between E n -algebra structures anditeration of the functor Ω is closely related to an additivity result of Dunn [26],who showed that E n +1 -algebras are equivalent to E -algebras in the categoryof E n -algebras. Example 3.4.
Associated to a topological monoid M , there is an operad O M whose only nonempty space is O M (1) = M . An algebra over this operad isprecisely an object with M -action. This operad is usually not perturbative.However, M can be resolved by a cellular topological monoid f M → M suchthat O f M -algebras are perturbative and can be rectified to O M -algebras. Thisconstruction is a recasting of Cooke’s obstruction theory for lifting homotopyactions of a group G to honest actions [25]; stronger versions of this were devel-oped by Dwyer–Kan and Badzioch [27, 4]. Example 3.5.
There is a free-forgetful adjunction between operads and sym-metric sequences. Given any sequence of spaces Z n with Σ n -actions, we canconstruct an operad Free( Z ) such that a Free( Z )-algebra structure is the sameas a collection of Σ n -equivariant maps Z n → Map C ( A ⊗ n , A ).If, further, Z is equipped with a chosen point e , we can construct an operadFree( Z, e ) such that a Free(
Z, e )-algebra structure is the same as a Free( Z )-algebra structure such that e acts as the identity: Free( Z, e ) is a pushout of adiagram Free( Z ) ← Free( { e } ) → Free( ∅ ) of operads. Example 3.6.
In the previous example, let Z be S with the antipodal actionof Σ and let all other Z n be empty, freely generating an operad Q that wecall the cup-1 operad . A Q -algebra is an object A with a Σ -equivariant map S → Map C ( A ⊗ , A ). The Σ -equivariant cell decomposition of S allows usto describe Q -algebras as objects with a binary multiplication m and a chosenhomotopy from the multiplication m to the multiplication in the opposite order m ◦ σ . In particular, any homotopy-commutative multiplication lifts to a Q -algebra structure.In the category Sp of spectra, one of the main applications of E n -algebras isthat they have well-behaved categories of modules, whose homotopy categoriesare triangulated categories. Theorem 3.7 (Mandell [59]) . An E -algebra R in Sp has a category of leftmodules LMod R . An E -algebra structure on R makes the homotopy categoriesof left modules and right modules equivalent, and gives the homotopy category ofleft modules a monoidal structure ⊗ R . An E -algebra structure on R extends thismonoidal structure to a braided monoidal structure. An E -algebra structure on R makes this braided monoidal structure into a symmetric monoidal structure. heorem 3.8. An E -algebra R in Sp has a monoidal category of bimodules.An E ∞ -algebra R in Sp has a symmetric monoidal category of left modules. If C is not just enriched, but is tensored over spaces, an O -algebra structure on X is expressible in terms internal to C . An O -algebra structure is equivalent tohaving action maps γ k : O ( k ) ⊗ X ⊗ k → X that are invariant under the action of Σ k and respect composition in the operad O . If C has colimits, we can define extended power constructions Sym k O ( X ) = (cid:0) O ( k ) ⊗ Σ k X ⊗ k (cid:1) , and an associated free O -algebra functorFree O ( X ) = a k ≥ Sym k O ( X ) . An O -algebra structure on X is then determined by a single map Free O ( X ) → X . To say more, we need C to be compatible with enriched colimits in the senseof [43, § Definition 3.9.
A symmetric monoidal category C is compatible with enrichedcolimits if the monoidal structure on C preserves enriched colimits in each vari-able separately.Compatibility with enriched colimits is necessary to give composite actionmaps O ( k ) ⊗ k O i =1 O ( n i ) ⊗ X ⊗ n i ! → X ⊗ Σ n i and make them assemble into a monad structure Free O ◦ Free O → Free O onthis free functor. In this case, O -algebras are equivalent to Free O -algebras, andSym k O and Free O are enriched functors.When these functors are enriched functors, they also give rise to a monadon the homotopy category h C . We refer to algebras over it as homotopy O -algebras . This is strictly stronger than being an O -algebra in the homotopycategory; the latter asks for compatible maps π O ( n ) → [ A ⊗ n , A ], whereas theformer asks for compatible elements in [ O ( n ) ⊗ Σ n A ⊗ n , A ] that use O beforepassing to homotopy. In the case of the E n -operads, such a structure in thehomotopy category is what is classically known as an H n -algebra [21].This type of structure can be slightly rigidified using pushouts of free alge-bras. For any operad O with identity e ∈ O (1), we can construct a homotopycoequalizer diagram Free(Free( O , e ) , e ) ⇒ Free( O , e ) → O h
5n the category of operads. An object A has an O h -algebra structure if andonly if there are Σ k -equivariant maps O ( k ) → Map C ( A ⊗ k , A ) so that the as-sociativity diagram homotopy commutes and so that e acts by the identity. Inparticular, A has an O h -algebra structure if and only if it has a homotopy O -algebra structure; the O h -structure has a chosen homotopy for the associativityof composition. For example, there is an operad parametrizing objects witha unital binary multiplication, a chosen associativity homotopy, and a chosencommutativity homotopy. In the category of spectra, the Eilenberg–Mac Lane spectra HA are character-ized by a useful mapping property. We refer to a spectrum as connective if it is( − X , the natural mapMap Sp ( X, HA ) → Hom Ab ( π X, A )is a weak equivalence.This has a number of strong consequences. For example, we get an equiva-lence of endomorphism operads End Sp ( HA ) → End Ab ( A ), obtained by taking π : End Sp ( HA ) k = Map( HA ⊗ k , HA ) ≃ End Ab ( A ) k = Hom( A ⊗ k , A ) . Thus, an action of an operad O on HA is equivalent to an action of π O on A , and this equivalence is natural. This technique also generalizes, using theequivalences Hom( Hπ R ⊗ n , Hπ R ) ∼ −→ Map( R ⊗ n , Hπ R ) . Proposition 3.10.
Suppose R is a connective spectrum and O is an operadacting on R . Then the map R → Hπ ( R ) can be given, in a functorial way, thestructure of a map of O -algebras. Example 3.11. If A is given the structure of a commutative ring, HA inheritsan essentially unique structure of an E ∞ -algebra. If R is a connective andhomotopy commutative ring spectrum, then it can be equipped with an actionof the cup-1 operad Q from 3.6. Any ring homomorphism π R → A lifts to amap of Q -algebras R → HA . Example 3.12.
There exist models for the category of spectra so that thefunction spectrum F (Σ ∞ + X, A ) = A X is a lax monoidal functor S op × Sp → Sp , with the homotopy groups of A X being the unreduced A -cohomology groups of X . The diagonal ∆ makes anyspace X into a commutative monoid in S op . If A is an O -algebra in Sp , then A X then becomes an O -algebra. 6 xample 3.13. For any spectrum E , composition of functions naturally givesthe endomorphism algebra spectrum End( E ) = F ( E, E ) the structure of an A ∞ -algebra, and E is a left module over End( E ). The homotopy groups of End( E )are sometimes called the E -Steenrod algebra and they parametrize operationson E -cohomology. Example 3.14.
The suspension spectrum functor X Σ ∞ + X = S [ X ]is strong symmetric monoidal. As a result, it takes O -algebras to O -algebras.For example, any topological group G has an associated spherical group algebra S [ G ]. Example 3.15.
For any pointed space X , the n -fold loop space Ω n X is an E n -algebra in spaces, and S [Ω n X ] is an E n -algebra. For any spectrum Y thespace Ω ∞ Y is an E ∞ -algebra in spaces, and S [Ω ∞ Y ] is an E ∞ -algebra. Example 3.16.
The Thom spectra
M O and
M U have E ∞ ring structures [65].At any prime p , M U decomposes into a sum of shifts of the Brown–Petersonspectrum BP , which has the structure of an E -ring spectrum [11]. Example 3.17.
The smash product being symmetric monoidal implies that itis also a strong symmetric monoidal functor Sp × Sp → Sp . If A and B are O -algebras then so is A ⊗ B . Example 3.18.
For a map Q → R of E ∞ ring spectra, there is an adjunction Mod Q ⇄ Mod R between the extension of scalars functor M R ⊗ Q M and the forgetful func-tor. The left adjoint is strong symmetric monoidal and the right adjoint is laxsymmetric monoidal, and hence both functors preserve O -algebras.This allows us to narrow our focus. For example, if E has an E ∞ -algebrastructure and we are interested in understanding operations on the E -homologyof O -algebras, we can restrict our attention to those operations on the homotopygroups of O -algebras in Mod E rather than considering all possible operationson the E -homology. A multicategory (or colored operad) encodes the structure of a category wherefunctions have multiple input objects. They serve as a useful way to encodemany multilinear structures in stable homotopy theory: multiplications, modulestructures, graded rings, and coherent structures on categories. In this sectionwe will give a quick introduction to them, and will return in § Definition 3.19.
A multicategory M consists of the following data:1. a collection Ob ( M ) of objects; 7. a set Mul M ( x , . . . , x d ; y ) of multimorphisms for any objects x i and y of M , or more generally a set Mul M ( { x s } s ∈ S ; y ) for any finite set S andobjects x s , y ;3. composition operations ◦ : Mul M ( { y t } t ∈ T ; z ) × Y t ∈ T Mul M ( { x s } s ∈ f − ( t ) ; y t ) → Mul M ( { x s } s ∈ S ; z )for any map f : S → T of finite sets and objects x s , y t , and z of M ; and4. identity morphisms id X ∈ Mul M ( x ; x ) for any object x .These are required to satisfy two conditions:1. unitality: id y ◦ g = g ◦ (id x s ) = g for any g ∈ Mul M ( { x s } s ∈ S ; Y ); and2. associativity: h ◦ ( g u ◦ ( f t )) = ( h ◦ ( g u )) ◦ f t for any S → T → U of finitesets.The underlying ordinary category of M is the category with the same objectsas M and Hom M ( x , y ) = Mul M ( x ; y ).If the sets of multimorphisms are given topologies so that composition iscontinuous, we refer to M as a topological multicategory .A (topological) multifunctor F : M → N is a map F : Ob ( M ) → Ob ( N ) onthe level of objects, together with (continuous) mapsMul M ( x , . . . , x d ; y ) → Mul M ( F x , . . . , F x d ; F y )that preserve identity morphisms and composition. Example 3.20.
An operad is equivalent to a single-object multicategory. Forany object x in a multicategory M , the full sub-multicategory spanned by x isan operad called the endomorphism operad of x . Example 3.21.
A symmetric monoidal topological category M can be regardedas a multicategory by definingMul M ( X , . . . , X d ; Y ) = Map M ( X ⊗ · · · ⊗ X d , Y ) . This recovers the definition of the endomorphism operad of an object X .The notion of an algebra over a multicategory will extend the notion of analgebra over an operad. Definition 3.22.
For (topological) multicategories M and C , the category Alg M ( C ) of M -algebras in C is the category of (topological) multifunctors M → C and natural transformations.For any object x ∈ M , the evaluation functor ev x : Alg M ( C ) → C sends analgebra A to the value A ( x ). 8 xample 3.23. The multicategory
Mod parametrizing “ring-module pairs”has two objects, a and m , andMul Mod ( x , . . . , x d ; y ) = ∗ if y = a and all x i are a , ∗ if y = m and exactly one x i is m , ∅ otherwise.A multifunctor Mod → C is equivalent to a pair ( A, M ) of a commutativemonoid A of C and an object M with an action of A . Example 3.24.
A commutative monoid Γ can be regarded as a symmetricmonoidal category with no non-identity morphisms, and in the associated mul-ticategory we have Mul Γ ( g , . . . , g d ; g ) = ( ∗ if P g s = g, ∅ otherwise.A multifunctor Γ → C determines objects X g of C , a map from the unit to X ,and multiplication maps X g ⊗ · · · ⊗ X g d → X g + ··· + g d : these multiplicationsare collectively unital, symmetric, and associative. We refer to such an objectas a Γ-graded commutative monoid. Example 3.25.
The addition of natural numbers makes the partially orderedset ( N , ≥ ) into a symmetric monoidal category. In the associated multicategorywe have Mul N ( n , . . . , n d ; m ) = ( ∗ if P n i ≥ m, ∅ otherwise.A multifunctor Γ → C determines a sequence of objects · · · → X → X → X of C and multiplication maps X n ⊗ · · · ⊗ X n d → X n + ··· + n d : these multi-plications are collectively unital, symmetric, and associative, as well as beingcompatible with the inverse system. We refer to such an object strongly filtered commutative monoid in C . Remark 3.26. If M and M are multicategories, there is a product multi-category M × M , obtained by taking products of objects and products ofmultimorphism spaces. Products allow us to extend the above constructions.For example, taking the product of an operad O with the multicategories of theprevious examples, we construct multicategories that parametrize: pairs ( A, M )of an O -algebra and an O -module; Γ-graded O -algebras; and strongly filtered O -algebras. Example 3.27.
Let M be the multicategory whose objects are integers, anddefine Mul M ( m , . . . , m d ; n ) to be the set of natural transformations θ : H m ( X ) × · · · × H m d ( X ) → H n ( X )9f contravariant functors on the category S of spaces; composition is composi-tion of natural transformations. The category M is a category of multivariatecohomology operations. Any fixed space X determines an evaluation multifunc-tor ev X : M → Sets , sending n to H n ( X ); any homotopy class of map X → Y of spaces determines a natural transformation of multifunctors in the oppositedirection. Stated concisely, this is a functor h S op → Alg M ( Sets )that takes a space to an encoding of its cohomology groups and cohomologyoperations.More generally, a category D with a chosen set of functors D → Sets deter-mines a multicategory M spanned by them: we can define Mul( F , . . . , F d ; G )to be the set of natural transformations Q F i → G , so long as there is always aset (rather than a proper class) of natural transformations. If we view a functor F as assigning an invariant to each object of D , a multimorphism Q F i → G is anatural operation of several variables on such invariants. Evaluation on objectsof D takes the form of a functor D → Alg M ( Sets ) , encoding both the invariants assigned by these functors and the natural opera-tions on them. These are examples of multi-sorted algebraic theory in the senseof Bergner [14], closely related to the work of [18, 89]. We will return to thediscussion of this structure in § C and algebras over a multicategory. Proposition 3.28.
Suppose that M is a small topological multicategory andthat C is a symmetric monoidal topological category with compatible colimits inthe sense of Definition 3.9.1. For objects x and y of M , there are extended power functors Sym k M , x → y : C x → C y , given by Sym k M , x → y ( X ) = Mul M ( x , x , . . . , x | {z } k ; y ) ⊗ Σ k X ⊗ k .
2. The evaluation functor ev x : Alg M ( C ) → C has a left adjoint Free M , x : C → Alg M ( C ) . The value of
Free M , x ( X ) on any object y of M is ev y (Free M , x ( X )) = a k ≥ Sym k M , x → y ( X ) . emark 3.29. These generalize the constructions of extended powers and freealgebras from § M has a single object x , encoding an operad O , thenSym k M , x → x = Sym k O and Free M , x encodes Free O . Example 3.30.
The free Z -graded commutative monoid on an object X indegree n = 0 is equal to the symmetric product Sym k ( X ) in degree kn for k ≥
0. All other gradings are the initial object.
Example 3.31.
The free strongly filtered commutative monoid on an object X in degree 1 is a filtered object of the form · · · → a k ≥ Sym k X → a k ≥ Sym k X → a k ≥ Sym k X . If we have a strongly filtered commutative algebra · · · → X → X → X , thenthis gives action maps Sym k X → X k . More generally, there are action mapsSym k X n → X kn that are compatible in n . In this section we will fix a spectrum E , viewed as a coefficient object. E -homology and E -modules We can study O -algebras through their E -homology. Definition 4.1.
Given a spectrum E , an E -homology operation for O -algebrasis a natural transformation of functors θ : E m ( − ) → E m + d ( − ) of functors onthe homotopy category of O -algebras.Such operations can be difficult to classify in general. However, if E has acommutative ring structure then we can do more. In this case, any O -algebra A has an E -homology object E ⊗ A which is an O -algebra in Mod E , and any space X has an E -cohomology object E X which is an E ∞ -algebra object in Mod E .By definition, we have E m ( A ) = [ S m , E ⊗ A ] Sp and E m ( X ) = [ S − m , E X ] Sp . Therefore, we can construct natural operations on the E -homology of O -algebrasor the E -cohomology of spaces by finding natural operations on the homotopygroups of O -algebras in Mod E . Example 4.2. If X is an O -algebra in spaces, then E [ X ] = E ⊗ Σ ∞ + X is an O -algebra in Mod E . 11 .2 Multiplicative operations In this section we will construct our first operations on the homotopy groups of O -algebras over a fixed commutative ring spectrum E .The functor π ∗ from the homotopy category of spectra to graded abeliangroups is lax symmetric monoidal under the Koszul sign rule. The inducedfunctor π ∗ from Alg O ( Sp ) or Alg O ( Mod E ) to graded abelian groups naturallytakes values in the category of graded abelian groups, or graded E ∗ -modules,with an action of the operad π O in sets. Example 4.3.
In the case of an E n -operad, π O is isomorphic to the associativeoperad when n = 1 and the commutative operad when n ≥
2. The E -homologygroups of an E n -algebra in Sp form a graded E ∗ -algebra. If n ≥
2, this algebrais graded-commutative.By applying E ∗ to the action maps in the operad, we stronger information. Proposition 4.4.
The homology groups E ∗ O ( k ) form an operad E ∗ O in graded E ∗ -modules, and the functor π ∗ from Alg O ( Mod E ) to graded abelian groups hasa natural lift to the category of graded E ∗ O -modules. Example 4.5.
The homotopy groups of E n -algebras have a natural bilinear Browder bracket [ − , − ] : π q ( A ) ⊗ π r ( A ) → π q +( n − r ( A ) . This satisfies the following formulas.
Antisymmetry: [ α, β ] = − ( − ( | α | + n − | β | + n − [ β, α ]. Leibniz rule: [ α, βγ ] = [ α, β ] γ + ( − | β | ( | α | + n − α [ β, γ ] . Graded Jacobi identity: − ( | α | + n − | γ | + n − [ α, [ β, γ ]]+ ( − ( | β | + n − | α | + n − [ β, [ γ, α ]]+ ( − ( | γ | + n − | β | + n − [ γ, [ α, β ]] . In the case of E -algebras, this reduces to the ordinary bracket[ α, β ] = αβ − ( − | α || β | βα in the graded ring π ∗ ( A ).The Browder bracket is defined, just as it was defined in homology [24], usingthe image of the generating class λ ∈ π n − E n (2) ∼ = π n − S n − coming from thelittle cubes operad. The antisymmetry and Jacobi identities are obtained byverifying identities in the graded operad π ∗ (Σ ∞ + E n ). For example, if σ is the2-cycle in Σ we have λ ◦ σ = ( − n λ, τ is a 3-cycle in Σ we have λ ◦ (1 ⊗ λ ) ◦ (1 + τ + τ ) = 0 . However, the signs indicate that there is some care to be taken. In particular,the Browder bracket of elements α ∈ π q ( A ) and β ∈ π r ( A ) is defined to be thefollowing composite: S q ⊗ S n − ⊗ S r → A ⊗ Σ ∞ + E n (2) ⊗ A → Σ ∞ + E n (2) ⊗ A ⊗ A → A This order is chosen because it is more consistent with writing the Browderbracket as an inline binary operation [ x, y ] than with writing it as an operator λ ( x, y ) on the left. The subscript on the range π q +( n − r ( A ) reflects this choice(cf. [74]). This gives us the definition[ α, β ] = ( − ( n − | α | γ ( λ ⊗ α ⊗ β ) , where γ is the action map of the operad π ∗ (Σ ∞ + E n ) on π ∗ A . Both the verificationof the identities on λ in the stable homotopy groups of configuration spaces, andthe verification of the consequent antisymmetry, Leibniz, and Jacobi identities,are reasonable but error-prone exercises from this point; compare [23]. We will ultimately be interested in natural operations on homotopy and homol-ogy groups. However, it is handy to use a more general definition that replaces S m by a general object. This accounts for the possibility of operations of severalvariables, and can also help reduce difficulties involving naturality in the input S m . Definition 4.6.
For spectra M and X , we define the M -indexed homotopy of X to be π M ( X ) = [ M, X ] Sp ∼ = [ E ⊗ M, X ] Mod E . For spectra M , X , and E we define the M -indexed E -homology of X to be E M ( X ) = π M ( E ⊗ X ) . If M is S m , we instead use the more standard notation π m ( − ) for π S m ( − ) or E m ( − ) for E S m ( − ). Definition 4.7.
Let E be a commutative ring spectrum. A homotopy operation for O -algebras over E is a natural transformation θ : π M → π N of functors on the homotopy category of Alg O ( Mod E ). When O and E are un-derstood, we just refer to such natural transformations as homotopy operations.We refer to the resulting operation E M ( − ) → E N ( − ) on the E -homologygroups of O -algebras as the induced E -homology operation.13s in Example 3.27, we can assemble operations with varying numbers ofinputs into an algebraic structure. Definition 4.8.
Fix an operad O and a commutative ring spectrum E . Themulticategory Op E O of operations for O -algebras in Mod E has, as objects, spec-tra N . For any M , . . . , M d and N , the group of multimorphismsOp E O ( M , . . . , M d ; N )is the group of natural transformations Q π M i → π N of functors h Alg O ( Mod E ) → Sets . If E or O are understood, we drop them from the notation.In the unary case, we write Op E O ( M ; N ) for the set of homotopy operations π M → π N for O -algebras in Mod E .The free-forgetful adjunction between spectra and O -algebras in Mod E al-lows us to exhibit the functor π M as representable. Proposition 4.9.
Suppose that E is a commutative ring spectrum, O is anoperad with associated free algebra monad Free O . Then there is a natural iso-morphism π M ( A ) ∼ = [ E ⊗ Free O ( M ) , A ] Alg O ( Mod E ) for A in the homotopy category of Alg O ( Mod E ) . In particular, the object E ⊗ Free O ( M ) is a representing object for the functor π M .Proof. The forgetful functor
Alg O ( Mod E ) → Sp can be expressed as a compos-ite Alg O ( Mod E ) → Alg O ( Sp ) → Sp , and as such has a composite left adjoint M Free O ( M ) E ⊗ Free O ( M ); this adjunction passes to the homotopycategory. Therefore, applying this adjunction we find π M ( A ) ∼ = [Free O ( M ) , A ] Alg O ( Sp ) ∼ = [ E ⊗ Free O ( M ) , A ] Alg O ( Mod E ) as desired. Remark 4.10.
It is possible to index more generally. Given an E -module L ,we also have functors π EL ( − ) = [ L, − ] Mod E ; the free O -algebra Free O ( L ) in thecategory of E -modules is then a representing object for π EL in Alg O ( Mod E ).We recover the above case by setting L = E ⊗ M .The Yoneda lemma now gives the following. Corollary 4.11.
Let F be a functor from h Alg O ( Mod E ) to the category of sets.Natural transformations of functors π M → F are in bijective correspondencewith F ( E ⊗ Free O ( M )) . In particular, there is an isomorphism Op E O ( M , . . . , M d ; N ) ∼ = E N (Free O ( ⊕ M i )) from the group of natural transformations Q π M i → π N to the E -homology groupof the free algebra. § O into extendedpowers gives us a canonical decomposition of operations. Definition 4.12.
For k ≥
0, the group of operations of weight k is the subgroupOp E O ( M , . . . , M d ; N ) h k i = E N (Sym k O ( ⊕ M i ))of Op E O ( M , . . . , M d ; N ) ∼ = E N (Free O ( ⊕ M i )).A power operation of weight k is a unary operation of weight k : an elementof the subgroup Op E O ( M, N ) h k i ∼ = E N (Sym k O ( M ))of Op E O ( M, N ). Remark 4.13.
Composition multiplies weight. Furthermore, if the object N isdualizable, the group of all operations is a direct sum: every operation decom-poses canonically as a sum of operations of varying weights. Even when restricted to ordinary homotopy groups, these operations betweenthe homotopy groups of O -algebras in Mod E form a rather rich algebraic struc-ture [14], whose characteristics should be discussed; we learned most of thisfrom Rezk [78, 77]. RecallOp( m , . . . , m d ; n ) = Op E O ( m , . . . , m d ; n ) ∼ = π n ( E ⊗ Free O ( ⊕ S m i )) . Here are some characteristics of this algebraic theory.1. We think of the elements in these groups as operators, in the sense thatthey can act . Given α ∈ Op( m , . . . , m d ; n ), an O -algebra R in Mod E and x i ∈ π m i R , we can apply α to get a natural element α ∝ ( x , . . . , x d ) ∈ π n R. This action is associative with respect to composition, but only distributesover addition on the left.2. For each 1 ≤ k ≤ d , there is a fundamental generator ι k ∈ Op( m , . . . , m d ; m k )that acts by projecting: ι k ∝ ( x , . . . , x d ) = x k .
3. These operators can compose : given α ∈ Op( m , . . . , m d ; n ) and β i ∈ Op( ℓ , . . . , ℓ c ; m i ), there is a composite operator α ∝ ( β , . . . , β d ) ∈ Op( ℓ , . . . , ℓ c ; n ) . Composition is unital. It is also associative, both with itself and withacting on elements. Again, it only distributes over addition on the left.15. Composition respects weight: if α is in weight a and β i are in weights b i ,then α ∝ ( β i ) is in weight a · ( P b i ). Example 4.14.
Take E = HR for a commutative ring R and let O to be theassociative operad. Then the graded group Op ( m , . . . , m d ; ∗ ) = ⊕ n Op( m , . . . , m d ; n ) ∼ = H ∗ (Free O ( ⊕ S m i ; R ))is the free associative graded R -algebra on the fundamental generators ι . . . ι d with ι i in degree m i , and the composition operations are substitution . For ex-ample, the element ι + ι ∈ Op( n, n ; n ) acts by the binary addition operation indegree n ; the elements ι ι and ι ι in Op( n , n ; n + n ) represent binary mul-tiplication in either order; the element ( ι ) ∈ Op( n ; 2 n ) represents the squaringoperation; for r ∈ R the element rι ∈ Op( n ; n ) represents scalar multiplicationby r ; combinations of these operations are represented by identities such as ι ∝ ( ι + ι ) = ι + ι ι + ι ι + ι . In this structure, each monomial has constant weight equal to its degree.
Example 4.15.
Take O to be an E n -operad. Then, for any p and q , theBrowder bracket is a natural transformation π p × π q → π p +( n − q , and it isrealized by an element [ ι , ι ] in Op( p, q ; p + ( n −
1) + q ) of weight two. Rela-tions between the product and the Browder bracket are expressed universallyby relations between compositions: for example, antisymmetry is expressed byan identity [ ι , ι ] = − ( − ( p + n − q + n − [ ι , ι ] . Remark 4.16.
Inside the collection of all unary operations, there is a subgroupof additive operations: those operations f that satisfy f ∝ ( ι + ι ) = f ∝ ι + f ∝ ι . Composition of such operations is bilinear, and so the collection of objects andadditive operations form a category enriched in abelian groups. In some cases,the additive operations can be used to determine the general structure [78].
We will begin to narrow our study of power operations and focus on unaryoperations, of fixed weight, between integer gradings.
Definition 4.17.
Fix an operad O and a commutative ring spectrum E . Thegroup of power operations of weight k on degree m for O -algebras in Mod E isthe graded abelian groupPow E O ( m, k ) = π ∗ ( F ( S m , E ⊗ Sym k O ( S m ))) ∼ = M r ∈ Z Op E O ( m, m + r ) h k i . If O or E are understood, we drop them from the notation.16n element of Pow( m, k ) in grading r represents a weight- k natural trans-formation π m → π m + r on the homotopy category of O -algebras in Mod E , andinduces a natural transformation E m → E m + r on the homotopy category of O -algebras. (While we index these group by integers, they depend on a choice ofrepresenting object and in particular on an orientation of S m ; making implicitidentifications will result in sign issues.) Remark 4.18.
These operations, and the relations between them, are stillpossessed by homotopy O -algebras in the sense of § Remark 4.19.
Suppose that Σ k acts freely and properly discontinuously on O ( k ). Let V ⊂ R k be the subspace of elements which sum to 0, with associatedvector bundle ρ → B Σ k of dimension k −
1. For any m there is an associatedvirtual bundle R m ⊗ ρ . If we define P ( k ) = O ( k ) / Σ k , then there is a virtual bundle mρ on P ( k ). The Thom spectrum P ( k ) mρ ofthis virtual bundle is canonically equivalent to the spectrum Σ − m Σ ∞ + O ( k ) ⊗ Σ k ( S m ) ⊗ k that appears in the definition of Pow( m, k ).This allows us to give a more concise expressionPow( m, k ) = E ∗ ( P ( k ) mρ ) , which is particularly useful in cases where we can apply a Thom isomorphismfor E -homology. Example 4.20.
Consider the case of operations of weight 2 for E n -algebras.The space P (2) = C n (2) / Σ is homotopy equivalent to the real projective space RP n − , the line bundle ρ = σ is associated to the sign representation of Σ , andthe Thom spectrum ( RP n − ) mσ is commonly known as the stunted projectivespace RP m + n − m which has a cell decomposition with one cell in each dimensionbetween m and m + n −
1. (When m ≥ RP m + n − / RP m − .) Therefore, the operations of weight 2 on degree m areparametrized by the E -homology groupOp Em (2) = E ∗ ( RP m + n − m ) . Example 4.21.
When E = H F , we find H ∗ ( RP m + n − m ) is F in degrees m through ( m + n − Dyer–Lashof operations Q r for m ≤ r ≤ m + n − π m to elements in π m + r . Example 4.22.
Consider the cup-1 operad Q defined in Example 3.6. Thenthe weight-2 operations on the E -homology of Q -algebras are parametrized by E ∗ ( RP m +1 m ). This stunted projective space is the Thom spectrum of m timesthe M¨obius line bundle over S .For example, we can take E to be the sphere spectrum. If m = 2 k there isa splitting RP k +12 k ≃ S k ⊕ S k +1 . π k ( S k ⊕ S k +1 ) and π k +1 ( S k ⊕ S k +1 ) give operationsthat increase degree by 2 k and 2 k +1, respectively. A choice of splitting S k +1 → RP k +12 k determines an operation Sq : π k ( − ) → π k +1 ( − ) called the cup-1square . It satisfies 2 Sq ( a ) = [ a, a ].In the case that we have an E ∞ ring spectrum, this has been studied in [21, § V] and [13], and can be chosen in such a way that it satisfies the followingaddition and multiplication identities on even-degree homotopy elements:2 Sq ( a ) = 0Sq ( a + b ) = Sq ( a ) + Sq ( b ) + ( | a | + 1) abη Sq ( ab ) = a Sq ( b ) + Sq ( a ) b + | ab | a b η. For example, Sq ( n ) = (cid:0) n (cid:1) η for n ∈ Z . In the absence of higher commutativity,these identities should have correction terms involving the Browder bracket. In this section we will consider compatibility relations between operations ondifferent homotopy degrees.Recall from § O decomposed into the homogeneousfunctors defined by Sym k O ( X ) = Σ ∞ + O ( k ) ⊗ Σ k X ⊗ k . In particular, these functors are continuous : they induce functionsMap(
X, Y ) → Map(Sym k O ( X ) , Sym k O ( Y ))between mapping spaces, and for k > pointed : Sym k O ( ∗ ) = ∗ and hence the functor Sym k O induces continuous maps of pointed mapping spaces. Definition 4.23.
For any spectrum M , any pointed space Z , and any k > assembly map Sym k O ( M ) ⊗ Σ ∞ Z → Sym k O ( M ⊗ Σ ∞ Z )is adjoint to the composite map of pointed spaces Z → Map Sp ( S , Σ ∞ Z ) → Map Sp ( M, M ⊗ Σ ∞ Z ) → Map Sp (Sym k O ( M ) , Sym k O ( M ⊗ Σ ∞ Z )) . The suspension map σ n : Pow( m, k ) → Pow( m + n, k )is induced by the composite map of function spectra F ( S m , E ⊗ Sym k O ( S m )) → F ( S m ⊗ S n , E ⊗ Sym k O ( S m ) ⊗ S n ) → F ( S m ⊗ S n , E ⊗ Sym k O ( S m ⊗ S n )) . emark 4.24. The operation σ = σ has a concrete meaning: it is designedfor compatibility with the Mayer–Vietoris sequence . To illustrate this, first recallthat for a homotopy commutative diagram A / / (cid:15) (cid:15) B (cid:15) (cid:15) C / / D of spectra, we have natural maps A → P ← Σ − D where P is the homotopypullback.Now suppose that we are given a diagram of O -algebras as above which is ahomotopy pullback, inducing a boundary map ∂ : Σ − D → P ≃ A . Given maps θ : N → E ⊗ Sym k O ( M ) and α : Σ M → D , we can map in a trivial homotopypullback diagram to the above, then apply action maps and naturality of theconnecting homomorphisms. We get a commuting diagram: N θ / / ∼ (cid:15) (cid:15) E ⊗ Sym k O M ∂α / / (cid:15) (cid:15) A ∼ (cid:15) (cid:15) Σ − Σ N / / P ′ / / P Σ − Σ N ∼ O O σθ / / Σ − E ⊗ Sym k O (Σ M ) O O Σ − α / / Σ − D ∂ O O Therefore, for an operation θ : [ M, − ] → [ N, − ] for O -algebras in Mod E , wefind that ∂ ◦ σθ ∼ θ ◦ ∂. This description makes implicit choices about the orientation of the circle thatappears in the operation Ω when taking homotopy pullbacks, and this can resultin sign headaches.
Proposition 4.25.
For k, r > , the suspension σ r : Pow( m, k ) → Pow( m + r, k ) is the map E ∗ ( P ( k ) mρ ) → E ∗ ( P ( k ) ( m + r ) ρ ) on E -homology induced by the inclusion of virtual bundles mρ → mρ ⊕ rρ .Proof. The assembly map Sym k O ( S m ) ⊗ S n → Sym k O ( S m + n ) is the map(Σ ∞ + O ( k ) ⊗ Σ k S mρ ) ⊗ S n → (Σ ∞ + O ( k ) ⊗ Σ k S ( m + n ) ρ ) , which is the map P ( k ) mρ ⊗ S r → P ( k ) ( m + r ) ρ induced by the direct sum inclusion mρ ⊕ r → ( mρ ⊕ r ) ⊕ rρ of virtual bundles.The map σ r is obtained by desuspending both sides ( m + r ) times, which givesthe map induced by the direct sum inclusion mρ → mρ ⊕ rρ of virtual bundles.19 xample 4.26. The Dyer–Lashof operations for E n -algebras are explicitly un-stable . For example, in weight two the n -fold suspension maps RP m + n − m → RP ( m + n )+ n − m + n are trivial, and so the map Op Em (2) → Op Em + n (2) is trivial. Thisrecovers the well-known fact that all Dyer–Lashof operations for E n -algebrasmap to zero under n -fold suspension.By contrast, the Dyer–Lashof operations for E ∞ -algebras are stable : themaps H ∗ RP ∞ m → H ∗ RP ∞ m +1 are surjections, and so the quadratic operations alllift to elements in the homotopy oflim m ( H ⊗ RP ∞ m ) . By [33, 16.1], this is the desuspended Tate spectrum (Σ − H ) t Σ . Remark 4.27.
More generally, the fully stable operations of prime weight p onthe homotopy of E ∞ E -algebras are detected by the p -localized Tate spectrum(Σ − E ( p ) ) t Σ p . See [21, II.5.3] and [31].
Suppose that E = colim E α is an expression of E as a filtered colimit of finitespectra. Then there is an identification E m A = colim α [ S m , E α ⊗ A ] = colim α [ S m ⊗ DE α , A ] , where D is the Spanier–Whitehead dual. We cannot move the colimit inside,but we can view { S m ⊗ DE α } as a pro-object in the category of spectra. Thismakes the functor E m representable by embedding the category of spectra intothe category of pro-spectra.For algebras over an operad O , we can go even further and find that E m ( A ) = [ { Free O ( S m ⊗ DE α ) } , A ] pro - O is now a representable functor in the homotopy category of pro- O -algebras, andin this category we can determine all the natural operations E m → E n : N at pro - O ( E m ( − ) , E n ( − )) = [ { Free O ( S n ⊗ DE α ) } , { Free O ( S m ⊗ DE β ) } ] pro - O = π lim β colim α Map O (Free O ( S n ⊗ DE α ) , Free O ( S m ⊗ DE β ))= π lim β colim α Map Sp ( S n ⊗ DE α , Free O ( S m ⊗ DE β ))= π lim β Map Sp ( S n , E ⊗ Free O ( S m ⊗ DE β ))= π n lim β E ⊗ (Free O ( S m ⊗ DE β )) . S m ⊗ E, S n ⊗ E ]of cohomology operations (and these maps are isomorphisms if O is trivial), andit has a natural map to the limitlim β E n (Free O ( S n ⊗ DE β )) . This map to the limit is an isomorphism if no higher derived functors intrude.We can think of this as the algebra of continuous operations on E -homology. E n Dyer–Lashof operations at p = 2 We will now specialize to the case of ordinary mod-2 homology. When wedo so, we have Thom isomorphisms for many bundles and we have explicitcomputations of the homology of configuration spaces due to Cohen [24]. Similarresults with more complicated identities hold at odd primes.
Proposition 5.1.
Let H = H F be the mod- Eilenberg–Mac Lane spectrum.Then the group Op Hm (2) of weight-2 operations for E n -algebras has exactly onenonzero operation in each degree between m and m + n − , and no others.Proof. By Remark 4.19, this is a calculation H ∗ ( RP n + m − m ) of the mod-2 ho-mology of stunted projective spaces. Theorem 5.2 ([21, III.3.1, III.3.2, III.3.3]) . Let H = H F be the mod- Eilenberg–Mac Lane spectrum. Then E n -algebras in Mod H have Dyer–Lashofoperations Q i : π m → π m + i for ≤ i ≤ n − . These satisfy the following formulas. Additivity: Q r ( x + y ) = Q r ( x ) + Q r ( y ) for r < n − . Squaring: Q x = x . Unit: Q j for j > . Cartan formula: Q r ( xy ) = P p + q = r Q p ( x ) Q q ( y ) for r < n − . Adem relations: Q r Q s ( x ) = P (cid:0) j − s − j − r − s (cid:1) Q r +2 s − j Q j ( x ) for r > s . Stability: σQ = 0 , and σQ r = Q r − for r > . Extension:
If an E n -algebra structure extends to an E n +1 -algebrastructure, the operations Q r for E n +1 -algebras coincide with theoperations Q r for E n -algebras. here is also a bilinear Browder bracket[ − , − ] : π r ⊗ π s → π r +( n − s satisfying the following formulas. Antisymmetry: [ x, y ] = [ y, x ] and [ x, x ] = 0 . Unit: [ x,
1] = 0 . Leibniz rule: [ x, yz ] = [ x, y ] z + y [ x, z ] . Jacobi identity: [ x, [ y, z ]] + [ y, [ z, x ]] + [ z, [ x, y ]] = 0 . Dyer–Lashof vanishing: [ x, Q r y ] = 0 for r < n − . Top additivity: Q n − ( x + y ) = Q n − x + Q n − y + [ x, y ] . Top Cartan formula: Q n − ( xy ) = P p + q = n − Q p ( x ) Q q ( y )+ x [ x, y ] y . Adjoint identity: [ x, Q n − y ] = [ y, [ y, x ]] . Extension:
If an E n -algebra structure extends to an E n +1 -algebrastructure, the bracket is identically zero. E -bracket: [ x, y ] = xy + yx if n = 1 . Remark 5.3.
There are two common indexing conventions for the Dyer–Lashofoperations. This lower-indexing convention is designed to emphasize the rangewhere the operations are defined, and is especially useful for E n -algebras. Theupper-indexing convention defines Q s x = Q s −| x | x so that Q s is always a naturaltransformation π m → π s + m , with the understanding that Q s x = 0 for s < | x | . Example 5.4.
Suppose that X is an n -fold loop space, so that H [ X ] is an E n -algebra in left H -modules. Then we recover the classical Dyer–Lashof operations Q r : H n ( X ) → H n + r ( X )in the homology of iterated loop spaces. Theorem 5.5 ([21, IX.2.1], [24, III.3.1]) . For any spectrum X and any ≤ n ≤∞ , H ∗ (Free E n ( X )) is the free object Q E n ( H ∗ X ) in the category of graded F -algebras with Dyer–Lashof operations and Browder bracket satisfying the iden-tities of Theorem 5.2. Remark 5.6.
This theorem is the analogue of the calculation of the cohomologyof Eilenberg–Mac Lane spaces as free algebras in a category of algebras withSteenrod operations. As such, it means that we have a complete theory ofhomotopy operations for E n -algebras over H . Example 5.7.
In the case n < ∞ we can give a straightforward description of Q E n V if V has a basis with a single generator e . In this case, the antisymmetry,unit, and Dyer–Lashof vanishing axioms can be used to show that the freealgebra has trivial Browder bracket, and so the free algebra Q E n ( V ) is a gradedpolynomial algebra F [ Q J e ]as we range over generators Q J e = ( Q ) j ( Q ) j . . . ( Q n − ) j n − e .22 .2 E ∞ Dyer–Lashof operations at p = 2 When n = ∞ , the results of the previous section become significantly simpler,and it is worth expressing using the upper indexing for Dyer–Lashof operations. Theorem 5.8 ([21, III.1.1]) . Let H = H F be the mod- Eilenberg–Mac Lanespectrum. Then E ∞ -algebras in Mod H have Dyer–Lashof operations Q r : π m → π m + r for r ∈ Z . These satisfy the following formulas. Additivity: Q r ( x + y ) = Q r ( x ) + Q r ( y ) . Instability: Q r x = 0 if r < | x | . Squaring: Q r x = x if r = | x | . Unit: Q r for r = 0 . Cartan formula: Q r ( xy ) = P p + q = r Q p ( x ) Q q ( y ) . Adem relations: Q r Q s = P (cid:0) i − s − i − r (cid:1) Q s + r − i Q i for r > s . Stability: σQ r = Q r . Example 5.9.
For any space X , H X is an E ∞ -algebra in the category of left H -modules, and hence it has Dyer–Lashof operations Q i : H n ( X ) → H n − i ( X ) . It turns out that these are precisely the
Steenrod operations : Sq i = Q − i . From this point of view, the identity Q x = x is not obvious. In fact, Mandellhas shown that this identity is characteristic of algebras that come from spaces:the functor X ( H F p ) X from spaces to E ∞ -algebras over the Eilenberg–MacLane spectrum H F p is fully faithful, and the essential image is detected interms of the coefficient ring being generated by classes that are annihilated bythe analogue at arbitrary primes of the identity ( Q −
1) [58].
Example 5.10.
In the case n = ∞ there is always a straightforward basis forthe free algebra. If { e i } is a basis of a graded vector space V over F , then thefree algebra Q E ∞ ( V ) is a graded polynomial algebra F [ Q J e i ]as we range over generators Q J e i = Q j . . . Q j p e i such that j i ≤ j i +1 and j − j − · · · − j p > | e i | . 23 .3 Iterated loop spaces The following is an unpointed group-completion theorem for E n -spaces. Theorem 5.11 ([24, III.3.3]) . For any space X and any ≤ n ≤ ∞ , the map X → Ω n Σ n X + induces a map Free E n ( X ) → Ω n Σ n X + , and the resulting ringmap Q E n ( H ∗ X ) = H ∗ (Free E n ( X )) → H ∗ (Ω n Σ n X + ) is a localization which inverts the images of π ( X ) . Remark 5.12.
A pointed version of the group-completion theorem, involvingΩ n Σ n X , is much more standard and implies this one. This theorem holds forΩ n Σ n if we replace Free E n with a version that takes the basepoint to a unitand we replace Q E n ( H ∗ X ) with either Q E n ( e H ∗ X ) a reduced version e Q E n thatsends a chosen element to the unit. However, we wanted to give a version thatde-emphasizes implicit basepoints for comparison with § Proposition 5.13.
Suppose Y is a pointed space. Then the suspension map σ : e H ∗ (Ω n Y ) → e H ∗ +1 (Ω n − Y ) , induced by the map ΣΩ n Y → Ω n − Y , is compatible with the Dyer–Lashof oper-ations and the Browder bracket: σ ( Q r x ) = Q r ( σx ) σ [ x, y ] = [ σx, σy ] In particular, in the bar spectral sequence
Tor H ∗ Ω n Y ∗∗ ( F , F ) ⇒ H ∗ Ω n − Y, the operations on the image e H ∗ Ω n Y ։ Tor H ∗ Ω n Y ( F , F ) are representativesfor the operations on H ∗ Ω n − Y . This provides some degree of conceptual interpretation for the bracket andthe Dyer–Lashof operations. Since H ∗ Ω Y is commutative, the Tor-algebra isalso commutative even though it is converging to the possibly noncommutativering H ∗ Ω Y , and so the noncommutativity is tracked by multiplicative exten-sions in the spectral sequence [74]. The Browder bracket in H ∗ Ω n Y exists toremember that, after n − xy ± yx in H ∗ Ω Y .Similarly, elements in positive filtration in the Tor-algebra of a commutativering always satisfy x = 0, even though this may not be the case in H ∗ Ω n − Y .The element Q x is x ; the elements Q x, Q x, . . . , Q n − x determine the lineof succession for the property of being x as the delooping process is iterated. Remark 5.14.
The group-completion theorem allows us to relate the homologyof a delooping to certain nonabelian derived functors [69]. Similar spectralsequences computing E n -homology of chain complexes have been studied byRichter and Ziegenhagen [79]. 24ssociated to the n -fold loop space Ω n Y of an ( n − E n -algebra (or an infinite loop space Ω ∞ Y associated to a connectivespectrum), we can construct three augmented simplicial objects: · · · Free E n Free E n Free E n Ω n Y ⇛ Free E n Free E n Ω n Y ⇒ Free E n Ω n Y → Ω n Y · · · Ω n Σ n + Free E n Free E n Ω n Y ⇛ Ω n Σ n + Free E n Ω n Y ⇒ Ω n Σ n + Ω n Y → Ω n Y · · · Σ n + Free E n Free E n Ω n Y ⇛ Σ n + Free E n Ω n Y ⇒ Σ n + Ω n Y → Y These are, respectively, two-sided bar constructions: B (Free E n , Free E n , Ω n Y ), B (Ω n Σ n + , Free E n , Ω n Y ), and B (Σ n + , Free E n , Ω n Y ).The first augmented bar construction B (Free E n , Free E n , Ω n Y ) has an extradegeneracy, and so its geometric realization is homotopy equivalent to Ω n Y as E n -spaces. Therefore, it is a group-complete E n -space.There is a natural map B (Free E n , Free E n , Ω n Y ) → B (Ω n Σ n + , Free E n , Ω n Y )which is, levelwise, a group-completion map [30, Appendix Q], [67], and in-duces a group-completion map on geometric realization. However, the source isalready group-complete, and so this map is an equivalence on geometric real-izations. Thus, the augmentation | B (Ω n Σ n + , Free E n , Ω n Y ) | → Ω n Y is an equiv-alence.The bar construction B (Σ n + , Free E n , Ω n Y ) is a simplicial diagram of ( n − n across geometric realization. The natural augmentationΩ n | B (Σ n + , Free E n , Ω n Y ) | → | B (Ω n Σ n + , Free E n , Ω n Y ) | → Ω n Y is an equivalence. By assumption, Y is ( n − | B (Σ n + , Free E n , Ω n Y ) | → Y is also an equivalence. Therefore, the simplicial object B (Σ n + , Free E n , Ω n Y )can be used to compute H ∗ Y .Let A = H ∗ (Ω n Y ). The reduced homology of B (Σ n + , Free E n , Ω n Y ) is · · · Σ n Q E n Q E n A ⇛ Σ n Q E n A ⇒ Σ n A, which is a bar complex Σ n B ( Q, Q E n , A ) computing nonabelian derived functors.These are specifically the derived functors of an indecomposables functor Q ,which takes an augmented Q E n -algebra A → F and returns the quotient of theaugmentation ideal by all products, brackets, and Dyer–Lashof operations. Theresult is a Miller spectral sequence that begins with nonabelian derived functorsof Q on H ∗ (Ω n Y ) and converges to e H ∗ Y . The Dyer–Lashof operations on the homology of the spaces BO and BU , andhence on the homology of the Thom spectra M O and
M U , was determined bywork of Kochman [44]; here we will state a form due to Priddy [75].25 heorem 5.15.
The ring H ∗ M O ∼ = H ∗ BO is a polynomial algebra on classes a i in degree i . The Dyer–Lashof operations are determined by the identities offormal series X j Q j a k = ∞ X n = k k X u =0 (cid:18) n − k + u − u (cid:19) a n + u a k − u ! ∞ X n =0 a n ! − , where a = 1 by convention. In particular, Q n a k ≡ (cid:0) n − k (cid:1) a n + k mod decompos-able elements.The ring H ∗ M U ∼ = H ∗ BU is a polynomial algebra on classes b i in degree i ,The Dyer–Lashof operations are determined by the identities of formal series X j Q j b k = ∞ X n = k k X u =0 (cid:18) n − k + u − u (cid:19) b n + u b k − u ! ∞ X n =0 b n ! − , where b = 1 by convention. In particular, Q n b k ≡ (cid:0) n − k (cid:1) b n + k mod decompos-able elements, and Q n +1 b k = 0 . Remark 5.16.
Implicit in this calculation is the fact that the Thom isomor-phisms H ∗ M O ∼ = H ∗ BO and H ∗ M U ∼ = H ∗ BU preserve Dyer–Lashof opera-tions. Lewis showed that, for an E n -map f : X → BGL ( S ), the Thom isomor-phism H ∗ X ∼ = H ∗ M f lifts to an equivalence of E n ring spectra H [ X ] → H ⊗ M f called the Thom diagonal [50, 7.4]. As a result, the Thom isomorphism isautomatically compatible with Dyer–Lashof operations for H -algebras. Example 5.17.
We have explicit calculations of the first few Dyer–Lashof op-erations in H ∗ M O : Q a = a Q a = a + a a + a Q a = a Q a = a + a a + a a + a a + a a + a a + a Q a = a + a a + a a + a a These same formulas hold for the b i in H ∗ M U . Recall that, if R is an E n -algebra in Sp , H ⊗ R is an E n -algebra in Mod H whose homotopy groups are the homology groups of R . As a result, there aretwo types of operations on H ∗ R : 26 The E n -algebra structure gives H ∗ ( R ) Dyer–Lashof operations Q , . . . , Q n − and a Browder bracket. • The property of being homology gives H ∗ ( R ) Steenrod operations P d : H m R → H m − d R . To make these dual to the Steenrod operations Sq d in cohomol-ogy, P d ( x ) is defined as a composite S m − d Σ − d x −−−→ (Σ − d H ) ⊗ R χSq d −−−→ H ⊗ R. This implicitly reverses multiplication order: for example, the Adem rela-tion Sq = Sq Sq becomes P = P P .The Nishida relations express how these structures interact. Theorem 5.18 ([21, III.1.1, III.3.2]) . Suppose that R is an E n -algebra in Sp .Then the Steenrod operations in homology satisfy relations as follows. Cartan formula: P r ( xy ) = P p + q = r P p ( x ) P q ( y ) . Browder Cartan formula: P r [ x, y ] = P p + q = r [ P p x, P q y ] . Nishida relations: P r Q s = P (cid:0) s − rr − i (cid:1) Q s − r + i P i if s < n − . Top Nishida relation: P r Q n − ( x ) = X (cid:18) n − − rr − i (cid:19) Q n − − r + i P i + X p + q = r,p By contrast with the Adem relations, the Nishida relationsbehave very differently if we use lower indexing. We find P r Q s ( x ) = X (cid:18) | x | + s − rr − i (cid:19) Q s − r + i P i ( x ) . In particular, the lower-indexed Nishida relations depend on the degree of x [22]. Remark 5.20. If we use the pro-representability of homology as in § Q r and the P d together with the Nishida relations. If the homology H ∗ R is easily described a module over the Steenrod algebra,the Nishida relations can completely determine the Dyer–Lashof operations.This was applied by Steinberger to compute the Dyer–Lashof operations in thedual Steenrod algebra explicitly. (Conversely, Baker showed that the Nishidarelations themselves are completely determined by the Dyer–Lashof operationstructure of the dual Steenrod algebra [5].)27 heorem 5.21 ([21, III.2.2, III.2.4]) . Let A ∗ be the dual Steenrod algebra F [ ξ , ξ , . . . ] where | ξ i | = 2 i − , with conjugate generators ξ i (here ξ i is denoted by ζ i in[71]). Then the Dyer–Lashof operations on the generators are determined bythe following formulas.1. There is an identity of formal series (1 + ξ + Q ξ + Q ξ + Q ξ + . . . ) = (1 + ξ + ξ + ξ + . . . ) − . 2. For any i , we have Q s ξ i = ( Q s +2 i − ξ if s ≡ , − i , otherwise.3. In particular, Q i − ξ = ξ i , and Q ξ i = ξ i +1 . Remark 5.22. This allows us to say that the dual Steenrod algebra can bere-expressed as follows: A ∗ ∼ = F [ x, Q x, ( Q ) x, . . . ]This is the same as the homology of Ω S : both are identified with the homologyof the free E -algebra on a generator x = ξ in degree 1. Mahowald showed thatit was possible to realize this isomorphism of graded algebras: he constructeda Thom spectrum over Ω S such that the Thom isomorphism realizes theisomorphism A ∗ ∼ = H ∗ Ω S [56]. This has a rather remarkable interpretation:there exists a construction of the Eilenberg–Mac Lane spectrum H as the free E -algebra R such that the unit map S → R has a chosen nullhomotopy of theimage of 2. This result has been extended to odd primes by Blumberg–Cohen–Schlichtkrull [15]. Proposition 5.23. Let Hk be the Eilenberg–Mac Lane spectrum for an algebra k over F . Then there is an isomorphism H ∗ H F ∼ = A ∗ ⊗ k of graded rings, and under this identification the Dyer–Lashof operation Q r on H ∗ Hk is given by Q r ⊗ ϕ , where ϕ is the Frobenius on k .Proof. For any H -module N , the action H ⊗ N → N induces an isomorphism H ∗ H ⊗ π ∗ N → H ∗ N . We already know Q (1 ⊗ α ) = 1 ⊗ α , and so by theCartan formula it suffices to show that Q s (1 ⊗ α ) = 0 for s > Q t (1 ⊗ α ) = 0 for 0 < t < s , we find that for all r > P r Q s (1 ⊗ α ) = X (cid:18) s − rr − i (cid:19) Q s − r + i P i (1 ⊗ α )= (cid:18) s − rr (cid:19) Q s − r (1 ⊗ α ) . 28y the inductive hypothesis, this vanishes unless s = r , but in the case s = r the binomial coefficient vanishes. However, the only elements in H ∗ Hk that areacted on trivially by all the Steenrod operations are the elements in the imageof π ∗ Hk , and those are concentrated in degree zero. Thus Q s (1 ⊗ α ) = 0. Remark 5.24. The same proof can be used to show that the Browder bracketis trivial on H ∗ Hk . Example 5.25. The composite map M U → M O → H , on homology, is givenin terms of the generators of Theorem 5.15 by b a ξ and b 0. Theimage of H ∗ M U in A ∗ is F [ ξ , ξ , . . . ], the homology of the Brown–Petersonspectrum BP .In H ∗ M U , Example 5.17 implies we have the identities Q b = b + b b + b b + b b = Q b + b Q ( b ) . By contrast, in the dual Steenrod algebra we have the identity 0 = Q ( ξ ) + ξ Q ( ξ ). Even though the map H ∗ M U → H ∗ BP splits as a map of algebras,and the target is closed under the Dyer–Lashof operations, we have Q b + b Q ( b ) = Q ( b ) + b Q ( b ) = 0but its image is zero. This implies that the map H ∗ M U → H ∗ BP does nothave a splitting that respects the Dyer–Lashof operations for E -algebras. Asa result, there exists no map BP → M U (2) of E -algebras. This result, and itsanalogue at odd primes, is due to Hu–Kriz–May [40]. The tremendous amount of structure present in the homology of a ring spectrumallows us to produce a rather large number of nonexistence results. The followingis a generalization of the classical result that the mod-2 Moore spectrum does notadmit a multiplication due to the existence of a nontrivial Steenrod operation Sq in its cohomology; we learned this line of argument from Charles Rezk. Proposition 5.26. Suppose that R is a homotopy associative ring spectrumcontaining an element u in nonzero degree such that P k ( u ) vanishes either inthe range k > | u | or in the range < k < | u | . Then either P | u | ( u ) is nilpotentor H ∗ R is nonzero in infinitely many degrees.Proof. We find, by the Cartan formula, that P d | u | ( u d ) = ( P | u | u ) d . Therefore, either the elements u d are nonzero for all d or the element P | u | u isnilpotent. Corollary 5.27. Suppose that R is a connective homotopy associative ringspectrum such that H ( R ) = π ( R ) / has no nilpotent elements. If any nonzeroelement in H ( R ) is in the image of the Steenrod operations, then H ∗ R must benonzero in infinitely many degrees. orollary 5.28. Suppose that R is a homotopy associative ring spectrum andthat some Hopf invariant element , η , ν , or σ maps to zero under the unit map S → R . Then either H ∗ R = 0 or H ∗ R is infinite-dimensional.Proof. Writing h for Hopf invariant element in degree 2 k − S → R extends to a map f : C ( h ) → R from the mapping cone. Thehomology of C ( h ) has a basis of elements 1 and v with one nontrivial Steenrodoperation acting via P k v = 1, and u = f ∗ ( v ) has the desired properties.Recall from § R connective, a map π R → A of commutativerings automatically extends to a map R → HA compatible with the multiplica-tive structure that exists on R ; e.g., if R is homotopy commutative then themap R → HA has the structure of a map of Q -algebras. This has the followingconsequence. Proposition 5.29. Suppose that R is a connective ring spectrum with a ringhomomorphism π R → k where k is an F -algebra (equivalently, a map H R → k ). Then there is a map R → Hk which induces a homology map H ∗ R → A ∗ ⊗ k with the following properties.1. The map H ∗ R → A ∗ ⊗ k is a map of rings which is surjective in degreezero.2. If R is homotopy commutative, then there is an operation Q on H ∗ R thatis compatible with the operation Q on A ∗ ⊗ k .3. If R has an E n -algebra structure, the map R → Hk is a map of E n -algebras and so H ∗ R → A ∗ ⊗ k is compatible with the Dyer–Lashof oper-ations Q , . . . , Q n − .In particular, the image of H ∗ R in A ∗ ⊗ k is a subalgebra B ∗ ⊂ A ∗ closed undermultiplication and some number of Dyer–Lashof operations. Example 5.30. For n > K -theories k ( n ), withcoefficient ring F [ v n ], that have homology F [ ξ , . . . , ξ n , ξ n +1 , ξ n +2 , . . . ]as a subalgebra of the dual Steenrod algebra. This subring is not closed underthe Dyer–Lashof operation Q unless n = 0, and so the connective Morava K -theories are not homotopy–commutative. (By convention we often define theconnective Morava K -theory k (0) to be H Z , which is commutative.)Similarly, for n > K -theories k Z ( n ), withcoefficient ring Z [ v n ], have homology F [ ξ , ξ , . . . , ξ n , ξ n +1 , ξ n +2 , . . . ]as a subalgebra of the dual Steenrod algebra. This subring is not closed underthe Dyer–Lashof operation Q unless n = 1, and so the only possible homotopy-commutative integral Morava K -theory is k Z (1)—the connective complex K -theory spectrum. 30here are obstruction-theoretic proofs which show that all of these have A ∞ structures [3, 49]. Example 5.31. The Dyer–Lashof operations satisfy Q ( ξ i ) = ξ i +1 , and sothe smallest possible subring of A ∗ that contains ξ = ξ and is closed under Q is an infinite polynomial algebra F [ ξ i ] = F [ ξ i ]. If R is a connective ringspectrum with a quotient map π R → F such that the Hopf element η ∈ π ( S )maps to zero in π ∗ R , then there is a commutative diagram C ( η ) / / (cid:15) (cid:15) R (cid:15) (cid:15) H Z / / / H F . We conclude that ξ is in the image of the map H ∗ R → H ∗ H F .The spectra X ( n ) appearing in the nilpotence and periodicity theorems ofDevinatz–Hopkins–Smith fit into a sequence X (1) → X (2) → X (3) → . . . of Thom spectra on the spaces Ω SU ( n ). They have E -ring structures, andeach ring H ∗ X ( n ) is a polynomial algebra F [ x , . . . , x n − ] on finitely manygenerators. For n = 2 the map H ∗ X (2) → A ∗ is the map F [ ξ ] → A ∗ , andthis implies that each X ( n ) has ξ in the image of its homology. As H ∗ X ( n ) isfinitely generated as an algebra, its image in the dual Steenrod algebra is toosmall to be closed under the operation Q . This excludes the possibility that X ( n ) has an E -structure. Associated to an E ∞ ring spectrum E , there is a sequence of infinite loop spaces { E n } n ∈ Z in an Ω-spectrum representing E . These spaces are extremely stronglystructured: they inherit both additive structure from the spectrum structure on E , and multiplicative structure from the E ∞ ring structure. In the case of thesphere spectrum, these operations were investigated in-depth in relationship tosurgery theory [68, 55, 63]. Ravenel and Wilson discussed the structure comingfrom a ring spectrum E extensively in [76], encoding it in the structure of a Hopf ring , and the interaction between additive and multiplicative operationsis developed in-depth in [24, § II]. These structures are very tightly wound.1. Because the E n are spaces, the diagonals E n → E n × E n gives rise to a coproduct ∆ : H ∗ ( E n ) → H ∗ ( E n ) ⊗ H ∗ ( E n ) , For an element x we write P x ′ ⊗ x ′′ for its coproduct. The path compo-nents E n = π E n also give rise to elements [ α ] ∈ H E n .31. The homology groups H ∗ E n have Steenrod operations P r .3. The suspension maps Σ E n → E n +1 in the spectrum structure give stabi-lization maps H m ( E n ) → H m +1 ( E n +1 ) . 4. The infinite loop space structure on E n gives H ∗ E n an additive Pontrjaginproduct H ∗ ( E n ) ⊗ H ∗ ( E n ) → H ∗ ( E n )making it into a Hopf algebra, and it has additive Dyer–Lashof operations Q r : H m ( E n ) → H m + r ( E n ) . 5. If E has a ring spectrum structure, the multiplication E ⊗ E → E gives multiplicative Pontrjagin products ◦ : H ∗ ( E n ) ⊗ H ∗ ( E m ) → H ∗ ( E n + m ) . These are appropriately unital, associative, or graded-commutative if E has these properties.6. If E has an E ∞ ring spectrum structure, there are multiplicative Dyer–Lashof operations e Q r : H m ( E ) → H m + r ( E )on the homology of the 0’th space. In general, we cannot say more. If E has further structure—an H d ∞ -structure—there are also multiplicativeDyer–Lashof operations outside degree zero [48, § Distributive rule: ( x y ) ◦ z = P ( x ◦ z ′ ) y ◦ z ′′ ) Projection formula: x ◦ Q s y = P Q s + k ( P k x ◦ y ) Mixed Cartan formula: e Q n ( x y ) = X p + q + r = n e Q p ( x ′ ) Q q ( x ′′ ◦ y ′ ) e Q r ( y ′′ ) Mixed Adem relations: e Q r Q s x = X i + j + k + l = r + s (cid:18) r − i − l − j + s − i − l (cid:19) Q i e Q j x ′ Q k e Q l x ′′ Example 5.32. There is an identity Q [ a ] − a ] = η · a η map π R → π R → H Ω ∞ R from the additive Dyer–Lashof structure. Similarly e Q determines information about its multiplicative version η m : π ( R ) → π ( R ).For example, the mixed Cartan formula implies that η m ( x + y ) = η m ( x ) + η m ( y ) + η · xy in H ( R ). In particular, e Q [ n ] = (cid:0) n (cid:1) η n ] for n ∈ Z (cf. Example 3.11). Secondary operations, at their core, arise when there are relations between re-lations. Suppose that we are a sequence X f −→ X g −→ X h −→ X of mapssuch that the double composites are nullhomotopic. Then hgf is nullhomo-topic for two reasons. Choosing nullhomotopies of gf and hg , we can glue thenullhomotopies together to determine a loop in the space of maps X → W : a value of the associated secondary operation. Because we must make choices ofnullhomotopy, there is some natural indeterminacy in this construction, and soit typically takes a set of values h h, g, f i . To construct secondary operations,we minimally need to work in a category C with mapping spaces; we also needcanonical basepoints of the spaces Map C ( X i , X j ) for j ≥ i +2 that are preservedunder composition [48, § Example 6.1. Suppose that A is a subspace of X and α ∈ H n ( X, A ) is acohomology element that restricts to zero in H n ( X ). Then the long exactsequence in cohomology implies that we can lift α to an element in H n − ( A ),but there are multiple choices of lift. This can be represented by a sequence ofmaps A → X → X/A → K ( Z , n )where the double composites are nullhomotopic; the secondary operation is thena map A → Ω K ( Z , n ) = K ( Z , n − h C with extra structure.1. Every test object T ∈ C represents a functor [ T, − ] = π Map C ( T, − ) on h C , and if T has an augmentation T → T, − ] differ on X and Y , X and Y cannot be equivalent in h C .2. Every map of test objects Θ : S → T determines an operation: a naturaltransformation of functors θ : [ T, − ] → [ S, − ] on h C . If S and T areaugmented and the map Θ is compatible with the augmentations, then θ preserves the null element. If θ has different behaviour for X and Y , X and Y cannot be equivalent. 33. Given an augmented map Φ : R → S and a map Θ : S → T such thatthe double composite ΘΦ : R → T is trivial, we get an identity ϕθ =0 of associated operations. There is an associated secondary operation h− , Θ , Φ i . It is only defined on those elements α ∈ [ T, X ] with θ ( α ) = 0; ittakes values in π Map C ( R, X ); it is only well-defined up to indeterminacy.4. We can also associate information to maps in the same way. Supposewe have an augmented map Θ : S → T of test objects representing anoperation θ . Given any map f : X → Y , there is an associated functionaloperation h f, − , Θ i . It is only defined on those elements α ∈ [ T, X ] suchthat f ( α ) = 0 and θ ( α ) = 0; it takes values in π Map C ( S, Y ); it is onlywell-defined up to indeterminacy.Applying this to the test objects S n in the category of pointed spaces, we getToda’s bracket construction that enriches the homotopy groups of spaces withsecondary composites. Applying this to the test objects K ( A, n ) in the oppositeof the category of spaces, we get Adams’ secondary operations that enrich thecohomology groups of spaces with secondary cohomology operations. We recall from § M and an O -algebra in Mod E , wehave π M ( A ) = [ M, A ] Sp ∼ = [ E ⊗ Free O ( M ) , A ] Alg O ( Mod E ) . Using free algebras as our test objects, we already used this representabilityof homotopy groups to classify the natural operations on the homotopy groupsof O -algebras in Mod E . The space of maps now means that we can constructsecondary operations. Proposition 6.2. Suppose that we have zero-preserving operations θ : π M → π N and ϕ : π N → π P on the homotopy category of O -algebras in Mod E , andthat there is a relation ϕ ◦ θ = 0 . Then there exists a secondary operation h− , Θ , Φ i : π M ( A ) ⊃ ker θ → π P +1 ( A ) /Im ( σϕ ) , where σ ( ϕ ) is a suspended operation (see § Such a secondary operation is constructed from a sequence E ⊗ Free O ( P ) Φ −→ E ⊗ Free O ( N ) Θ −→ E ⊗ Free O ( M ) → A where the double composites are null; the nullhomotopy of Θ ◦ Φ is chosen onceand for all, while the second nullhomotopy is allowed to vary. This produceselements in π Map Alg O ( Mod E ) ( E ⊗ Free O ( P ) , A ) ∼ = π Map Sp ( P, A ) ∼ = [Σ P, A ] . xample 6.3. Every Adem relation between Dyer–Lashof operations producesa secondary Dyer–Lashof operation. For example, the relation Q n +2 Q n + Q n +1 Q n +1 = 0 produces a natural transformation π m ( A ) ⊃ ker( Q n , Q n +1 ) → π m +2 n +3 ( A ) /Im ( Q n +2 , Q n +1 )on the homotopy of H -algebras. Example 6.4. Relations involving operations other than composition and addi-tion can also produce secondary operations, and the canonical examples of theseare Massey products . An A -algebra R has a binary multiplication operation R ⊗ R → R , and if R is an A -algebra it has a chosen associativity homotopy.As a result, if we have elements x , y , and z in π ∗ R such that xy = yz = 0, thenwe can glue together two nullhomotopies of xyz to obtain a bracket h x, y, z i thatspecializes to definitions of Massey products or Toda brackets.In trying to express nonlinear relations as secondary operations, however, werapidly find that we want to move into a relative situation. A Massey product isdefined on the kernel of the map π p × π q × π r → π p + q × π q + r sending ( x, y, z ) to( xy, yz ). However, the relation x ( yz ) = ( xy ) z is not expressible solely as someoperation on xy and yz : we need to remember x and z as well, but we do not want to enforce that they are zero.We find that the needed expression is homotopy commutativity of the fol-lowing diagram:Free A ( S p ⊕ S p + q + r ⊕ S r ) Φ / / (cid:15) (cid:15) Free A ( S p ⊕ S p + q ⊕ S q + r ⊕ S r ) Θ (cid:15) (cid:15) Free A ( S p ⊕ S r ) / / Free A ( S p ⊕ S q ⊕ S r )The right-hand map classifies the operation θ ( x, y, z ) = ( x, xy, yz, z ), and thetop map classifies the operation ϕ ( x, u, v, z ) = ( x, xv − uz, z ). The bottom-leftobject is not the initial object in the category of A -algebras, so we enforce thisby switching to the category C of A -algebras under Free A ( S p ⊕ S r ). In thiscategory, we genuinely have augmented objects with a nullhomotopic doublecomposite Free C ( S p + q + r ) → Free C ( S p + q ⊕ S q + r ) → Free C ( S q )that defines a Massey product. Secondary operations are part of the homotopy theory of C , and there is typicallyno method to determine secondary operations purely in terms of the homotopycategory. However, there are many composition-theoretic tools that use onesecondary operation to determine information about another: typically, onestarts with a 4-fold composite X f −→ Y g −→ Z h −→ U k −→ V, juggling , andlearning to juggle secondary operations is one of the main steps in applyingthem. For instance, one of the main juggling formulas—the Peterson–Steinformula—asserts that the sets h k, h, g i f and k h h, g, f i are inverse in π whenboth sides make sense. Example 6.5. The Adem relations Q n +1 Q n and Q n +3 Q n +1 give rise to a sec-ondary operation h Q n , Q n +1 , Q n +3 i , an element of π n +5+ m ( H ⊗ Free E ∞ ( S m ))representing an operation that increases degree by 7 n + 5. The juggling formulasays that, for any element α ∈ π m ( A ) with Q n ( A ) = 0, we have Q n +3 h α, Q n , Q n +1 i = h Q n , Q n +1 , Q n +3 i ( α ) . In other words, this secondary composite of operations gives a universal formulafor how to apply Q n +3 to this secondary operation. In this section we will give a brief account of the main result of [48], which usessecondary operatons to show that the 2-primary Brown–Peterson spectrum BP does not admit the structure of an E ring spectrum. These results have beengeneralized by Senger to show that, at the prime p , BP does not have an E p +4 ring structure [87].As in § E n -algebra structurethen the map BP → H Z (2) → H can be given the structure of a map of E n -algebras. On homology, this wouldthen induce a monomorphism F [ ξ , ξ , . . . ] → F [ ξ , ξ , . . . ]of algebras equipped with E n Dyer–Lashof operations and secondary Dyer–Lashof operations. The dual Steenrod algebra, on the right, has operations thatare completely forced. Therefore, if we can calculate enough to show that thesubalgebra H ∗ BP is not closed under secondary operations for E n -algebras, wearrive at an obstruction to giving BP the structure of E n ring spectrum.The calculation of secondary operations in H ∗ H is accomplished with judi-cious use of juggling formulas, ultimately reducing questions about secondaryoperations to questions about primary ones. • There is a pushout diagram of E ∞ ring spectra H ⊗ M U i / / (cid:15) (cid:15) H ⊗ H j (cid:15) (cid:15) H / / H ⊗ MU H. H ⊗ M U into an augmented H -algebra, and gives a nullho-motopy of the composite H ⊗ M U → H ⊗ H → H ⊗ MU H . The elements α in H ∗ M U that map to zero in H ∗ H are then candidates for secondaryoperations: we can construct h j, i, α i in the M U -dual Steenrod algebra π ∗ ( H ⊗ MU H ). • These elements are concretely detected: they have explicit representativeson the 1-line of a two-sided bar spectral sequenceTor H ∗ MU ( H ∗ H, F ) ⇒ π ∗ ( H ⊗ MU H ) . • If we can determine primary operations θ ( h i, j, α i ) in the M U -dual Steen-rod algebra, the juggling formulas of § h j, α, Θ i in the ordinary dual Steenrod algebra. • Steinberger’s calculations of primary operations ϕ in the dual Steenrodalgebra then allow us to determine the values of ϕ h j, α, Θ i , and jugglingformulas again allow us to determine information about secondary opera-tions h α, Θ , Φ i in the dual Steenrod algebra.This method, then, reduces us to carrying out some key computations.We must determine primary operations in the M U -dual Steenrod algebra.Some of these, by work of Tilson [94], are determined by Kochman’s calculationsfrom Theorem 5.15: the K¨unneth spectral sequenceTor π ∗ MU ( F , F ) ⇒ π ∗ ( H ⊗ MU H )calculating the M U -dual Steenrod algebra is compatible with Dyer–Lashof op-erations. However, there are remaining extension problems in the Tor, and theseturn out to be precisely what we are interested in when juggling.The M U -dual Steenrod algebra is an exterior algebra, whose generatorscorrespond to the indecomposables in π ∗ M U . The extension problems in theTor spectral sequence arise because some generators in π ∗ M U have nontrivialimage in H ∗ M U and are detected by Tor , while others have trivial image in H ∗ M U and are detected by Tor . The solution is to find an algebra R mappingto M U that does not have this problem. If we can find one so that the map π ∗ R → π ∗ M U is surjective, the map from the R -dual Steenrod algebra to the M U -dual Steenrod algebra is surjective. If the generators of π ∗ R have nontrivialimage in H ∗ R , then the spectral sequenceTor H ∗ R ( H ∗ H, F ) ⇒ π ∗ ( H ⊗ R H )detects all needed classes with Tor and hence eliminates the extension problem.For this purpose, we used the spherical group algebra S [ SL ( M U )]. TheDyer–Lashof operations in H ∗ SL ( M U ) are derived from the multiplicativeDyer–Lashof operations e Q n in Ω ∞ M U . This is a lengthy calculation of poweroperations in the Hopf ring, and it is ultimately determined by calculations ofJohnson–Noel of power operations in the formal group theory of M U [41].37inally, we must determine a candidate secondary operation in H ∗ H towhich we can apply this procedure—there are many candidate operations andmany dead ends. The secondary operation is rather large: it was found using acalculation in Goerss–Hopkins obstruction theory that is detailed at length in[47]. In § M , including extended powerand free algebra functors. The definitions we used made heavy use of a strictsymmetric monoidal structure on the category of spectra.In this section we will discuss the coherent viewpoint on these constructionsthat makes use of the machinery of Lurie [54], and with the goal of connectingdifferent strata in the literature. To begin, we should point some of the problemsthat this discussion is meant to solve.We would like to demonstrate that our constructions are model-independent.There are several different symmetric monoidal categories of spectra [28, 39, 83]with several different model structures, and there is a nontrivial amount ofwork involved in showing that an equivalence between two different categoriesof spectra gives an equivalence between categories of algebras [85]. These issuesare compounded when we attempt to relate notions of commutative algebras indifferent categories, even if they have equivalent homotopy theory [95].We would also like to allow weaker structure than a symmetric monoidalstructure. For example, given a fixed E n -algebra R we will use this to discussthe classification of power operations on E n -algebras under R . Our natural homefor this discussion will be the category of E n R -modules (as in Example 6.4). As discussed in § 1, classical symmetric monoidal categories are analogues of com-mutative monoids with the difference that they require natural isomorphismsto express associativity, commutativity, and the like. We can express this struc-ture using simplicial operads. For any categories C and D , there is a groupoidFun( C , D ) ≃ of functors and natural isomorphisms. Taking the nerve, we get asimplicially enriched category Cat , and it makes sense to ask whether C has thestructure of an algebra over a simplicial operad O . Example 7.1. A symmetric monoidal category can be expressed as an algebraover the Barratt–Eccles operad [6]. Example 7.2. In classical category theory, a braided monoidal category in thesense of [42] can be encoded by a sequence of maps N P n → Fun( C n , C )38rom the nerves of the pure braid groups to the categories of functors C n → C .The required compatibilities between these maps can be concisely expressedby noting that these nerves assemble into an E -operad, and that a braidedmonoidal category is an algebra over this operad.We would like to discuss E n -analogues of these structures in the context ofcategories with morphism spaces. We will give some definitions in this section,on the point-set level, with the purpose of interpolating the older and newerdefinitions. We would like to say that an O -monoidal category is an algebra overthe operad O in Cat , but this requires us to be clever enough to have a well-behaved definition of a space of functors between two enriched categories; thefailure of enriched categories to have a well-behaved enriched functor categoryis a principal motivation for the use of quasicategories.Until further notice, all categories and multicategories are assumed to beenriched in spaces and all functors are functors of enriched categories. Definition 7.3. Suppose that p : C → M is a multifunctor, and write C x forthe category p − ( x ). Given objects X i ∈ C x i and a map α : A → Mul M ( x , . . . , x d ; y )of spaces, an α -twisted product is an object Y ∈ C y and a map A → Mul C ( X , . . . , X d ; Y )such that, for any Z ∈ C with p ( Z ) = z , the diagramMap C ( Y, Z ) / / (cid:15) (cid:15) Map( A, Mul C ( X , . . . , X d ; Z )) (cid:15) (cid:15) Map M ( y , z ) / / Map( A, Mul M ( x , . . . , x d ; z ))is a pullback. If it exists, we denote it by A ⋉ α ( X , . . . , X d ). Definition 7.4. An weakly M -monoidal category is a multifunctor p : C → M that has α -twisted product for any inclusion α : { f } ⊂ Mul M ( x , . . . , x d ; y ) . A strongly M -monoidal category is a category that has α -twisted products forall α . Remark 7.5. In particular, for any point f ∈ Mul M ( x , . . . , x d ; y ), this uni-versal property can be used to produce a functor { f } ⋉ ( − ) : C x × · · · × C x d → C y , and these are compatible with composition (up to natural isomorphism). Aweakly M -monoidal category determines, up to natural equivalence, a multi-functor M → Cat . 39 xample 7.6. Every multicategory C has a multifunctor to the one-objectmulticategory C omm associated to the commutative operad. The multicategoryis C omm -monoidal if and only if multimorphisms ( X , . . . , X d ) → Y are alwaysrepresentable by an object X ⊗ · · · ⊗ X d , which is precisely when C comesfrom a symmetric monoidal category. It is strongly C omm -monoidal only if itis also tensored over spaces in a way compatible with the monoidal structure asin Definition 3.9. Example 7.7. Associated to a monoidal category C we can build a multicate-gory: multimorphisms ( X , . . . , X d ) → Y are pairs of a permutation σ ∈ Σ d anda map f : X σ (1) ⊗ · · · ⊗ X σ ( d ) → Y . There is a multifunctor from this categoryto the multicategory A ssoc corresponding to the associative operad: it sendsall objects to the unique object, and sends each multimorphism ( σ, f ) as aboveto the permutation σ . Conversely, an A ssoc -monoidal category comes from amonoidal category. Example 7.8. Suppose that A is a commutative ring and B is an A -algebra.Then there is a multicategory C as follows.1. An object of C is either an A -module or a right B -module.2. The set Mul C ( M , . . . , M d ; N ) of multimorphisms is Hom A ( M ⊗ A · · · ⊗ A M d , N ) if N and all M i are A -modules,Hom B ( M ⊗ A · · · ⊗ A M d , N ) if N and exactly one M i are B -modules, ∅ otherwise.This comes equipped with a functor from C to the multicategory Mod fromExample 3.23 that parametrizes ring-module pairs: any A -module is sent to a and any B -module is sent to m . This makes C into a Mod -monoidal category,expressing the fact that Mod A has a tensor product and that objects of RMod B can be tensored with objects of Mod A . This makes RMod B left-tensored over Mod A . Example 7.9. Fiberwise homotopy theory studies the category S /B of spacesover B . Let O be an operad and B be a space with the structure of an O -algebra.Then S /B has the structure of a strongly O -monoidal category in the followingway. For spaces X , . . . , X d and Y over B , the space of multimorphisms is thepullbackMul /B ( X , . . . , X d ; Y ) / / (cid:15) (cid:15) Map( X × · · · × X d , Y ) (cid:15) (cid:15) O ( d ) / / Map( B d , B ) / / Map( X × · · · × X d , B ) . That is, a multimorphism consists of a point f ∈ O ( d ) and a commutative40iagram X × · · · × X d / / (cid:15) (cid:15) Y (cid:15) (cid:15) B d f / / B. With this definition, it is straightforward to verify that for α : A → O ( d ), the α -twisted product A ⋉ α ( X , . . . , X d ) is the following space over B : A × X × · · · × X d → O ( d ) × B d → B. In general, this should not be expected to be part of a symmetric monoidalstructure on the category of spaces over B , even up to equivalence. Example 7.10. Let L be the category of universes : an object is a countablyinfinite dimensional inner product space U . These objects have an associatedmulticategory: the space Mul L ( U , . . . , U d ; V ) of multimorphisms is the (con-tractible) space of linear isometric embeddings U ⊕ · · ·⊕ U d ֒ → V . Over L , thereis a category Sp L of indexed spectra . An object is a pair ( U, X ) of a universe U and a spectrum X (in the Lewis–May–Steinberger sense [50]) indexed on U ; amultimorphism (( U , X ) , . . . , ( U d , X d )) → ( V, Y ) is a pair of a linear isometricembedding i : U ⊕ · · · ⊕ U d → V and a map i ∗ ( X ∧ · · · ∧ X d ) → Y of spectraindexed on V .This does not describe the topology on the multimorphisms in this category.Given a map A → L ( U , . . . , U d ; V ) and spectra X i indexed on U i , there is a twisted half-smash product A ⋉ ( X , . . . , X d ) indexed on V [50, § VI], equivalentto the smash product A + ∧ X ∧ · · · ∧ X d . There exists a topology on themultimorphisms so that a continuous map in from A is equivalent to a map A → L ( U , . . . , U d ; V ) and a map A ⋉ ( X , . . . , X d ) → Y . By design, then, theprojection Sp L → L makes the category of indexed spectra strongly L -monoidal. Example 7.11. Fix an E n -algebra A in Sp , and consider the category of E n -algebras R with a factorization A → R → A of the identity map. This has anassociated stable category , serving as the natural target for Goodwillie’s calculusof functors: the category of E n A -modules [29]. This category should also notbe expected to have a symmetric monoidal structure, but the tensor productover R does give it the structure of an E n -monoidal category. For example, foran associative algebra A in Sp , the tensor product over A gives the category of A -bimodules a monoidal structure. Just as we cannot make sense of a commutative monoid in a nonsymmetricmonoidal category, we need relationships between an operad O and any multi-plicative structure on a category C before O can act on objects.41 efinition 7.12. Suppose p : C → N and M → N are multifunctors. An M -algebra in C is a lift in the diagram C p (cid:15) (cid:15) M > > / / N of multifunctors. We write Alg M / N ( C ) for this category of M -algebras. Example 7.13. If C and M are arbitrary multicategories, then using the uniquemaps from C and M to the terminal multicategory C omm we recover the defi-nition of Alg M ( C ), the category of M -algebras in C from Definition 3.22. Example 7.14. Let the space B be an algebra over an operad O and considerthe fiberwise category S /B of spaces over B with the strongly O -monoidal struc-ture from Example 7.9. An O -algebra in S /B is an O -algebra X with a map of O -algebras X → B . Example 7.15. Consider the category of indexed spectra Sp L from Exam-ple 7.10. The fact that the external smash product ( X ∧ · · · ∧ X n ) is naturallyindexed on the direct sum of the associated universes obstructed making thecategory of spectra indexed on any individual universe Sp strictly symmetricmonoidal, and so we cannot ask about commutative monoids in Sp L —but thestructure available is still enough to do multiplicative homotopy theory. An L -algebra in Sp L recovers the classical definition of an E ∞ ring spectrum from[65]. Similarly we can define O -algebras for any operad O with an augmentationto L [50, VII.2.1]. Proposition 7.16. Suppose that C is strongly N -monoidal and that M → N isa map of multicategories. In addition, suppose that C has enriched colimits andthat formation of α -twisted products preserves enriched colimits in each variable.1. For objects x and y of M , there are extended power functors Sym k M , x → y : C x → C y , given by Sym k M , x → y ( X ) = Mul M ( x , x , . . . , x | {z } k ; y ) ⋉ ( X, X, . . . , X ) / Σ k . 2. The evaluation functor ev x : Alg M ( C ) → C x has a left adjoint Free M , x : C x → Alg M ( C ) . The value of Free M , x ( X ) on any object y of M is ev y (Free M , x ( X )) = a k ≥ Sym k M , x → y ( X ) . xample 7.17. Let B be a space with an action of an operad O , and let X aspace over B . Then the extended powers areSym k O ( X ) = (cid:0) O ( k ) × Σ k X k → O ( k ) × Σ k B k → B (cid:1) . Example 7.18. Suppose that Γ is a commutative monoid and that X is aΓ-graded E ∞ ring spectrum, as in Example 3.24. Then there are action mapsSym k X g → X kg . These give rise to Dyer–Lashof operations Q i : H ∗ X g → H ∗ + i ( X g ). Example 7.19. Suppose that · · · → X → X → X is a strongly filtered E ∞ ring spectrum, as in Example 3.25. Then there are action maps Sym k X n → X kn that are compatible. These give rise to power operations Q i : H ∗ X n → H ∗ + i X n that are compatible as n varies, and there are induced power operations on theassociated spectral sequence. Example 7.20. Given a spectrum X indexed on a universe U as in Exam-ple 7.10, the extended powers are modeled by twisted half-smash products:Sym kU → U ( X ) ≃ E Σ k ⋉ Σ k ( X ∧ k )This recovers the machinery that was put to effective use in the 1970s and 1980sfor studying E ∞ ring spectra and H ∞ -ring spectra, before the development ofstrictly monoidal categories of spectra. ∞ -operads The point-set discussion of the previous sections provides a library of exam-ples. As the basis for a theory it relies on the existence of rigid models andpreservation of colimits. Example 7.21. Consider the fiberwise category of spaces over a fixed basespace B . This category has a symmetric monoidal fiber product X × B Y .The fiber product typically needs fibrant input to represent the homotopy fiberproduct; the fiber product typically does not produce cofibrant output. Thismakes it difficult to use the standard machinery to study algebras and modules inthis category. These problems have received significant attention in the settingof parametrized stable homotopy theory [62, 52, 53]. Example 7.22. The category of nonnegatively graded chain complexes over acommutative ring R is equivalent to the category of simplicial R -modules viathe Dold–Kan correspondence. This correspondence is lax symmetric monoidalin one direction, but only lax monoidal in the other. Moreover, while both sideshave morphism spaces, the Dold–Kan correspondence only preserves these upto weak equivalence, even for fibrant-cofibrant objects. Example 7.23. In the standard models of equivariant stable homotopy theorythe notion of strict G -commutativity is equivalent to one encoded by equivariantoperads rather than ordinary ones [57, 36, 16]. This means that an E ∞ -algebra43 (in the sense of an ordinary E ∞ operad) may not have a strictly commuta-tive model [66, 35], and this makes it more difficult to construct a symmetricmonoidal model for the category of A -modules.The framework of ∞ -operads [54] (or, alternatively, that of dendroidal sets[72]) is one method to express coherent multiplicative structures. Here are someof the salient points. • This generalization takes place in the theory of ∞ -categories (specificallyquasicategories), equivalent to the study of categories enriched in spaces.Every category enriched in spaces gives rise to an ∞ -category; every ∞ -category has morphism spaces between its objects. • In this framework, for ∞ -categories C and D there is a space Fun( C , D )encoding the structure of functors and natural equivalences. • In an ∞ -category, homotopy limits and colimits are intrinsic notions ratherthan arising from a particular construction. Many common constructionsproduce presentable ∞ -categories, which have all homotopy limits andcolimits. • Multicategories generalize to so-called ∞ -operads . These have an under-lying ∞ -category, and there are spaces of multimorphisms to an objectfrom a tuple of objects. Every topological multicategory gives rise to an ∞ -operad; every ∞ -operad can be realized by a topological multicate-gory. The precise definitions are similar in spirit to Segal’s encoding of E ∞ -spaces [86]. • An ∞ -operad O has an associated notion of an O -monoidal ∞ -category.An O -monoidal ∞ -category is expressed in terms of maps C → O of ∞ -operads with properties analogous to that from Definition 7.4, with themain difference that spaces of morphisms are respected. An O -monoidal ∞ -category is also equivalent to a functor from O to a category of cat-egories: each object x of O has an associated category C x , and one canassociate a mapMul O ( x , . . . , x d ; y ) → Fun( C x , . . . , C x d ; C y )of spaces. • We can discuss algebras and modules in terms of sections, just as in Defi-nition 7.12.All of this structure is systematically invariant under equivalence. Equivalent ∞ -operads give rise to equivalent notions of an O -algebra structure on C ; ∞ -categories equivalent to C have equivalent notions of O -algebra structures tothose on C ; equivalent O -monoidal ∞ -categories have equivalent categories of M -algebras for any map M → O of ∞ -operads.44 xample 7.24. An E n -operad has an associated ∞ -operad O , and as a resultwe can define an E n -monoidal ∞ -category C to be an O -monoidal ∞ -category.When n = 1, 2, or ∞ we can recover monoidal, braided monoidal, and symmetricmonoidal structures. Mandell’s theorem (3.7), which is about structure on the homotopy categoryof left modules over an E n -algebra, is a reflection of higher structure on thecategory of left modules itself. Theorem 7.25 ([54, 5.1.2.6, 5.1.2.8]) . Suppose that C is an E k -monoidal ∞ -category which has geometric realization of simplicial objects, and such that thetensor product preserves such geometric realizations in each variable separately.Then the category of left modules over an E k -algebra A is E k − -monoidal, andhas all colimits that exist in C . As previously discussed, the category of left modules over an associativealgebra R is not made monoidal under the tensor product over R , but thecategory of bimodules is. The generalization of this result to E n -algebras is thefollowing. Theorem 7.26 ([54, 3.4.4.2]) . Suppose C is an E n -monoidal presentable ∞ -category such that the monoidal structure preserves homotopy colimits in eachvariable separately. Then for any E n -algebra R in C , there is a category Mod E n R ( C ) of E n R -modules. This is a presentable E n -monoidal ∞ -category whose under-lying monoidal operation is the tensor product over R .In particular, if C is a presentable ∞ -category with a symmetric monoidalstructure that preserves colimits in each variable, and R is an E n -algebra in C ,the category of E n R -modules in C has an E n -monoidal structure that preservescolimits in each variable. Roughly, an E n R -module M has multiplication operations R ⊗ k ⊗ M → M parametrized by ( k + 1)-tuples of points of configuration space, where one pointis marked by M and the rest by R . This has the more precise description of E n -modules as left modules. Theorem 7.27 ([54, 5.5.4.16], [29]) . Suppose that C is a symmetric monoidal ∞ -category and that the monoidal product preserves colimits in each variableseparately. For an E n -algebra R in C , the factorization homology R D n \ R hasthe structure of an E -algebra, and the category of E n R -modules is equivalentto the category of left modules over R D n \ R . Remark 7.28. In the category of spectra, this could be regarded as a con-sequence of the Schwede–Shipley theorem [84] or its generalizations. Thereis a free-forgetful adjunction between E n R -modules and Sp , and the imageFree E n - R ( S ) of the sphere spectrum under the left adjoint is a compact genera-tor for the category of E n R -modules. Therefore, E n R -modules are equivalent45o the category of modules over the endomorphism ring F E n - R (Free E n - R ( S ) , Free E n - R ( S )) ≃ Free E n - R ( S ) . This theorem, then, is an identification of the free E n R -module. Example 7.29. When n = 1, the category of E R -modules is the category ofleft modules over R ⊗ R op . When n = 2, the category of E - R -modules is thecategory of left modules over the topological Hochschild homology THH( R ). In the classical case, we described an O -algebra structure on A in terms of actionmaps Sym k O ( A ) = O ( k ) ⊗ Σ k A ⊗ k → A from extended power constructions to A , and gave a formulaFree O ( X ) = a k ≥ Sym k O ( A )for the free O -algebra on an object in the case where the monoidal structure iscompatible with enriched colimits; we also discussed the multi-object analoguein § ∞ -operads are carried out in [54, § ∞ -operad O . For any objects x , . . . , x d , y of O , we can construct aspace Mul O ( x , . . . , x d ; y )of multimorphisms in O ; if the x i are equal, this further can be given a naturalaction of the symmetric group.Let C be an O -monoidal ∞ -category C . In particular, C encodes categories C x parametrized by the objects x of O , and functors f : C x × · · · × C x d → C y parametrized by the multimorphisms f : ( x , . . . , x d ) → y of O . Suppose thatthe categories C x have homotopy colimits and the functors preserve homotopycolimits in each variable. Then there exist extended power functors Sym k O , x → y : C x → C y , whose value on X ∈ C u is a homotopy colimit (cid:18) hocolim α ∈ Mul O ( x ,..., x ; y ) α ( X ⊕ · · · ⊕ X ) (cid:19) h Σ k . These extended powers have the property that an O -algebra A has naturalmaps Sym k O , x → y ( A ( x )) → A ( y ). Moreover, there is a free-forgetful adjunctionbetween O -algebras and C x , and the free object Free O , x ( X ) on X ∈ C U has theproperty that its value on y is exhibited as the coproductev y (Free O , x ( X )) ≃ a k ≥ Sym k O , x → y ( X ) . emark 7.30. Composing with the diagonal C x → Q C x gives a Σ k -equivariantmap Mul O ( x , . . . , x | {z } k ; y ) → Fun( C x × · · · × C x , C y ) → Fun( C x , C y )that factors through the homotopy orbit space P ( k ) = Mul O ( x , . . . , x ; y ) h Σ k . This space P ( k ) then serves as a parameter space for tensor-power functors C x → C y .In the case of an ordinary single-object ∞ -operad O such as an E n -operad,we can rephrase in terms of P ( k ). Such an ∞ -operad O is equivalent to anordinary operad in spaces and an O -monoidal ∞ -category is equivalent to an ∞ -category C with a map O → End( C ). We recover a formulaFree O ( X ) ≃ a k ≥ hocolim α ∈ P ( k ) α ( X, . . . , X )for the free algebra on X . When X = S m , this is the Thom spectrum a k ≥ P ( k ) mρ , closely related to Remark 4.19.When O is an E n -operad, the space P ( k ) is equivalent to the space C n ( k ) / Σ k ,a model for the space of unordered configurations of k points in R n . When n = ∞ the space P ( k ) is a model for B Σ k , and we find that the we recover theordinary homotopy symmetric power:Sym kE ∞ ( X ) ≃ ( X ⊗ k ) h Σ k . Example 7.31. Fix a space B and consider the fiberwise category S /B . Thehomotopy fiber product X × hB Y gives this the structure of a symmetric monoidal ∞ -category, breaking up independently over the components of B . If B is path-connected, then the extended power and free functors on ( X → B ) are thoseobtained by applying the extended power and free functors to the fiber. Example 7.32. Given an E n R -module M , the free E n R -algebra on an E n R -module M is a k ≥ hocolim α ∈ C n ( k ) / Σ k M ⊗ α k , where each point α of configuration space determines a functor M ⊗ α k ≃ M ⊗ R · · · ⊗ R M .More can be said under the identification between E n -modules and modulesover factorization homology. If M is the free E n R -module on S m , then weobtain an identification of the free E n -algebra under R on S m : R ∐ E n Free E n ( S m ) ≃ a k ≥ Z R k \{ p ,...,p k } R ! ⊗ Σ k S mρ k . emark 7.33. The interaction between connective objects and their Postnikovtruncations from § O -monoidal ∞ -category C with a compatible t -structure in the sense of [54, 2.2.1.3]. This meansthat the categories C x indexed by the objects x of O all have t -structures, andthe functors induced by the morphisms in O are all additive with respect toconnectivity. Then [54, 2.2.1.8] implies that connective O -algebras have Post-nikov towers: the collection of truncation functors τ ≤ n is compatible with the O -monoidal structure on C ≥ . Definition 8.1. For an E n -monoidal ∞ -category C with unit I , the Picardspace Pic( C ) is the full subgroupoid of C spanned by the invertible objects :objects X for which there exists an object Y such that Y ⊗ X ≃ X ⊗ Y ≃ I . Remark 8.2. The classical Picard group of the homotopy category h C is theset π Pic( C ) of path components.In particular, Pic( C ) is closed under the E n -monoidal structure on C , giv-ing it a canonical E n -space structure. Moreover, by construction π Pic( C ) =( π C ) × is a group, and so Pic( C ) is an n -fold loop space. The loop spaceΩ Pic( C ) is the space of homotopy self-equivalences of the unit I ; in the caseof the category LMod R of left modules, it is homotopy equivalent to the unitgroup GL ( R ) of R . Proposition 8.3 ([2, § . If R is an E n ring spectrum, then the space GL ( R ) of homotopy self-equivalences of the left module R has an n -fold delooping. If n ≥ , the space Pic( R ) = Pic( LMod R ) has an ( n − -fold delooping. Topological Andr´e-Quillen homology and cohomology are invariants of ringspectra developed by Kriz and Basterra [45, 8]. For a fixed map of E ∞ ringspectra A → B , we can define a topological Andr´e–Quillen homology objectTAQ( A → R → B ) for any object R in the category of E ∞ rings between A and B . This is characterized by the following properties [9]:1. It naturally takes values in the category of B -modules.2. It takes homotopy colimits of E ∞ ring spectra between A and B to ho-motopy colimits of B -modules.3. There is a natural map B ⊗ A ( R/A ) → TAQ( A → R → B ).4. For a left A -module X with a map X → B , the composite natural map B ⊗ A X → B ⊗ A Free AE ∞ ( X ) → TAQ( A → Free AE ∞ ( X ) → B )48f B -modules is an equivalence.5. Under the above equivalence, the natural mapTAQ( A → Free AE ∞ Free AE ∞ ( X ) → B ) → TAQ( A → Free AE ∞ ( X ) → B )is equivalent to the map B ⊗ A Free AE ∞ ( X ) → B ⊗ A X that collapses B ⊗ A ( ∐ Sym k ( X )) to the factor with k = 1.Topological Andr´e-Quillen homology measures how difficult it is to build R as an A -algebra: any description of R as an iterated pushout along maps of freeof E ∞ -algebras, starting from A , determines a description of the topologicalAndr´e–Quillen cohomology of R as an iterated pushout of B -modules. Basterrashowed that TAQ-cohomology groupsTAQ n ( R ; M ) = [TAQ( S → R → R ) , Σ n M ] Mod R plays the role for Postnikov towers of E ∞ ring spectra that ordinary cohomologydoes for spectra.From this point of view, TAQ also has natural generalizations to TAQ O foralgebras over an arbitrary operad [9, 34], although there may be a choice oftarget category that takes more work to describe. In particular, for E n -algebrasthese are related to an iterated bar construction [10].Topological Andr´e–Quillen homology also enjoys the following properties,proved in [8, 9]. Base-change: For a map B → C , we have a natural equivalence C ⊗ B TAQ( A → R → B ) ≃ TAQ( A → R → C ) . In particular, if we define Ω R/A = TAQ( A → R → R ), thenTAQ( A → R → B ) = B ⊗ R Ω R/A . Transitivity: For a composite A → R → S → B , there is a naturalcofiber sequenceTAQ( A → R → B ) → TAQ( A → S → B ) → TAQ( R → S → B ) . In particular, for A → R → S we have cofiber sequences S ⊗ R Ω R/A → Ω S/A → Ω S/R . Representability: Suppose that there is a functor h ∗ from the cat-egory of pairs ( R → S ) of E ∞ ring spectra between A and B tothe category of graded abelian groups. Suppose that this is acohomology theory on the category of E ∞ ring spectra between49 and B : it satisfies homotopy invariance, has a long exact se-quence, satisfies excision for homotopy pushouts of pairs, andtakes coproducts to products. Then there is a B -module M with a natural isomorphism h n ( S, R ) ∼ = TAQ n ( S, R ; M )= [TAQ( R → S → B ) , Σ n M ] Mod B of abelian groups.For any E ∞ ring spectrum B , algebras mapping to B have TAQ-homologyTAQ( S → R → B ), valued in the category of B -modules. The square-zeroalgebras B ⊕ M are representing objects for TAQ-cohomology TAQ ∗ ( R ; M ).Representability allows us to construct and classify operations in T AQ -cohomology by B -algebra maps between such square-zero extensions. Proposition 8.4. Any element in [Σ Sym M, N ] Mod B has a naturally associ-ated map B ⊕ M → B ⊕ N of augmented commutative B -algebras and hencegives rise to a natural TAQ -cohomology operation TAQ( − ; M ) → TAQ( − ; N ) for commutative algebras mapping to B .Proof. By viewing B as concentrated in grading 0 and M as concentrated ingrading 1, we can give a Z -graded construction (as in Example 3.24) of B ⊕ M as an iterated sequence of pushouts along maps of free algebras. The first suchpushout is Free BE ∞ ( M ) ← Free BE ∞ (Sym M ) → B Further pushouts only alter gradings 3 and higher.We now view B ⊕ N as graded by putting N in grading 2. We find thathomotopy classes of maps of graded algebras B ⊕ M → B ⊕ N are equivalentto maps Σ Sym M → N . Example 8.5. Letting M = B ⊗ S m , we haveΣ Sym ( M ) ≃ B ⊗ Σ m +1 RP ∞ m . Therefore, we get a map from the B -cohomology B n (Σ m +1 RP ∞ m ) of stunted pro-jective spaces to the group of natural cohomology operations T AQ m ( − ; B ) → T AQ n ( − ; B ). Remark 8.6. The fact that elements in the B -homology of stunted projectivespaces produce homotopy operations while elements in their B -cohomology pro-duce TAQ-cohomology operations with a shift is a reflection of Koszul duality. Example 8.7. Letting M = ( B ⊗ S q ) ⊕ ( B ⊗ S r ), and using the projectionΣ Sym ( B ⊗ ( S q ⊕ S r )) ≃ Σ Sym ( B ⊗ S q ) ⊕ Σ Sym ( B ⊗ S r ) ⊕ Σ( B ⊗ S q ⊗ S r ) → B ⊗ S q +1+ r , 50e get a binary operation[ − , − ] : TAQ q ( − ; B ) × TAQ r ( − ; B ) → TAQ q +1+ r ( − ; B )that (up to a normalization factor) we call the TAQ -bracket . Example 8.8. If B = H F , then there are TAQ-cohomology operations R a : TAQ m ( − ; H F ) → TAQ m + a ( − ; H F )for a ≥ m + 1, and a bracketTAQ q ( − ; H F ) × TAQ r ( − ; H F ) → TAQ q +1+ r ( − ; H F ) . In this form, the operation R a +1 is Koszul dual to Q a , in the sense that non-trivial values of R a +1 in TAQ-cohomology detect relations on the operator Q a in homology. Similarly, the bracket in TAQ is Koszul dual to the multiplication.The operations were constructed by to Basterra–Mandell [11]. In furtherunpublished work, they showed that these operations (and their odd-primaryanalogues) generate all the natural operations on TAQ-cohomology with valuesin H F p and determined the relations between them. In particular, the opera-tions R a above satisfy the same Adem relations that the Steenrod operations Sq a do; the TAQ-bracket has the structure of a shifted restricted Lie bracket,whose restriction is the bottommost defined operation R a .Basterra–Mandell’s proof uses a variant of the Miller spectral sequence from[69]. We will close out this section with a sketch of how such spectral sequencesare constructed, parallel to the delooping spectral sequence from Remark 5.14. Proposition 8.9. Suppose that R is an E ∞ ring spectrum with a chosen map R → H F p . Then there is a Miller spectral sequence AQ DL ∗ ( π ∗ ( H F p ⊗ R )) ⇒ TAQ ∗ ( S → R → H F p ) , where the left-hand side are the nonabelian derived functors of an indecompos-ables functor Q that sends an augmented graded-commutative F p -algebra withDyer–Lashof operations to the quotient of the augmentation ideal by all productsand Dyer–Lashof operations.Proof. We construct an augmented simplicial object: · · · Free E ∞ Free E ∞ Free E ∞ R ⇛ Free E ∞ Free E ∞ R ⇒ Free E ∞ R → R. If U is the forgetful functor, from commutative ring spectra mapping to H F p tospectra mapping to H F p , this is the bar construction B (Free E ∞ , U Free E ∞ , U R ).The underlying simplicial spectrum B ( U Free E ∞ , U Free E ∞ , U R ) has an extradegeneracy, so its geometric realization is equivalent to R . Moreover, the for-getful functor from E ∞ rings to spectra preserves sifted homotopy colimits, andhence geometric realization because the simplicial indexing category is sifted.51herefore, applying the homotopy colimit preserving functor TAQ = TAQ( S → ( − ) → H F p ) and the natural equivalence TAQ ◦ Free E ∞ ( R ) ≃ H F p ⊗ R , we getan equivalence | B ( H F p ⊗ ( − ) , U Free E ∞ , U R ) | ≃ TAQ( R ) . However, this bar construction is a simplicial object of the form · · · H F p ⊗ Free E ∞ Free E ∞ R ⇛ H F p ⊗ Free E ∞ R ⇒ H F p ⊗ R. Taking homotopy groups, we get a simplicial object Q E ∞ Q E ∞ H ∗ R ⇛ Q E ∞ H ∗ R ⇒ H ∗ R. Moreover, the structure maps make this the bar construction B ( Q, Q E ∞ , H ∗ R )that computes derived functors of Q on graded-commutative algebras withDyer–Lashof operations. Therefore, the spectral sequence associated to thegeometric realization computes TAQ ∗ ( S → R → H F p ) and has the desired E -term. Remark 8.10. We can also apply cohomology rather than homology and geta spectral sequence computing topological Andr´e–Quillen cohomology.This leaves open a hard algebraic part of Basterra–Mandell’s work: actuallycalculating these derived functors, and in particular finding relations amongstthe operations R a and the bracket [ − , − ] that give a complete description ofTAQ-cohomology operations. We will close this paper with some problems that we think are useful directionsfor future investigation. Problem 9.1. Develop useful obstruction theories which can determine theexistence of or maps between E n -algebras in a wide variety of contexts. The obstruction theory due to Goerss–Hopkins [32] is the prototype for theseresults. In unpublished work [88], Senger has given a development of this theoryfor E ∞ -algebras where the obstructions occur in nonabelian Ext-groups calcu-lated in the category of graded-commutative rings with Dyer–Lashof operationsand Steenrod operations satisfying the Nishida relations, and provided tools forcalculating with them. This played a critical role in [48, 47].In closely related situations, the tools available remain rudimentary. Forexample, there is essentially no workable obstruction theory for the constructionof commutative rings of any type in equivariant stable theory. Tools arisingfrom the Steenrod algebra have been essential in most of the deep results inhomotopy theory, such as the Segal conjecture [51] and the Sullivan conjecture[70]. Without the analogues, there is a limit to how much structure can berevealed. 52 roblem 9.2. Give a modern redevelopment of homology operations for E ∞ ring spaces and E n ring spaces. The observant reader may have noticed that, despite the rich structurepresent, the principal material that we have referenced for E ∞ ring spaces isseveral decades old. Several major advances have happened in multiplicativestable homotopy theory since then, and the author feels that there is still agreat deal to be mined. Having this material accessible to modern toolkitswould be extremely useful.For one example, the theory of E ∞ ring spaces from the point of view of sym-metric spectra has been studied in detail by Sagave and Schlichtkrull [82, 80, 81].For another, the previous emphasis on E ∞ ring spaces should be tempered bythe variety of examples that we now know only admit A ∞ or E n ring structures. Problem 9.3. Give a unified theory of graded Hopf algebras and Hopf rings,capable of encoding some combination of non-integer gradings, power operations,group-completion theorems, and the interaction with the unit. Ravenel–Wilson’s theory of Hopf rings is integer-graded. We now know manyexamples—motivic homotopy theory, equivariant homotopy theory, K ( n )-localtheory, modules over E n ring spectra—that may have natural gradings of amuch wider variety than this, such as a Picard group. Moreover, multiplicativetheory should involve much more structure: we should have a sequence of spacesgraded not just by a Picard group, but by the Picard space that also encodesstructure nontrivial higher interaction between gradings and the unit group. Problem 9.4. Give a precise general description of the Koszul duality rela-tionship between homotopy operations and TAQ -cohomology operations. Givea complete construction of the algebra of operations on TAQ -cohomology for E n -algebras with coefficients in Hk , for k a commutative ring. Give completedescriptions of the TAQ -cohomology for a large library of Eilenberg–Mac Lanespectra Hk and Morava’s forms of K -theory. Because TAQ-cohomology governs the construction of ring spectra via theirPostnikov tower, essentially any information that we can provide about theseobjects is extremely useful. Problem 9.5. Determine an algebro-geometric expression for power operationsand their relationship to the Steenrod operations. Do the same for the operationswhich appear in the Hopf ring associated to an E ∞ ring space. At the prime 2, it has been known for some time that the action of theSteenrod algebra can be concisely packaged as a coaction of the dual Steenrodalgebra, a Hopf algebra corresponding to the group scheme of automorphismsof the additive formal group over F . The Dyer–Lashof operations on infiniteloop spaces generate an algebra analogous to the Steenrod algebra, and its dualwas described by Madsen [55]; the result is closely related to Dickson invari-ants. However, the full action of the Dyer–Lashof operations or the interactionbetween the Dyer–Lashof algebra and the Steenrod algebra does not yet have ageometric packaging. 53 onjecture 9.6. For Lubin–Tate cohomology theories E and F of height n ,there is a natural algebraic structure parametrizing operations from continuous E -homology to continuous F -homology for certain E ∞ ring spectra, expressedin terms of the algebraic geometry of isogenies of formal groups.This is complete: there is an obstruction theory for the construction of andmapping between K ( n ) -local E ∞ ring spectra whose algebraic input is completed E -homology equipped with these operations. In this paper we have not really touched on the extensive study of poweroperations in chromatic homotopy theory (cf. [90, 7]). Given Lubin–Tate coho-mology theories E and F associated to formal groups of height n at the prime p , we have both cohomology operations and power operations. In [38] the alge-bra of cohomology operations is expressed in terms of isomorphisms of formalgroups. Extensive work of Ando, Strickland, and Rezk has shown that poweroperations are expressed in terms of quotient operations for subgroups of theformal groups. It has been known for multiple decades [93, § 28] that the natu-ral home combining these two types of operations is the theory of isogenies offormal groups. However, there are important details about formal topologieswhich have never been resolved. Problem 9.7. Determine the natural instability relations for operations in un-stable elliptic cohomology and in unstable Lubin–Tate cohomology. Strickland states that isogenies are a natural interpretation for unstable co-homology operations in E -theory. However, isogenies encode the analogue of thecohomological Steenrod operations, the multiplicative Dyer–Lashof operations,and the Nishida relations between them. They do not encode any analogue ofthe instability relation Sq n = Q − n that we see in the cohomology of spaces.In chromatic theory, our only accessible example so far is K -theory. For p -completed K -theory, the cohomology operations are generated by the Adamsoperations ψ k for k ∈ Z × p . For torsion-free algebras, the power operations arecontrolled by the operation ψ p and its congruences [37, 78]. The unstable opera-tions in the K -theory of spaces, by contrast, arise from the algebra of symmetricpolynomials and are essentially governed by the ψ n for n ∈ N ; the fact that theother ψ k are determined by these enforces some form of continuity. This is alsoclosely tied to the question of whether there are geometric interpretations ofsome type for elliptic cohomology theories or Lubin–Tate cohomology theories. Problem 9.8. Determine a useful way to encode secondary operation structureson E ∞ or E n rings. In the case of secondary Steenrod operations, there is a useful formulationdue to Baues of an extension of the Steenrod algebra that can be used to encodeall of the secondary operation structure [12, 73]. No such systematic descriptions The reader should be advised that, even at height 1, there are difficult issues with E -theoryhere involving left-derived functors of completion. One possible viewpoint is that we could interpret N as the monoid of endomorphisms ofthe multiplicative monoid M , which contains the unit group GL . Problem 9.9. Determine useful relationships between the homotopy types ofan E n ring spectrum, the unit group GL ( R ) and the Picard space Pic( R ) , andthe spaces BGL n ( R ) . This is closely tied to orientation theory, algebraic K -theory, and the studyof spaces involved in surgery theory.Investigations in these directions due to Mathew–Stojanoska revealed thatthere is a nontrivial relationship between the k -invariants for R and the unitspectrum gl ( R ) at the edge of the stable range at the prime 2 [60], and forth-coming work of Hess has shown that this relation can be recovered from themixed Cartan formula. The odd-primary analogues of this are not yet known. Problem 9.10. Find an odd-primary formula for the mixed Adem relationssimilar to the Kuhn–Tsuchiya formula. There is a description of the mixed Adem relations [24], valid at any prime,but it is difficult to apply in concrete examples. The 2-primary formula describedin § References [1] J. F. Adams. Stable homotopy and generalised homology . University ofChicago Press, Chicago, Ill., 1974. Chicago Lectures in Mathematics.[2] Matthew Ando, Andrew J. Blumberg, and David Gepner. 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