Recognition of commutative algebra spectra through an idempotent quasiadjunction
aa r X i v : . [ m a t h . A T ] J a n Recognition of commutative algebra spectrathrough an idempotent quasiadjunction
Renato Vasconcellos Vieira ∗ January 2021
Abstract
In this article a recognition principle for ∞ -loop pairs of spaces of con-nective commutative algebra spectra over connective commutative ringspectra is proved. This is done by generalizing the classical recognitionprinciple for connective commutative ring spectra using relative oper-ads. The machinery of idempotent quasiadjunctions is used to handlethe model theoretical aspects of the proof. The category Sp N of sequential prespectra [12] consists of sequences of spaces ⟨ Y N ⟩ ∈ ∏ N Top ∗ equipped with structural maps σ MN ∶ Y M ∧ S N − M → Y N for M ≤ N satisfying compatibility conditions. An Ω-spectrum is a prespectrumwhose adjoint structural maps ˜ σ MN are weak equivalences, which by Brown rep-resentability represents (co)homology theories [3]. Spectra are prespectra suchthat the ˜ σ NM are homeomorphisms (see for instance [5]). In this article we willwork exclusively in the category of prespectra, so from now on we will simplyrefer to prespectra as spectra. From Sp N we can define via filtered colimits overthe dual structural maps the ∞ -loop spaces functorΩ ∞ ∶ Sp N → Top ∗ ; Ω ∞ Y ∶ = colim N Y S N N . The ∞ -loop spaces Ω ∞ Y are homotopy commutative H -spaces, but suchdescription ignores a lot of information. In order to describe the algebraicstructure completely we require an E ∞ -operad E , a gadget used to describetopological spaces with operations that are associative and commutative up tocoherent homotopy [16]. ∗ This work was financed by the grant 2020/06159-5, S˜ao Paulo Research Foundation(FAPESP) S the category of finite sets a topological operad is a contravariant functorequipped with an abstract identity element and composition maps P ∶ S op → Top ; id ∈ P , ◦ ∶ P A × ∏ A P B a → P Σ A B a with P ∅ = ∗ satisfying invariance, associativity and unitary laws. We caninterpret points in the underlying spaces as abstract multivariable functionswith inputs indexed by the sets A . This structures allows us to define via thecoend construction [13] the monad P ∶ Top ∗ → Top ∗ , P X ∶ = ∫ S inj P A × X × A ; ηx ∶ = [ id , x ] , µ [ α, ⟨[ β a , ⟨ x a,b ⟩]⟩] ∶ = [ α ◦ ⟨ β a ⟩ , ⟨ x a,b ⟩] . The category P [ Top ] of P -spaces consists of pointed spaces X ∈ Top ∗ equipped with maps ξ ∶ P X → X compatible with the monad maps, whichwe interpret as an instantiation of the abstract operations of P .An important family of operads are the embeddings operads Emb N for N ∈ N with Emb N A ∶ = { α = ⟨ α a ⟩ ∈ ( R N ) ⊔ A R N ∣ α an embedding } . There are natural operad inclusions End M ↪ End N and we define End ∞ ∶ = colim N End N . All N -loop spaces are naturally End N -spaces with α ⟨ γ a ⟩ ∶ = ( ⃗ u ↦ { γ a α − a ⃗ u, ⃗ u ∈ α a R N ∗ , ⃗ u / ∈ α ⊔ A R N ) and these induce End ∞ -space structures on ∞ -loop spaces.An E ∞ -operad is an operad E with each underlying space E A a contractiblefree S A -space. For the purpose of studying ∞ -loop spaces we further require E ∞ -operads to be equipped with an operad map ψ ∶ E → End ∞ . This allowsus to define by pullback an E -algebra structure on ∞ -loop spaces Ω ∞ Y , whichinduces a functor Ω ∞ ∶ Sp → E [ Top ] . This functor is not a right adjoint sinceany abelian group G is an E -spaces and it can be shown that due to the strictnessof the operations in G any E -map ϕ ∶ G → Ω ∞ Y must be trivial, so no unit ofadjunction can be constructed.In May’s recognition theorem [16] the solution was to consider the resolutionof E -spaces by the bar construction B ∶ E [ Top ] → E [ Top ] ; BX ∶ = B ( E, E, X ) , which comes equipped with a natural weak equivalence η ′ ∶ B ⇒ Id .The maps ψ ∶ E → End ∞ induce by pullback a suboperad filtration E N on E . If each underlying space E N A is equivariantly homotopy equivalent to the2onfiguration space of A elements in R N then we can define the ∞ -deloopingfunctor B ∞ ∶ E [ Top ] → Sp ; B ∞ X ∶ = ⟨ B ( Σ N , E N , X )⟩ ;such that there is a natural transformation η ∶ B ⇒ Ω ∞ B ∞ , with η X a weakequivalence if and only if X is grouplike, meaning that π X is not only a monoidbut also a group.Dually there is no counit map. There is a spectrification functor ̃ Ω ∶ Sp → Sp ; ̃ Ω Y ∶ = ⟨ colim M ≤ N ̃ Y S N − M N ⟩ , where ̃ Y is a certain inclusion prespectrum constructed from Y , such that wehave natural inclusions ǫ ′ ∶ Id ⇒ ̃ Ω which are stable weak equivalences. Thisfunctor plays an important role in the construction of the stable model struc-tures of spectra. There is a natural transformation ǫ ∶ B ∞ Ω ∞ ⇒ ̃ Ω such thatthe equation Ω ∞ ǫη Ω ∞ = Ω ∞ ǫ ′ η ′ Ω ∞ holds in E [ Top ] and we have a homotopyequivalence ǫ B ∞ B ∞ η X ≃ ǫ ′ B ∞ X B ∞ η ′ X in Sp . B ∞ BX B ∞ η ′ X ∼ (cid:15) (cid:15) B ∞ η X / / B ∞ Ω ∞ B ∞ X ǫ B ∞ X (cid:15) (cid:15) B ∞ X ∼ ǫ ′ B ∞ X / / ̃ Ω B ∞ X B Ω ∞ Y ∼ η ′ Ω ∞ Y (cid:15) (cid:15) η Ω ∞ Y / / Ω ∞ B ∞ Ω ∞ Y Ω ∞ ǫ Y (cid:15) (cid:15) Ω ∞ Y ∼ Ω ∞ ǫ ′ Y / / Ω ∞ ̃ Ω Y Note the similarity of these equations to the ones for an adjunction. Indeed if B , ̃ Ω, η ′ and ǫ ′ were substituted by identities and both equations held strictly wewould have an adjunction in the regular sense. In [25] I defined a generalizationof Quillen adjunctions, called weak Quillen quasiadjunctions, that allowed forunits and counits to exist up to functorial resolutions. I proved that weakQuillen quasiadjunctions still induce adjunctions of the homotopy categories,generalizing the analogous result for Quillen adjunctions. In the same veinI defined a generalization of Quillen idempotent (co)monads that induce left(right) Bousfield localizations of model structures, and through these we havea natural definition of idempotent quasiadjunctions which induce equivalencesbetween the associated homotopy subcategories.Adapting May’s original proof of the recognition principle in [16, 14] we canshow that the weak Quillen quasiadjunction ( B ∞ ⊣ B, ̃ Ω Ω ∞ ) ∶ E [ Top ] ⇋ Sp N is idempotent and induces an equivalence between the homotopy category ofgrouplike E -spaces and the the homotopy category of connective spectra. Idem-potent quasiadjunctions provide a model categorical axiomatization of the essen-tial elements of May’s original proof, and it can be adapted to prove variationsof the recognition principle. For instance the relative recognition principle for The spectrification functor ̃ Ω is the left adjoint to the inclusion of the category of spectrain the sense used in [5] into the category of prespectra, hence the name. -loop pairs of spaces of spectra maps of degree 1 was proved using the abovemachinery and relative operads in [25].In this article I show that the machinery of quasiadjunctions and relativeoperads are also compatible with actions by a natural relative version L (cid:1) of thelinear isometries operad L , introducing in particular a relative version of actionsbetween operads which provides a natural definition of E ∞ -algebra spaces over E ∞ -ring spaces. The main theorem 4.5 is a recognition principle for ∞ -looppairs of spaces of commutative algebra spectra over commutative ring spectra.Explicitly it states that the homotopy category of algebralike E (cid:1) ∞ -algebras isequivalent to the homotopy category of connective commutative algebra spectraover connective commutative ring spectra. As in [5] this will require us to workon the more structured category Mod S of S -modules, which is monoidal and soprovides a convenient language to describe algebraic structures. In particular wewill work with the coordinate-free spectra of [11] which substitutes the naturalnumbers N by the set of finitely dimensional subspaces of some countably infinitedimensional inner product spaces such as R ∞ as the indexing set of spectra. Thisresult is a simple consequence of the intermediary theorems 4.2 and 4.3, whichis a recognition principle for ∞ -loop pairs of spaces of spectra maps. In section 2 we review the definition of weak Quillen quasiadjunctions, idempo-tent quasimonads and idempotent quasiadjunctions. Our main theorem will bea particular case of the fact that idempotent quasiajunctions induce equivalencesbetween the associated homotopy subcategories.In section 3 we present the definition of E ∞ -algebras over E ∞ -rings throughrelative operads. A detailed description of relative sets and filtered rooted rela-tive trees and operations on them will be required to construct bar resolutionsand delooping spectra, as well as describe their algebraic structures. We thengive a brief review on relative operads and E (cid:1) ∞ -operads and the bar resolutionof E (cid:1) ∞ -algebras. Relative operad actions are then introduced which providesan account of distributivity laws between multiplicative and additive relativeoperad actions and is central in the definition of the category ( E (cid:1) , L (cid:1) )[ Top ] of E ∞ -algebras. We also give a brief review of how the Quillen model structureon ( E (cid:1) , L (cid:1) )[ Top ] is transfered from the one on Top ∗ .The main theorems are in section 4. We review the basics of coordinate-free spectra and the construction of the stable mixed model structure. Therecognition principle for ∞ -loop pairs of spaces of spectra maps is proved via anidempotent quasiadjunction in theorems 4.2 and 4.3, which imply the homotopycategory of grouplike E (cid:1) ∞ -pairs is equivalent to the homotopy category of spectramaps between connective spectra. After a review of the basics of S -modules andcommutative algebra spectra, including the construction of stable mixed modelstructures, the main theorem 4.5 is proved. An E (cid:1) ∞ -algebra is algebralike if it admits additive inverses up to homotopy. .2 Notation and terminology We assume the theory of model categories in [7, 8, 10], and the theory ofmonoids, their algebras and the bar construction in [16, Section 9]. In dia-grams in a model category T the morphisms in the class of weak equivalences W are denoted by arrows marked with a tilde ∼ − → , the ones in the class of cofibra-tions C by hooked arrows ↪ and the ones in the class of fibrations F by doubleheaded arrows ↠ . The functorial weak factorization systems are denoted by ( Fat
C,F t , C − , F t − ) and ( Fat C t ,F , C t − , F − ) such that a morphism f ∈ T ( X, Y ) is factored for instance as X (cid:31) (cid:127) Cf / / Fat
C,F t f F t f ∼ / / / / Y .The notations C ∶ T → T and cof ∶ C ⇒ Id are used for the cofibrantresolution functor and the associated natural trivial fibration, and the notations F ∶ T → T and fib ∶ Id ⇒ F are used for the fibrant resolution functor and theassociated natural trivial cofibration. The homotopy category of T with objectsthe bifibrant objects of T and with morphisms between bifibrant objects X and Y the set T ( X, Y )/ ≃ of homotopy classes of maps [10, Section 1.2] is denotedby H o T .The monoidal category Top of compactly generated weakly Hausdorff spacesas presented in Strickland’s [23] admits two model structures, the cofibrantlygenerated Quillen model structure [17] with weak equivalences the weak ho-motopy equivalences ( q -equivalences), fibrations of this model structure theSerre fibrations ( q -fibrations), and cofibrations retracts of inclusions of wellpointed relative CW-complexes ( q -cofibrations), and the Hurewicz/Strøm modelstructure [24] with distinguished classes of maps the homotopy equivalences ( h -equivalences), the Hurewicz fibrations ( h -fibrations) and the Hurewicz cofibra-tions ( h -cofibrations). As Cole proved in [4] we can mix these model structuresinto one with weak equivalences the q -equivalences, fibrations the h -fibrationsand cofibrations the maps that can be factored as a q -cofibration followed byan h -equivalence. We use the notation K ⊂ cpct X to indicate K is a compactsubspace of X . We denote by I the interval [ , ] ⊂ R . We denote by Top ∗ the category of pointed spaces and for X ∈ Top we denote by X + ∈ Top ∗ thepointed space obtained by adjoining a disjoint base point.We denote by T (cid:1) the category of morphisms f ∶ X d → X c in T as objectsand commutative squares as morphisms. For notational convenience we denoteelements of categories of pairs T as X = ( X d , X c ) , and we will consider relativeoperads colored on the set { d, c } , with d being the “domain” color and c the“codomain” color.Let I denote the topological category of finite or countably infinite di-mensional real inner product spaces and linear isometries, with the topologydefined as the colimit of the finite dimensional sub-spaces. This category ismonoidal under direct sums. For U ∈ I we denote by A U the set of finitedimensional subspaces of U , partially ordered by inclusion, and for U ∈ A U wedefine A U ∶ = { V ∈ A U ∣ U ≤ V } . For U = R ∞ we simply write A ∶ = A R ∞ .For ⟨ f a ⟩ ∈ I ( ⊕ A U a , V ) and ⟨ ⃗ u a ⟩ ∈ ⊕ A U a we use the Einstein summationconvention f a ⃗ u a ∶ = ∑ A f a ⃗ u a . For U ∈ I and U ⊂ U any subspace we use the5otation U ⊥ ∶ = { ⃗ v ∈ U ∣ ∀ ⃗ u ∈ U ∶ ⃗ v ⋅ ⃗ u = } for the orthogonal complement.For any ⃗ u ∈ U and U ⊂ U we use the notation ⃗ u U to denote the projection of ⃗ u on U . For ⃗ v ∈ V and f ∈ I ( U , V ) we use the notation ⃗ v f ∶ = ⃗ v f U ∈ V forthe projection of ⃗ v onto the image of f and ⃗ u f ∶ = f − ⃗ v f ∈ U . For all U ∈ A U let S U be the one point compactification of U obtained by adding a point ∞ atinfinity and for ( U, V ) ∈ Σ A A U let V − U ∶ = U ∩ V ⊥ .We will make extensive use of mapping spaces Y X and will express theirelements as x ↦ Φ for some expression Φ which may use the variable x . For X a set (or space) equipped with an equivalence relation ∼ we will denote theequivalence classes of x ∈ X using square brackets [ x ] ∈ X / ∼ .We denote by Set the category of sets and functions, by S inj the subcategoryof finite sets and injections and by S the subcategory of finite sets and bijections.We will use the notation m for the sets { , . . . , m } , with 0 = ∅ .Given a class A and a family of classes ⟨ B a ⟩ indexed by A the dependentsum Σ A B a is the class of pairs ( a, b ) with a ∈ A and b ∈ B a and the dependentproduct Π A B a is the class of sequences ⟨ b a ⟩ indexed on A with b a ∈ B a foreach a ∈ A , or equivalently it is the class of sections of the natural surjectionΣ A B a → A .We denote by POSet the category of ordered sets and monotone functionsand ∆ the full subcategory on ⟨ m ⟩ = ⟨ < ⋯ < m ⟩ for m ∈ N . This categoryis generated by the coface injections ∂ i ∶ ⟨ m − ⟩ → ⟨ m ⟩ , with i / ∈ ∂ i ⟨ m − ⟩ ,and codegeneracy surjections δ i ∶ ⟨ m + ⟩ → ⟨ m ⟩ , with δ i i = δ i ( i + ) , for all i ∈ ⟨ m ⟩ .Consider the cosimplicial space of partitions of the interval Part − ∈ Top ∆ with Part ⟨ m ⟩ ∶ = POSet (⟨ m − ⟩ , I ) ; ∂ i ⋅ t ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩⟨{ , j = t j − , j > ⟩ , i = ⟨{ t j , j < it j − , j ≤ i ⟩ , < i < m ⟨{ t j , j < m − , j = m − ⟩ , i = m , δ i ⋅ t ∶ = ⟨{ t j , j < it j + , j ≥ i ⟩ , with Part ⟨ m ⟩ topologized as a subspace of I ⟨ m − ⟩ . For each ⟨ t a ⟩ ∈ Π A Part ⟨ m a ⟩ there is a unique ⊲ A t a ∈ Part ⟨ ∑ A m a ⟩ obtained by ordering the elements t a,i for a ∈ A and i ∈ ⟨ m a − ⟩ . For each a ∈ A and ⟨ t a ′ ⟩ ∈ Π A ∆ ⟨ m a ′ ⟩ we can define δ a ∈ ∆ (⟨ ∑ A m a ′ ⟩ , ⟨ m a ⟩) , δ a i ∶ = { min ( j ∣ ( ⊲ A t a ′ ) i ≤ t a,j ) , ( ⊲ A t a ′ ) i ≤ t a,m a − m a , ( ⊲ A t a ′ ) i > t a,m a − δ a ⋅ ⊲ A t a ′ = t a .For any simplicial space X − ∈ Top ∆ op its geometric realization ∣ X − ∣ is de-fined via the coend construction [13] as ∣ X − ∣ ∶ = ∫ ∆ X ⟨ m ⟩ × Part ⟨ m ⟩ . The reason we consider the geometric realization via the partitions cosimpli-cial space instead of the usual homeomorphic cosimplicial space of topologicalsimplexes is that this choice simplifies the algorithm in [16, Theorem 11.5].
The following definition introduced in [25] is a generalization of Quillen adjunc-tions between model categories. The basic idea is that to construct the unitand counit natural transformations of an adjuction between the homotopy cat-egories it suffices to construct a unit natural span and counit natural cospan atthe model categories level, plus some natural compatibility conditions with themodel structures.
Definition 2.1.
Let T and A be model categories. A weak Quillen quasiad-junction , or just quasiadjunction , between T and A , denoted by ( S ⊣ C , F Λ ) ∶ T ⇌ A , is a quadruple of functors T S / / C & & A Λ o o F f f with S the left quasiadjoint and Λ the right quasiadjoint , equipped with a nat-ural span in T and a natural cospan in A Id T C η ′ ∼ k s η + Λ S S Λ ǫ + F Id A ǫ ′ ∼ k s such that(i) S is left derivable;(ii) Λ is right derivable;(iii) C and F preserve cofibrant and fibrant objects;(iv) η ′ and ǫ ′ are natural weak equivalences;(v) If X ∈ T is cofibrant then ǫ SX Sη X ≃ ǫ ′ SX Sη ′ X ;7vi) If Y ∈ A is fibrant then Λ ǫ Y η Λ Y ≃ Λ ǫ ′ Y η ′ Λ Y . S C X Sη ′ X ∼ (cid:15) (cid:15) Sη X / / S Λ SX ǫ SX (cid:15) (cid:15) SX ∼ ǫ ′ SX / / F SX C Λ Y ∼ η ′ Λ Y (cid:15) (cid:15) η Λ Y / / Λ S Λ Y Λ ǫ Y (cid:15) (cid:15) Λ Y ∼ Λ ǫ ′ Y / / Λ F Y Theorem 2.2 ([25, Theorem 2.1.2]) . A quasiadjunction induces an adjunction ( L S ⊣ R Λ ) ∶ H o T ⇌ H o A ; Id H o T [ cof η ′ C ] − + L C [( Λ fib S η ) C ] + R Λ L S , L S R Λ [( ǫS cof Λ ) F ] + R F [ ǫ ′ F fib ] − + Id H o A between the homotopy categories. The following generalization of idempotent Quillen monads [2] was also intro-duced following the same principle of only requiring the existence of a unitnatural span, and they also induce Bousfield localizations.
Definition 2.3.
Let T be a right proper model category with distinguishedsubclassses of morphisms ( W, C, F ) . A Quillen idempotent quasimonad on T ,or simply an idempotent quasimonad , is a pair of endofunctors Q, C ∶ T → T equipped with a natural span Id T C η ′ ∼ k s η + Q such that:(i) Q preserves weak equivalences;(ii) Qη and η Q are natural weak equivalences;(iii) If f ∈ T ( X, B ) , p ∈ F ( E, B ) and η E , η B , Qf ∈ W then Q ( f ∗ p ) ∈ W ; X × B E p ∗ f (cid:15) (cid:15) f ∗ p / / E p (cid:15) (cid:15) (cid:15) (cid:15) C E C p (cid:15) (cid:15) η E ∼ / / η ′ E ∼ o o QE Qp (cid:15) (cid:15) Q ( X × B E ) Q ( p ∗ f ) (cid:15) (cid:15) Q ( f ∗ p ) ∼ o o X f / / B C B η ′ B ∼ o o η B ∼ / / QB QX Qf ∼ o o (iv) η ′ is a natural weak equivalence; 8v) If ι ∈ C ( C X, K ) then ι ∗ η ′ ∈ W . C X (cid:127) _ ι (cid:15) (cid:15) η ′ ∼ / / X η ′∗ ι (cid:15) (cid:15) K ∼ ι ∗ η ′ / / K ⊔ C X X Theorem 2.4 ([25, Theorems 2.3.5 and 2.3.6]) . An idempotent quasimonadinduces a left Bousfield localization T Q = ( T ; W Q ∶ = Q − W, C Q ∶ = C, F Q ∶ = { p ∈ F ∣( . ) a homotopy pullback }) E p (cid:15) (cid:15) (cid:15) (cid:15) i E / / E ⊔ C E QE ( p,Qp ) (cid:15) (cid:15) B i B / / B ⊔ C B QB (2.1) The resulting homotopy category is the reflective subcategory H o T Q ∶ = { X ∈ H o T ∣ ( i X ∶ X → X ⊔ C X QX ) ∈ W } of Q -fibrant objects. The above definition can be dualized and the resulting idempotent quasi-comonads induce right Bousfield localizations and associated coreflective homo-topy subcategories.
A quasiadjunction ( S ⊣ C , F Λ ) ∶ T ⇌ A induces the following natural span on T and natural cospan on A : Id T C C cof η ′ C ∼ k s ( Λfib S η ) C + Λ F S C S C Λ F ( ǫS cof Λ ) F + F F Id A ǫ ′ F fib ∼ k s Definition 2.5. An idempotent quasiadjunction is a quasiadjunction such thatthe induced span and cospan are respectively an idempotent quasimonad andan idempotent quasicomonad. Theorem 2.6 ([25, Theorem 2.3.8]) . An idempotent quasiadjunction ( S ⊣ C , F Λ ) ∶ T ⇌ A induces an equivalence between the associated (co)reflective homo-topy subcategories. H o T L Id / / ⊥ H o T Λ F S C L S / / ⊥ _? R Id o o H o A S C Λ F (cid:31) (cid:127) L Id / / ⊥ R Λ o o H o A R Id o o E (cid:1) ∞ -algebras Relative operads are abstract operations with entries indexed by relative sets.We now give the basic definitions and constructions on these colored sets. Wewill also require filtered rooted relative trees in the construction of the barresolutions and delooping spectra, and we provide here the relevant definitionsand constructions.Let
Set { d,c } be the category of relative sets composed of sets equipped witha coloring on the colors { d, c } , ie the class of objects {( A, c ) ∈ Σ Set
Set ({ A } ⊔ A, { d, c }) ∣ c A = d ⟹ ∀ a ∈ A ∶ c a = d } , with ( A, c ) usually being denoted simply as A or explicitly as a set of elementsin brackets with coloring given by subscripts, eg { d , d , c , d , c } c . The mor-phisms sets are Set { d,c } ( A, A ′ ) ∶ = {{ σ ∈ Set ( A, A ′ ) ∣ c a = c ⟹ c ′ σa = c } , c A = d or c ′ A ′ = c ∅ , c A = c and c ′ A ′ = d For ⋆ ∈ { d, c } we denote by Set ⋆ ⊂ Set { d,c } the full subcategory of relativesets A such that c A = ⋆ .Given (( A, c ) , ⟨( B a , c a )⟩) ∈ Σ Set { d,c } Π A Set c a we have the dependent sum ( Σ A B a , Σ A c a ) ∈ Set { d,c } , Σ A c a ( Σ A B a ) = c A, Σ A c a ( a, b ) = c a b. For σ ∈ Set { d,c } ( A, A ′ ) let σ ( B a ) ∈ Set { d,c } ( Σ A B a , Σ A ′ B σ − a ′ ) ; σ ( B a )( a ′ , b ) ∶ = ( σa ′ , b ) and for ⟨ τ a ⟩ ∈ Π A Set c ( a ) ( B a , B ′ a ) letΣ A τ a ∈ Set { d,c } ( Σ A B a , Σ A B ′ a ) ; Σ A τ a ( a ′ , b ) ∶ = ( a ′ , τ a ′ b ) . We also have the dependent productΠ A B a ∈ Set { d,c } , Π A c a Π A B a = c A, Π A c a ⟨ b a ⟩ = { d, ∀ a ∈ A ∶ c a b a = d ; c, ∃ a ∈ A ∶ c a b a = c. For σ ∈ Set { d,c } ( A, A ′ ) let σ ⟨ B a ⟩ ∈ Set { d,c } ( Π A B a , Π A ′ B σ − a ′ ) ; σ ⟨ B a ⟩⟨ b a ⟩ ∶ = ⟨ b σ − a ′ ⟩ , and for ⟨ τ a ⟩ ∈ Π A Set c a ( B a , B ′ a ) letΠ A τ a ∈ Set { d,c } ( Π A B a , Π A B ′ a ) ; Π A τ a ⟨ b a ⟩ ∶ = ⟨ τ a b a ⟩ . ⟨ b a ⟩ ∈ Π A B a we can form a new relative set A ⟨ b a ⟩ composed ofthe pairs ( a, b a ) with coloring c ⟨ b a ⟩ A ⟨ b a ⟩ = Π A c a ⟨ b a ⟩ and c ⟨ b a ⟩ ( a, b a ) = c a b a .This relative set is naturally equipped with π ⟨ b a ⟩ ∈ Set { d,c } ( A ⟨ b a ⟩ , A ) with π ⟨ b a ⟩ ( a, b a ) = a . Let ν ∈ Set { d,c } ( Π A Σ B a C a,b , Σ Π A B a Π A ( ba ) C a,b a ) ; ν ⟨( b a , c a )⟩ ∶ = (⟨ b a ⟩ , ⟨ c a ⟩) . This is a key element in distributivity properties.Let S inj { d,c } ⊂ Set { d,c } be the subcategory of Set { d,c } composed of the finiterelative sets and the injective functions that preserve coloring, i.e. S inj { d,c } ( A, A ′ ) = {{ σ ∈ Set { d,c } ( A, A ′ ) ∣ σ is injective , c ′ σa = c a } , c A = c ′ A ′ ∅ , c A ≠ c ′ A ′ Let S { d,c } ⊂ S inj { d,c } be the subcategory with the same objects and bijections thatpreserve coloring as morphisms. For ⋆ ∈ { d, c } we denote by S inj ⋆ and S ⋆ thefull subcategories of S inj { d,c } and S { d,c } respectively composed of relative sets A such that c A = ⋆ . Define also the subcategory S ⟨ d < c ⟩ ⊂ Set { d,c } with objectsthe finite relative sets and with morphisms the bijections (that don’t necessarilypreserve coloring). Note that S { d,c } is a subcategory of S ⟨ d < c ⟩ .Many spaces of interest are built via the two sided bar construction formonads induced by operads, which can be described using filtered rooted relativetrees. Definition 3.1.
The simplicial category T { d,c } ∈ Cat ∆ op of filtered rooted rel-ative trees has as objects quintuples T = (⟨ V i ⟩ , ⟨ E i ⟩ , ⟨ s i ⟩ , ⟨ t i ⟩ , c ) ∈ T { d,c } ⟨ m ⟩ composed of- A sequence of nonempty finite sets ⟨ V i ⟩ ∈ S ⟨ m − ⟩ . We also set V − ∶ = { v r } , and call v r the root vertex of T . We set V ∶ = ⊔ ⟨ m − ⟩ V i and V ∗ ∶ = V − ⊔ V . For v ∈ V we will denote by ∣ v ∣ ∈ ⟨ m − ⟩ the element such that v ∈ V ∣ v ∣ .- A sequence of finite sets ⟨ E i ⟩ ∈ S ⟨ m ⟩ . We also set E − ∶ = { e r } , and call e r the root edge of T . We set E ∶ = E − ⊔ ( ⊔ ⟨ m ⟩ E i ) . The edges in E m arecalled the leaves of T .- A sequence of bijections ⟨ s i ⟩ ∈ Π ⟨ m − ⟩ S ( E i , V i ) , called the start of theedges. We sometimes omit the subscript and write simply se ∶ = s i e . Notethat the leaves don’t have a source.- A sequence of functions ⟨ t i ⟩ ∈ Π ⟨ m ⟩ S ( E i , V i − ) , the target of the edges.We sometimes omit the subscript and write simply te ∶ = t i e . Note thatthe root edge doesn’t have a target.11 A function c ∈ Set ( E, { d, c }) , the coloring of the edges, such that if te ′ = se and c e = d then c e ′ = d .We sometimes just write T = (⟨ V i ⟩ , ⟨ E i ⟩) and leave the mappings implicit.Morphisms σ ∈ T { d,c } ⟨ m ⟩( T, T ′ ) are pairs of sequences of bijections (⟨ σ iV ⟩ , ⟨ σ iE ⟩) ∈ Π ⟨ m − ⟩ S inj ( V i , V ′ ,i ) × Π ⟨ m ⟩ S inj ( E i , E ′ ,i ) that commute with the structural functions.The simplicial structural functors are defined on objects as: T ⋅ ∂ i ∶ = (⟨ V ∂ i j ⟩ , ⟨ E ∂ i j ⟩ , ⟨ s ∂ i j ⟩ , ⟨{ t ∂ i j , j ≠ it i s − i t i + , j = i. ⟩ , c ⋅ ∂ i ) T ⋅ δ i ∶ = (⟨ V δ i j ⟩ , ⟨ E δ i j ⟩ , ⟨ s δ i j ⟩ , ⟨{ t δ i j , j ≠ i + s i , j = i + . ⟩ , c ⋅ δ i ) with the coloring maps induced naturally from the ones in T .Define also T { d,c } ∈ Cat ∆ op as the full simplicial subcategory of relative treessuch that ∣ V ∣ =
1. Define also the simplicial full subcategories T ⋆ ⊂ T { d,c } for ⋆ ∈ { d, c } of the trees such that c e r = ⋆ . We similarly define the simplicial fullsubcategories T ⋆ ⊂ T { d,c } .Note that any T ∈ T { d,c } ⟨ m ⟩ has a natural partial order structure on theunion of the set of vertices and edges induced by the start and target maps suchthat e r is the unique minimal element. For each e ∈ E let T ≥ e ∈ T { d,c } ⟨ m ⟩ bethe sub-tree composed of the root vertex, root edge and the vertices and edgesgreater than e .For all T ∈ T { d,c } ⟨ m ⟩ and v ∈ V define the relative setin v ∶ = { e ∈ E ∣ te = v } c s − v ∈ S { d,c } . Note that S { d,c } is isomorphic to T { d,c } ⟨ ⟩ .We have natural dependent sums and dependent products of filtered rootedtrees of a fixed height ⟨ T a ⟩ ∈ Π A T { d,c } ⟨ m ⟩ defined asΣ A T a ∶ = (⟨ Σ A V a,i ⟩ , ⟨ Σ A E a,i ⟩ , ⟨ Σ A s ai ⟩ , ⟨ Σ A t ai ⟩ , Σ A c a ) , Π A T a ∶ = (⟨ Π A V a,i ⟩ , ⟨ Π A E a,i ⟩ , ⟨ Π A s ai ⟩ , ⟨ Π A t ai ⟩ , Π A c a ) . We also have for all T = (⟨ V i ⟩ , ⟨ E i ⟩) ∈ T { d,c } ⟨ m ⟩ and ⟨ S e ⟩ = ⟨(⟨ W e,i ⟩ , ⟨ F e,i ⟩)⟩ ∈ Π E m T c e ⟨ n ⟩ the grafting T ◦ ⟨ S e ⟩ ∶ = (⟨{ V i , i < m Σ E m W e,i − m − , i ≥ m ⟩ , ⟨{ E i , i ≤ m Σ E m F e,i − m − , i > m ⟩) in T { d,c } ⟨ m + n + ⟩ , with the obvious start, target and coloring maps.12 r v v v v v v v v v v v v v e e e e e e e e e E − E E E E V − V V V Figure 1: A filtered rooted relative tree in T c ⟨ ⟩ with wiggled edges representing“domain” edges and straight edges “codomain” edges. The leaves are the onlyedges that are labeled. We now give a brief review of relative operads, a kind of colored operad intro-duced by Voronov in [26].
Definition 3.2.
The category of S { d,c } -spaces is the contravariant functor cat-egory Top S op { d,c } . A topological relative operad is an S { d,c } -space P ∈ Top S op { d,c } equipped with elements id ⋆ ∈ P { ⋆ } ⋆ for ⋆ ∈ { d, c } and structural maps ⟨ ◦ A, ⟨ B a ⟩ ⟩ ∈ Π Σ S { d,c } Π A S c a Top ( P A × Π A P B a , P Σ A B a ) such that P ∅ ⋆ = ∗ for ⋆ ∈ { d, c } and, using the notation α ⟨ β a ⟩ ∶ = ◦ A, ⟨ B a ⟩ ( α, ⟨ β a ⟩) , the following equations are satisfied: α ⟨ β a ⟨ γ a,b ⟩⟩ = α ⟨ β a ⟩⟨ γ a,b ⟩ ; id c A α = α = α ⟨ id c a ⟩ ; α ⋅ σ ⟨ β a ⟩ = α ⟨ β σ − a ′ ⟩ ⋅ σ ( B a ) ; α ⟨ β a ⋅ τ a ⟩ = α ⟨ β a ⟩ ⋅ Σ A τ a . Operad morphisms are natural transformations that preserve the unit andcompositions, and we denote the category of topological relative operad as Op { d,c } [ Top ] .For X = (( X d , e d ) , ( X c , e c )) ∈ Top ∗ defineΠ − X ∶ S inj { d,c } → Top ; Π A X ∶ = Π A X c a , σ ⋅ ⟨ x a ⟩ ∶ = ⟨{ e c a ′ , a ′ / ∈ Im σ ; x σ − a ′ , a ′ ∈ Im σ. ⟩ . P can be extended to afunctor on S inj , op { d,c } . For σ ∈ S inj { d,c } ( A, A ′ ) the right action ⋅ σ ∈ Top ( P A ′ , P A ) isdefined as α ⋅ σ ∶ = α ⟨{ ∗ c a ′ , a ′ / ∈ Im σ ; id c a ′ , a ′ ∈ Im σ. ⟩ . These morphisms are the degenerations of the relative operad.A relative operad P induces a monad ( P ; η, µ ) on Top ∗ with P X ⋆ ∶ = ∫ S inj ⋆ P A × Π A X ; η ⋆ x ∶ = [ id ⋆ , x ] , µ ⋆ [ α, ⟨[ β a , ⟨ x a,b ⟩]⟩] ∶ = [ α ⟨ β a ⟩ , ⟨ x a,b ⟩] . Definition 3.3.
Let P be a relative operad. A P -space is a P -algebra, i.e. apair of pointed spaces X ∈ Top ∗ equipped with structural maps ⟨ θ A ⟩ ∈ Π S { d,c } Top ( P A × Π A X, X c A ) , satisfying, using the notation α ⟨ x a ⟩ = θ A ⟨ α, ⟨ x a ⟩⟩ , the following equations: α ⟨ β a ⟨ x a,b ⟩⟩ = α ◦ ⟨ β a ⟩ ⟨ x a,b ⟩ ; id ⋆ x = x ; α ⋅ σ ⟨ x a ⟩ = α ( σ ⋅ ⟨ x a ⟩) . The category of P -spaces is denoted P [ Top ] .The following are the relative operads relevant to the main result.The terminal relative operad is Com (cid:1) with underlying S { d,c } -space givenby Com (cid:1) ( A ) ∶ = ∗ . The S { d,c } right actions, units and compositions are theunique terminal maps. The Com (cid:1) -spaces are pairs ( M d , M c ) of topologicalcommutative monoids equipped with a continuous homomorphism ι ∶ M d → M c induced by the unique element in Com (cid:1) { c } o .For U ∈ A let the relative operad of U -embeddings Emb (cid:1) U isEmb (cid:1) U A ∶ = { α = ⟨ α a ⟩ ∈ U ⊔ A U ∣ ⟨ α a ⟩ is an embedding } ; ⟨ α a ′ ⟩ ⋅ σ ∶ = ⟨ α σa ⟩ , id ⋆ ∶ = id U , α ⟨ β a ⟩ ∶ = ⟨ α a β ab ⟩ and degenerations deleting embeddings.For U ∈ A the loop space map functors image has natural Emb (cid:1) U -pairsstructure, giving us the functorΩ U ∶ Top (cid:1) ∗ → Emb (cid:1) U [ Top ] ; Ω U ( ι ∶ Y d → Y c ) ∶ = ( Y S U d , Y S U c ) ; α ⟨ γ a ⟩ ∶ = ( ⃗ u ↦ { γ a α − a ⃗ u, c a = c Aιγ a α − a ⃗ u, c a ≠ c A ) (3.1)For ( U, V ) ∈ Σ A A U we have natural inclusion of relative operads i UV ∶ Emb (cid:1) U ⇒ Emb (cid:1) V ; i UV α ∶ = ⟨ ⃗ v ↦ ⃗ v V − U + α a ⃗ v U ⟩ (cid:1) ∞ ∶ = colim A Emb (cid:1) U .The embeddings operad contains embeddings of configuration spaces, andthese embeddings are relevant to the definition of E (cid:1) -operads we give here. Foreach U ∈ A define the configurations S { d,c } -spaceConf (cid:1) U ∶ S op { d,c } → Top ; Conf (cid:1) U A ∶ = { ⃗ x = ⟨ ⃗ x a ⟩ ∈ U A ∣ a ≠ a ′ ⟹ ⃗ x a ≠ ⃗ x a ′ } . Note that Conf (cid:1) U is m -cofibrant, since it is h -equivalent to the underlyingspace of the Fulton-MacPherson operads which are q -cofibrant [18, 9]. We candefine the S { d,c } -space maps χ U ∶ Conf (cid:1) U ⇒ Emb (cid:1) U ; χ U ⃗ x ∶ = ⟨ ⃗ u ↦ ⃗ x a + min a ′ ≠ a ′′ ∥ ⃗ x a ′ − ⃗ x a ′′ ∥ ⃗ u min a ′ ≠ a ′′ ∥ ⃗ x a ′ − ⃗ x a ′′ ∥ + ∥ ⃗ u ∥ ⟩ . Definition 3.4. An E (cid:1) ∞ -operad is an operad E (cid:1) ∈ Op { d,c } [ Top ] equipped with a relative operad mapΨ ∈ Op { d,c } [ Top ]( E (cid:1) , Emb (cid:1) ∞ ) and, for the induced A -filtration E (cid:1) U ∶ = Ψ − Emb U , a S { d,c } -space homotopyequivalence Φ U Top S op { d,c } ( Conf U , E (cid:1) U ) for each U ∈ A such that Ψ ↾ U Φ U = χ U .By this definition the E (cid:1) U are m -cofibrant as S { d,c } -spaces and E (cid:1) is con-tractible and free. One of the main examples of E (cid:1) ∞ -operads we will consider isthe Steiner relative operad, composed of paths of embeddings [22].For all U ∈ A define the relative operad H (cid:1) U as H (cid:1) U A ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ α = ⟨ α a ⟩ ∈ U ⊔ A I × U »»»»»»»»»»»»»»»»»»» ∀ a ∈ A, t ∈ I ∶ ( ⃗ u ↦ α a ( t, ⃗ u )) ∈ Emb (cid:1) U { a } ; ∀ a ∈ A, ∀ t ∈ I, ∀ ⃗ u, ⃗ v ∈ U ∶ ∥ α a ( t, ⃗ u ) − α a ( t, ⃗ v )∥ ≤ ∥ ⃗ u − ⃗ v ∥ ; ∀ a ∈ A, ⃗ u ∈ U ∶ α a ( , ⃗ u ) = ⃗ u ; ⟨ ⃗ u ↦ α a ( , ⃗ u )⟩ ∈ Emb (cid:1) U A. ⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭⟨ α a ′ ⟩ ⋅ σ ∶ = ⟨ α σa ⟩ , id ⋆ ∶ = (( t, ⃗ u ) ↦ ⃗ u ) , α ⟨ β a ⟩ ∶ = ⟨( t, ⃗ u ) ↦ α a ( t, β ab ( t, ⃗ u ))⟩ and degenerations deleting paths of embeddings.We have natural inclusions ι UV ∶ H (cid:1) U ⇒ H (cid:1) V for all ( U, V ) ∈ Σ A A U with ι UV α ∶ = ⟨( t, ⃗ v ) ↦ ⃗ v V − U + α a ( t, ⃗ v U )⟩ and we define H (cid:1) ∞ ∶ = colim A H (cid:1) U . 15he E (cid:1) ∞ -structural transformations areΨ ∶ H (cid:1) ⇒ Emb (cid:1) ; Ψ U α ∶ = ⟨ ⃗ u ↦ α a ( , ⃗ u )⟩ ;Φ U ∶ Conf (cid:1) U ⇒ H (cid:1) U ; Φ U ⃗ x ∶ = ⟨( t, ⃗ u ) ↦ ( − t ) (( χ U ⃗ x ) a ⃗ u ) + t ⃗ u ⟩ . The homotopy inverses of the Φ U are¯Φ U ∶ H (cid:1) U ⇒ Conf (cid:1) U ; ¯Φ U α = ⟨ α a ( , ⃗ )⟩ . See [22] for the construction of the homotopies.
For the construction of the quasiadjunctions in our main theorems we will re-quire the bar resolution of E (cid:1) -pairs. Recall from [16, Construction 9.6] that fora monad ( C, η, µ ) in the category T , a C -functor ( F, λ ) in the category A anda C -algebra ( X, ξ ) the two sided bar construction B − ( F, C, X ) ∈ A ∆ with B ⟨ m ⟩ ( F, C, X ) ∶ = F C m X ; δ i ∶ = F C i η C m − i , ∂ i ∶ = ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ λ C m , i = F C i − µ C m − i + , < i < m ; F C m − ξ, i = m. In particular for a relative operad P and C = P = F we have a naturalisomorphism B ⟨ m ⟩ ( P, P, X ) ⋆ ≅ ∫ T ⋆ ⟨ m ⟩ Π V ∗ P in v × Π E m X c e , [ α r , ⟨ α v ⟩ , ⟨ x e ⟩] T ⋅ ∂ i ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩[ α r ⟨ α se ′ ⟩ , ⟨ α v ⟩ , ⟨ x e ⟩] T ⋅ ∂ , i = [ α r , ⟨{ α v ⟨ α se ′ ⟩ , ∣ v ∣ = i − α v , ∣ v ∣ ≠ i − ⟩ , ⟨ x e ⟩] T ⋅ ∂ i , < i < m [ α r , ⟨ α v ⟩ , ⟨ α s − e ⟨ x e ′ ⟩⟩] T ⋅ ∂ m , i = m , [ α r , ⟨ α v ⟩ , ⟨ x e ⟩] T ⋅ δ i ∶ = [ α r , ⟨{ id c s − v , ∣ v ∣ = iα v , ∣ v ∣ ≠ i ⟩ , ⟨ x e ⟩] T ⋅ δ i . The E (cid:1) -pair structural maps in each dimension are α ⟨[ β a,r , ⟨ β a,v ⟩ , ⟨ x a,e ⟩] T a ⟩ ∶ = [ α ⟨ β a,r ⟩ , ⟨ β a,v ⟩ , ⟨ x a,e ⟩] Σ A T a (3.2)The bar resolution of E (cid:1) -pairs is then the geometric realization of this sim-plicial E (cid:1) -pair functor B ∶ E (cid:1) [ Top ] → E (cid:1) [ Top ] , B X ⋆ ∶ = ∣ B − ( E (cid:1) , E (cid:1) , X ) ⋆ ∣ . By the above isomorphism we can intuitively think of points in B X asequivalence classes of filtered rooted relative trees with vertices decorated with16 β ,r β , β , x , x , x , x , , β ,r β , β , ∗ β , β , x , x , x , = α ⟨ β ,r,β ,r ⟩ id c id cβ , β , β , β , id d id d id d id c id c id d id c ∗ β , β , x , x , x , x , x , x , x , Figure 2: E (cid:1) ∞ -structure of B X elements of E (cid:1) , leaves decorated with elements of X and we associate an orderedpartition with the filtration of the inner vertices.It is not the case in general that the geometric realization of a simplicial C -algebra for a topological monad C is a C -algebra. This is however the casewhen the monad is the one induced by an operad. The structural maps areinduced by the algorithm described in [16, Theorem 11.5]. For a sequenceof elements with representatives of distinct dimensions we can systematicallydetermine equivalent representatives of the same dimension, and then apply theformula 3.2, so that the E (cid:1) -pair structural maps of B X are defined by theformula α ⟨[[ β a,r , ⟨ β a,v ⟩ , ⟨ x a,e ⟩] T a , t a ]⟩ ∶ = [[ α ⟨ β a,r ⟩ , ⟨{ id c s − v , ∃ i > ∣ v ∣ ∶ δ a i = δ a ∣ v ∣ β a,v , otherwise ⟩ , ⟨ x a,e ⟩] Σ A T a ⋅ δ a , ⊲ A t a ] (3.3)which is illustrated in figure 2.This functor can be equipped with the natural transformation η ′ ∶ B ⇒ Id, η ′⋆ [[ α r , ⟨ α v ⟩ , ⟨ x e ⟩] T , t ] ∶ = ◦ T α v ⟨ x e ⟩ . (3.4)where ◦ T α v is the composition of all the α v , including α r , induced by theoperadic composition and the structure of T . Operad actions encodes distributive laws between operations defined by operads[15, Definition VI.1.6]. The following definition is a relative version of thisnotion.
Definition 3.5. A relative operad pair is a pair of relative operads ( P , G ) equipped with an extension of G to S op ⟨ d < c ⟩ and structural maps ⟨ ⋉ A, ⟨ B a ⟩ ⟩ ∈ Π Σ S { d,c } Π A S c a Top ( G A × Π A P B a , P Π A B a ) action of G on P , such that, using the notations f ⋉ ⟨ α a ⟩ ∶ = ⋉ ⟨ f, ⟨ α a ⟩⟩ , f ⋉ ⟨ β a,b a ⟩ ∶ = f ⋅ π ⟨ b a ⟩ ⋉ ⟨ β a,b a ⟩ the following equations are satisfied: f ⋉ ⟨ g a ⋉ ⟨ α a,b ⟩⟩ = f ◦ ⟨ g a ⟩ ⋉ ⟨ α a,b ⟩ ; (3.5) f ⋉ ⟨ α a ◦ ⟨ β a,b ⟩⟩ = f ⋉ ⟨ α a ⟩ ◦ ⟨ f ⋉ ⟨ β a,b a ⟩⟩ ⋅ ν ; (3.6) id c A ⋉ α = α ; (3.7) f ⋉ ⟨ id c a ⟩ = id c A ; (3.8) f ⋅ σ ⋉ ⟨ α a ⟩ = f ⋉ ⟨ α σ − a ′ ⟩ ⋅ σ ⟨ B a ⟩ ; (3.9) f ⋉ ⟨ α a ⋅ τ a ⟩ = f ⋉ ⟨ α a ⟩ ⋅ Π A τ a . (3.10)We refer to the operad P as the additive relative operad and G as the multi-plicative relative operad of the pair.For X = (( X d , d , d ) , ( X c , c , c )) ∈ Top S define X ∧− ∶ S inj { d,c } → Top ; X ∧ A ∶ = ∧ A X c a , σ ⋅ [ x a ] ∶ = [{ c a ′ , a ′ / ∈ Im σ ; x σ − a ′ , a ′ ∈ Im σ. ] with the zeros as base points for the wedge products. We can then define themonad ( G ; η, µ ) on Top S with G X ⋆ ∶ = ∫ S inj ⋆ G A + ∧ X ∧ A ; η ⋆ x ∶ = [ id ⋆ , x ] , µ ⋆ [ f, [[ g a , [ x a,b ]]]] ∶ = [ f ⟨ g a ⟩ , [ x a,b ]] . Definition 3.6. A G -space is a G -algebra, i.e. a pair of S -spaces X ∈ Top S equipped with a structural map χ ∶ G X → X satisfying, using the notation f [ x a ] = χ A [ f, [ x a ]] similar equations as in 3.3 and also that 0 is an absorbing element, ie ∃ a ∈ A ∶ x a = c a ⟹ f [ x a ] = c A . The category of G -spaces is denoted G [ Top ] .If G acts on P then the functor P induces a monad on G [ Top ] . Definition 3.7.
Let ( G , P ) be a relative operad pair. A ( G , P ) -space is a P -algebra in G [ Top ] . Equivalently a ( G , P ) -space is a pair of S -spaces X ∈ Top S equipped with a G -space structure and a P -space structure with neutralelements the zeros such that f [ α a ⟨ x a,b ⟩] = f ⋉ ⟨ α a ⟩⟨ f [ x a,b a ]⟩ . The category of ( P , G ) -spaces is denoted ( P , G )[ Top ] .18here is a natural operad pair structure on ( Com (cid:1) , Com (cid:1) ) . Set the notation ∑ A ∈ Com (cid:1) ( A ) for the additive copy of Com (cid:1) and ∏ A ∈ Com (cid:1) ( A ) for themultiplicative copy of Com (cid:1) . Then in a ( Com (cid:1) , Com (cid:1) ) -space the distributivityequations and the equality of the additive and multiplicative homomorphisms ∏ A ∑ B a x a,b = ∑ Π A B a ∏ A ⟨ ba ⟩ x a,b a φ + x = ∏ { c , c } c ⟨ φ + x, c ⟩ = ∏ { c , c } c ⋉ ⟨ φ + , id c ⟩ ∏ { d , c } c ⟨ x, c ⟩ = φ ⋅ x hold. This means that ( Com (cid:1) , Com (cid:1) )[ Top ] is isomorphic to the category oftopological commutative semi-algebras over commutative semi-rings .The main example of multiplicative relative operad we will consider is the relative linear isometries operad L (cid:1) with L (cid:1) A ∶ = I ( ⊕ A R ∞ , R ∞ ) ; f ⋅ σ ∶ = ⟨ f σa ⟩ , id ∶ = id R ∞ , f ◦ ⟨ g a ⟩ ∶ = ⟨ f a g ab ⟩ . The classical action of the linear isometries operad on the Steiner operadinduces an action on the relative versions. The extension of L (cid:1) to S op ⟨ d < c ⟩ isgiven by identity maps and the action maps are given by the formula f ⋉ ⟨ α a ⟩ ∶ = [( t, f a ⃗ u a ) ↦ f a ( α ab a ( t, ⃗ u a ))] . Definition 3.8.
The category of E (cid:1) ∞ -algebras is ( E (cid:1) , L (cid:1) )[ Top ] for an E (cid:1) ∞ -operad E (cid:1) equipped with an action by L (cid:1) . .Although we give this general definition we note that there is no knownnon-trivial example of an E ∞ operad equipped with an L -action other then theSteiner operad H (cid:1) ∞ . Having a q -cofibrant, not just mixed Σ-cofibrant examplewould be interesting and useful, but since we can work in the mixed modelstructure of spectra it is not necessary.The images of B X are also L (cid:1) -pairs with structural maps defined as f [[[ α a,r , ⟨ α a,v ⟩ , ⟨ x a,e ⟩] T a , t a ]] ∶ = (3.11) ⎡⎢⎢⎢⎢⎢⎢⎢⎣[ f ⋉ ⟨ α a,r ⟩ , ⟨ f ⋉ ⟨{ id c s − v a , ∃ i > ∣ v a ∣ ∶ δ a i = δ a ∣ v a ∣ α a,v a , otherwise ⟩⟩ , ⟨ f [ x a,e a ]⟩] Π A T a ⋅ δ a , ⊲ A t a ⎤⎥⎥⎥⎥⎥⎥⎥⎦ which is illustrated in figure 3. Semi-algebras and semi-rings are like algebras and rings without the assumption thatadditive inverses exist, ie we have an additive commutative monoid instead of an additiveabelian group α ,r α , α , x , x , x , x , , α ,r α , α , ∗ α , α , x , x , x , = f ⟨ α ,r,α ,r ⟩ f ⟨ id c,α , ⟩ f ⟨ id c,α , ⟩ f ⟨ id c,α , ⟩ f ⟨ id c,α , ⟩ f ⟨ α , , id d ⟩ f ⟨ α , , id d ⟩ f ⟨ α , , id d ⟩ f ⟨ α , , id d ⟩ f ⟨ α , , id d ⟩ f ⟨ α , , id d ⟩ ∗∗ f ⟨ id c,α , ⟩ f ⟨ id c,α , ⟩ f ⟨ id c,α , ⟩ f ⟨ id c,α , ⟩ ∗∗ f ⟨ id d,α , ⟩ f ⟨ id c,α , ⟩ f ⟨ id d,α , ⟩ f ⟨ id c,α , ⟩ f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] f [ x , ,x , ] Figure 3: L (cid:1) -structure of B X .5 Model structure of E (cid:1) ∞ -algebras The model structure of E (cid:1) ∞ -pairs is transferred from the q -model structure of Top ∗ by the adjunction ( E (cid:1) L (cid:1) ⊣ U ) ∶ Top ∗ ⇋ ( E (cid:1) , L (cid:1) )[ Top ] , so the weak equivalences and fibrations are respectively the maps that are q -equivalences and q -fibrations as topological space maps [17]. All objects arefibrant and cofibrant algebras are retracts of cellular E (cid:1) ∞ -algebras, with cellsthe E (cid:1) L (cid:1) -images of the cells in Top ∗ . We give a brief review of coordinate-free spectra [11] and give some examples.Let U ∈ I be countably infinite dimensional (In the context of coordinate-free spectra we refer to U as a universe ). The topological category Sp U ofcoordinate-free U -spectra is composed of the class of objects { Y = (⟨ Y U ⟩ , ⟨ σ UV ⟩) ∈ Σ Π A Top ∗ Π Σ A A U Top ∗ ( Y U ∧ S V − U , Y V ) ∣ σ UU [ y, ⃗ ] = y, σ VW [ σ UV [ y, ⃗ v ] , ⃗ w ] = σ UW [ y, ⃗ v + ⃗ w ]} and the morphisms spaces Sp U ( Y, Z ) defined as { f = ⟨ f U ⟩ ∈ Π A Top ∗ ( Y U , Z U ) ∣ σ UV [ f U y, ⃗ v ] = f V σ UV [ y, ⃗ v ]} . We are particularly interested here in the case U = R ∞ and in this case weuse the notation Sp ∶ = Sp R ∞ . Example 4.1.
Interesting coordinate-free spectra to keep in mind are the fol-lowing, with details similar to the equivalent symmetric examples in [20, SectionI.2]: • For each p ∈ Z the p -sphere spectrum is defined as S p ∶ = ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩⟨ S U − R − p ⟩ , σ UV [ ⃗ u, ⃗ v ] ∶ = ⃗ u + ⃗ v V − R − p , p < ⟨ S U ⟩ , σ UV [ ⃗ u, ⃗ v ] ∶ = ⃗ u + ⃗ v, p = ⟨ S U ⊕ R p ⟩ , σ UV [( ⃗ u, ⃗ w ) , ⃗ v ] ∶ = ( ⃗ u + ⃗ v, ⃗ w ) , p > S ∶ = S . • For each G ∈ AbGrp define the
Eilemberg-MacLane spectrum HG ∶ = ⟨ G ⊗ F [ S U ] ∗ ⟩ ; σ UV [ g a ⊗ ⃗ u a , ⃗ v ] ∶ = g a ⊗ ⃗ u a + ⃗ v. F [ S U ] ∗ denotes the quotient of the free abelian group generatedby the points of the U -sphere by the subgroup generated by ∞ , and as inthe Einstein convention g a ⊗ ⃗ u a indicates a finite sum of elements. Notethat g ⊗ ∞ = • For each U ∈ A let O U be the orthogonal group of isometric automor-phisms of U . The total space EO U of the universal principal O U -bundleis the geometric realization of the simplicial space O − U ∈ Top ∆ op with ⟨ f j ⟩ ⋅ ∂ i ∶ = ⟨⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ f j , j < i − f j f j + , j = i − f j + , j > i − ⟩ , ⟨ f j ⟩ ⋅ δ i ∶ = ⟨⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ f j , j < iid, j = if j − , j > i ⟩ . The U -spheres admit a left O U -action by evaluation f ⋅ ⃗ u ∶ = f ⃗ u and EO U, + admits the right O U -action [⟨ g i ⟩ , t ] ⋅ f ∶ = [⟨{ g i , i < mg m f, i = m ⟩ , t ] . For ( U, V ) ∈ Σ A A U we have a natural inclusion ι UV ∶ O U → O V , ι UV f ⃗ v ∶ = ⃗ v V − U + f ⃗ v U . We can define the
Thom spectrum as M O ∶ = ⟨ EO U, + ∧ O U S U ⟩ ; σ UV [[⟨ f i ⟩ , t, ⃗ u ] , ⃗ v ] ∶ = [⟨ ι UV f i ⟩ , t, ⃗ u + ⃗ v ] . An Ω-spectrum is a spectrum Y ∈ Sp such that the adjoint structural maps ̃ σ UV ∈ Top ∗ ( Y U , Y S V − U V ) are q -equivalences.The stable homotopy groups of spectra are defined as π Sp Y ∶ = π Sp ( S p , Y ) .If Y is an Ω-spectrum then π Sp Y ≅ { π Y R ∣ p ∣ , p < π p Y , p ≥ stable weak equivalences , and spectra Y ∈ Sp with π Sp Y trivial for p < connective .We base space functorΛ ∞ ∶ Sp → Top ∗ ; Λ ∞ Y ∶ = Y which is right adjoint to the suspension spectrum functorΣ ∞ ∶ Top ∗ → Sp ; Σ ∞ X ∶ = ⟨ X ∧ S U ⟩ ; σ UV [[ x, ⃗ u ] , ⃗ v ] ∶ = [ x, ⃗ u + ⃗ v ] . with unit and counit of the adjunction ηx ∶ = [ x, ⃗ ] ; ǫ U [ y, ⃗ u ] ∶ = σ U [ y, ⃗ u ] . .2 Stable mixed model structure of spectra For any spectrum Y ∈ Sp the cylinder spectrum is defined as Y ∧ I + ∶ = ⟨ Y U ∧ I + ⟩ and the cone spectrum is CY ∶ = Y ∧ I + / [ y, ] ∼ [ y ′ , ] .In the strict Quillen model structure on Sp a morphism f ∈ Sp ( X, Y ) isa weak equivalence if each f U is a q -equivalence, a fibration if it is a Serrefibration, ie if it has the homotopy lifting property with respect to the cylinderinclusions of cones of sphere spectra in ∈ Sp ( C S q , C S q ∧ I + ) for all q ∈ Z ,and a cofibration if it is a retract of a relative cell-spectrum, with cells givenby cones of sphere spectra and domain of the attaching maps the boundarysphere spectra [5, Section VII.4]. This is a cofibrantly generated model structurewith factorization systems induced by the small object argument. The weakequivalences, fibrations and cofibrations of this model structure are referred toas q -equivalences, q -fibrations and q -cofibrations respectively.Homotopy equivalences in Sp are spectra maps that admit an inverse upto homotopy, with homotopies defined via the cylinder spectra in the usualway. In the strict Hurewicz/Strøm model structure f is a weak equivalence ifit is a homotopy equivalence, a fibration if it is a Hurewicz fibration, ie if ithas the homotopy lifting property with respect to all cylinder inclusions in ∈ Sp ( X, X ∧ I + ) , and a cofibration if it has the left lifting property against trivialHurewicz fibrations.The weak factorization system can be constructed through (co)monads asdescribed in [1]. For any Y ∈ Sp let the spectrum of Moore paths in Y be M Y ∶ = ⟨ Σ [ , ∞ ) { Y [ , ∞ ] + U ∣ s ≥ t ⟹ γs = γt }⟩ ,σ UV [( t, γ ) , ⃗ v ] ∶ = ( t, r ↦ σ UV [ γr, ⃗ v ]) . The factorization systems are then defined asΓ f ∶ = X × Y M Y ; C t f x ∶ = ( x, , r ↦ f x ) , F f ( x, t, γ ) ∶ = γt.E f ∶ = Γ f ∧ [ , ∞ ] + ⊔ Γ f Y ; C f x ∶ = ( x, , r ↦ f x, ) , F f ( x, t, γ, s ) ∶ = γs. The weak equivalences, fibrations and cofibrations of this model structureare referred to as h -equivalences, h -fibrations and h -cofibrations respectively.We then equip Sp with the mixed model structure as described in [4, Prop 3.6].Since the point of spectra is to study stabilization phenomena we are actu-ally interested in inverting the stable weak homotopy equivalences. The stablemodel structure with stable weak homotopy equivalences as weak equivalences isobtained from the strict model structure by the process of Bousfield localizationthrough the following idempotent monad [2, 19]. For every spectrum Y ∈ Sp we can functorialy define an inclusion spectrum ̃ Y equipped with a quotientmap Y → ̃ Y , so we may think of points in ̃ Y as equivalence classes of pointsin Y (see [11, Ap1] for a detailed construction). If Y is already an inclusionspectrum then ̃ Y = Y . We may then define the spectrification functor ̃ Ω ∶ Sp → Sp ; ̃ Ω Y ∶ = ⟨ colim A U ̃ Y S V − U V ⟩ ; σ UW [ γ, ⃗ w ] ∶ = [ ⃗ v ↦ γ ( ⃗ v + ⃗ w )] Inclusion spectra are those with adjoint structural maps ˜ σ all inclusions. σ and with the formula for the structuralmaps determined for a choice of representative γ with domain V ∈ A W . Thisis a Quillen idempotent monad with structural natural map ǫ ′ ∶ Id ⇒ ̃ Ω; ǫ ′ U y ∶ = [ ⃗ v ↦ σ UV [ y, ⃗ v ]] (4.1)The stable model structure on spectra Sp ̃ Ω has as weak equivalences thestable weak equivalences and stable fibrations are p ∈ Sp ( E, B ) composed ofindexwise Hurewicz fibrations such that the maps ( ˜ σ UV , p U ) ∶ E U → E S V − U V × B S V − UV B U are q -equivalences. The fibrant spectra are the Ω-spectra, and the cofibrantspectra are those homotopy equivalent to retracts of q -cofibrant spectra. Withthe induced stable model structure the adjunction ( Σ ∞ ⊣ Λ ∞ ) is a Quillenadjunction.The morphisms category Sp (cid:1) admits a projective stable model structurewith ( f d , f c ) ∈ Sp (cid:1) ( i ∶ Y d → Y c , j ∶ Z d → Z c ) a weak equivalence or fibration if f c and f o are both stable weak equivalences or stable fibrations respectively, andit is a cofibration if both f d and ( f c , j ) ∶ Y c ∨ Y d Z d → Z c are stable cofibrations. ∞ -loop maps We can now prove the recognition principle for ∞ -loop pairs of spaces of spectramaps. The base pair of spaces functor isΛ ∞ ∶ Sp (cid:1) → Top ∗ , Λ ∞ i ∶ = ( Y d, , Y c, ) and the relative suspension functor isΣ ∞ (cid:1) ∶ Top ∗ → Sp (cid:1) , Σ ∞ (cid:1) X ∶ = Σ ∞ ( in d ∶ X d → X d ∨ X c ) . We have a Quillen adjunction ( Σ ∞ (cid:1) ⊣ Λ ∞ ) ∶ Top ∗ ⇋ Sp (cid:1) ; η ⋆ x ∶ = [ x, ⃗ ] , ǫ ⋆ ,U [ y, ⃗ u ] ∶ = [{ σ U [ y, ⃗ u ] , c y = ⋆ σ U [ i y, ⃗ u ] , c y ≠ ⋆ ] . The spectrification functor ̃ Ω induces ̃ Ω (cid:1) ∶ Sp (cid:1) → Sp (cid:1) ; ̃ Ω (cid:1) i ∶ = ( ̃ Ω i ∶ ̃ Ω Y d → ̃ Ω Y c ) . The ∞ -loop pair of spaces functor is defined asΩ ∞ ∶ Sp (cid:1) → E (cid:1) [ Top ] ; Ω ∞ i ∶ = Λ ∞ ̃ Ω (cid:1) i with structural maps induced by the formula 3.1 by taking representatives ofthe γ a with a common domain. 24his functor is not a right adjoint, but it is a weak Quillen right quasiadjoint.The left quasiadjoint functor is defined as follows: We have simplicial pointedmaps B − ( Σ U (cid:1) , E (cid:1) U , X ) ∈ ( Top (cid:1) ∗ ) ∆ with B ⟨ m ⟩ ( Σ U (cid:1) , E (cid:1) U , X ) ⋆ ≅ ( ∫ T ⋆ ⟨ m ⟩ Π V E (cid:1) in v × Π E m X c e ) ∧ S U , [⟨ α v ⟩ , ⟨ x e ⟩ , ⃗ u ] T ⋅ ∂ i ∶ = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩{[⟨ α v ⟩ , ⟨ x e ⟩ , α , − e ′ ⃗ u ] T ≥ e ′ , ⃗ u ∈ α e ′ U ∞ , ⃗ u / ∈ α ⊔ E U , i = [⟨{ α v ⟨ α se ′ ⟩ , ∣ v ∣ = i − α v , ∣ v ∣ ≠ i − ⟩ , ⟨ x e ⟩ , ⃗ u ] T ⋅ ∂ i , < i < m [⟨ α v ⟩ , ⟨ α s − e ⟨ x e ′ ⟩⟩ , ⃗ u ] T ⋅ ∂ m , i = m [⟨ α v ⟩ , ⟨ x e ⟩ , ⃗ u ] T ⋅ δ i ∶ = [⟨{ id c s − v , ∣ v ∣ = iα v , ∣ v ∣ ≠ i ⟩ , ⟨ x e ⟩ , ⃗ u ] T ⋅ δ i . Define the relative ∞ -delooping functor as B ∞ (cid:1) ∶ E (cid:1) [ Top ] → Sp (cid:1) ; B ∞ (cid:1) X ⋆ ∶ = ⟨∣ B − ( Σ U (cid:1) , E (cid:1) U , X ) ⋆ ∣⟩ ,σ UV [[[⟨ α v ⟩ , ⟨ x e ⟩ , ⃗ u ] T , t ] , ⃗ v ] ∶ = [[⟨ α v ⟩ , ⟨ x e ⟩ , ⃗ u + ⃗ v ] T , t ] . Points in B ∞ (cid:1) X ⋆ ,U are equivalence classes of decorated filtered rooted relativetrees as in the description of the bar resolution B X , except the root vertex isdecorated with a vector in U and the relative operad points decorating the innervertices must be contained in the suboperad E (cid:1) U of the A -filtration of E (cid:1) . Theorem 4.2.
For E (cid:1) an E (cid:1) ∞ -operad there is an idempotent quasiadjunction ( B ∞ (cid:1) ⊣ B , ̃ Ω (cid:1) Ω ∞ ) ∶ E (cid:1) [ Top ] ⇋ Sp (cid:1) Proof:
The unit span and cospan has η ′ the natural weak equivalence 3.4, ǫ ′ induced by the idempotent monad transformation 4.1 and η and ǫ are definedby the following formulas: η ∶ B ⇒ Ω ∞ B ∞ (cid:1) , ǫ ∶ B ∞ (cid:1) Ω ∞ ⇒ ̃ Ω (cid:1) ,η ⋆ [[ α r , ⟨ α v ⟩ , ⟨ x e ⟩] T , t ] ∶ = [ ⃗ u ↦ {[[⟨ α v ⟩ , ⟨ x e ⟩ , α r, − e ′ ⃗ u ] T ≥ e ′ , t ] , ⃗ u ∈ α re ′ U ∞ , ⃗ u / ∈ α r ⊔ E U ] ,ǫ ⋆ ,U [[⟨ α v ⟩ , ⟨ γ e ⟩ , ⃗ u ] T , t ] ∶ = [ ⃗ v ↦ ◦ T α v ⟨ γ e ⟩( ⃗ u + ⃗ v )] . We verify that the conditions for definition 2.1 are satisfied.(i): By the assumptions on E (cid:1) and [25, Prop. 3.2.3] the functor B ∞ (cid:1) is leftderivable.(ii): Trivially Ω ∞ preserves fibrant objects. Since Ω ∞ = Λ ∞ ̃ Ω and stableweak equivalences are by definition maps whose images under ̃ Ω are strict weakequivalences we have that Ω ∞ preserves weak equivalences.25 u ↦ t α α α t α α α α t α α α α α α x x x x x x x x x α r, − ⃗ uα r, − ⃗ uα r, − ⃗ u Figure 4: Representative U -loop of η c [[ α r , ⟨ α v ⟩ , ⟨ x e ⟩] T , t ] ⃗ v ↦ ⃗ u ⃗ v ⃗ u + ⃗ vγ d γ c γ d γ c Figure 5: Representative V -loop of ǫ c,U [[⟨ α v ⟩ , ⟨ γ e ⟩ , ⃗ u ] T , t ] B preserves cofibrant objects by [25, Prop. 3.2.3] andtrivially preserves fibrant objects. The functor ̃ Ω (cid:1) preserves cofibrant objectsby the results in [6, Sec. 5.3] and the fact that we are using the mixed stablemodel structure on spectra, and it trivially preserves fibrant objects since it isthe fibrant replacement functor of the stable model structure.(iv): As a map of topological spaces η ′ is a realization of a simplicial strongdeformation retract, so it is itself a strong deformation retract of topologicalspaces and therefore in particular a q -equivalence [16, theorems 9.10, 9.11 and11.10]. The map ǫ ′ is a weak equivalence by the definition of the stable modelstructure.(v): The natural homotopy H which gives the homotopy commutativity in Sp (cid:1) ǫ B ∞ (cid:1) X B ∞ (cid:1) η X [[⟨ α v ⟩ , ⟨[ β e,r , ⟨ β e,w ⟩ , ⟨ x e,f ⟩] S e , s e ]⟩ , ⃗ u ] T , r ] = [ ⃗ v ↦ {[[⟨ β e,w ⟩ , ⟨ x e,f ⟩ , ( ◦ < e α ve ′ β e,rf ′ ) − ⃗ u + ⃗ v ] S e ≥ f ′ , s e ] , ⃗ u + ⃗ v ∈ ◦ < e α ve ′ β e,rf ′ V ∞ , ⃗ u + ⃗ v / ∈ ◦ T α v ⟨ β e,r ⟩ ⊔ Σ Em F e, V ] ≃ H X [ ⃗ v ↦ [[⟨ α v ⟩ , ⟨ ◦ S e β e,w ⟨ x e,f ⟩⟩ , ⃗ u + ⃗ v ] S e , r ]] = ǫ ′ B ∞ (cid:1) X B ∞ (cid:1) η ′ X [[⟨ α v ⟩ , ⟨[ β e,r , ⟨ β e,w ⟩ , ⟨ x e,f ⟩] S e , s e ]⟩ , ⃗ u ] T , r ] is H ∶ B ∞ (cid:1) B ∧ I + ⇒ ̃ Ω B ∞ (cid:1) ,H X,U ([[⟨ α v ⟩ , ⟨[ β e,r , ⟨ β e,w ⟩ , ⟨ x e,f ⟩] S e , s e ]⟩ , ⃗ u ] T , r ] , t ) ∶ = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⃗ v ↦ ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⟨⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ α v id β e,w ⟩ , ⟨ x e,f ⟩ , ⃗ u + ⃗ v ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ T ◦ ⟨ S e ⋅ δ e ⟩ , Φ t ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ where Φ t ∶ = ( − t )( ◦ m + n + j = m + ∂ j ⋅ r ) i + t ( ◦ mj = ∂ ⋅ ⊲ E m s e ) i , with the conditions in the formula similar to the ones in 3.3.(vi): In E (cid:1) [ Top ] we have strict commutativityΩ ∞ ǫη Ω ∞ [[ α r , ⟨ α v ⟩ , ⟨ γ e ⟩] T , t ] = [ ⃗ v ↦ ◦ T α v ⟨ γ e ⟩ ⃗ v ] = Ω ∞ ǫ ′ η ′ Ω ∞ [[ α r , ⟨ α v ⟩ , ⟨ γ e ⟩] T , t ] . (cid:4) Theorem 4.3.
The quasiadjunction in theorem 4.2 is idempotent and inducesan equivalence ( L B ∞ (cid:1) ⊣ R Ω ∞ ) ∶ H o E (cid:1) [ Top ] grp ⇋ H o Sp (cid:1) con . between the homotopy categories of grouplike E (cid:1) -pairs and maps between con-nective spectra. roof: In E (cid:1) [ Top ] the conditions for definition 2.3 are satisfied and theresulting reflective homotopy subcategory is composed of the grouplike E (cid:1) -pairs:(i) As we have seen η ′ is a natural weak equivalence and by definition cof isa natural trivial fibration, so cof η ′ C is a weak equivalence.(ii) Since Ω ∞ preserves weak equivalence between fibrant objects and B ∞ (cid:1) preserves weak equivalences between cofibrant objects we have that Ω ∞ F B ∞ (cid:1) C preserves weak equivalences.(iii) The natural transformation η is a natural group completion, since it isa realization of a simplicial group completion map (see [16, Theorems 2.7, 9.10and 9.11], and [14, Theorem 2.2]), and the images of Ω ∞ F B ∞ (cid:1) C are grouplike,therefore η Ω ∞ F B ∞ (cid:1) C is a natural weak equivalence. By naturality Ω ∞ F B ∞ (cid:1) C η isalso a group completion, and since the domain and codomain are grouplike thisis a natural weak equivalence.(iv) This condition holds since fibrations are preserved by pullbacks, fi-brations induce long exact sequences of homotopy groups and for a fibration p ∶ E ↠ B and a map f ∶ X → B the fibers of the pullback f ∗ p ∶ X × B E → X are homeomorphic to the fibers of p .(v) Pushouts in E (cid:1) [ Top ] by a cofibration whose domain is m -cofibrant in Top ∗ is a retract of a transfinite composition of pushouts by m -cofibrations in Top ∗ (see [21, I.4]), hence this condition holds since Top ∗ with the mixed modelstructure is left proper and the underlying functor of E (cid:1) is an m -cofibrant S { d,c } -space.By the characterization of fibrations in the resulting Bousfield localizationin [25, Prop. 2.3.6] the fibrations are the group completions and fibrant objectsare the grouplike E (cid:1) -pairs.The dual conditions for definition 2.3 are also satisfied in Sp (cid:1) and the re-sulting coreflective homotopy subcategory is composed of the maps betweenconnective spectra. Note that conditions (i), (ii) and (iii) are self dual.(i) By definition of the stable model structure ǫ ′ is a natural stable weakequivalence and by definition fib is a natural trivial cofibration, so η ′ F fib is aweak equivalence.(ii) That B ∞ (cid:1) C Ω ∞ F preserves weak equivalences follows by the same argu-ment for Ω ∞ F B ∞ (cid:1) C .(iii) We have that η Ω ∞ is a natural weak equivalence, and since Ω ∞ ǫη Ω ∞ = Ω ∞ ǫ ′ η ′ Ω ∞ and Ω ∞ ǫ ′ η ′ Ω ∞ is a natural weak equivalence by the 2-out-of-3 propertyΩ ∞ ǫ is a natural weak equivalence. Since the images of ̃ Ω are Ω-spectra bythe formula for stable homotopy groups of Ω-spectra we have that ǫ inducesisomorphisms on the non-negative stable homotopy groups, and is therefore astable weak equivalence on the maps between connective spectra. The images of B ∞ (cid:1) are connective by [16, 11.12] and [14, A5]. Therefore ǫ B ∞ (cid:1) C Ω ∞ F is a naturalweak equivalence. By naturality B ∞ (cid:1) C Ω ∞ F ǫ also induces isomorphisms on thenon-negative stable homotopy groups and so is also a natural weak equivalence.(iv) This condition holds since cofibrations are preserved by pullbacks, spec-tra cofibrations induce long exact sequences of stable homotopy groups and forany cofibration i ∶ A ↪ X and map f ∶ A → Y the cofiber of the pushout28 ∗ i ∶ Y → X ⊔ A Y are homeomorphic to the cofibers of i .(v) The stable model structure of spectra is right proper so the dual of (v)holds.By the dual of the characterization in [25, Prop. 2.3.6] the cofibrant objectsare the spectra maps such thatΓ (( ǫB ∞ (cid:1) cof Ω ∞ ) F i ) × ̃ Ω Fi i → i are weak equivalences, which is equivalent to ι being a map of connectivespectra. (cid:4) S -modules and commutative algebra spectra In order to define a monoidal category of spectra, so that we get natural defini-tions of spectral algebraic structures, we need to work on the more structuredcategory of sphere modules
Mod S [5]. As a first step consider for A ∈ S the external smash product functor ∧ A ∶ Π A Sp → Sp ⊕ A R ∞ , ∧ A ⟨ Y a ⟩ ∶ = ⟨ ∧ A Y aU a ⟩ ,σ ⟨ U a ⟩⟨ V a ⟩ [[ y a ] , ⟨ ⃗ v a ⟩] ∶ = [ σ U a V a [ y a , ⃗ v a ]] . The change of universe in this product is formally problematic, and thefollowing construction is used to internalize the smash product in Sp . For K ⊂ cpct L A define the monotone functions µ ∈ POSet ( A ⊕ A R ∞ , A ) , µ ⟨ U a ⟩ ∶ = ∑ K f ⟨ U a ⟩ ν ∈ POSet ( A , A ⊕ A R ∞ ) , νU ∶ = ∩ K f − U which satisfy µνU ⊂ U, νµνU = νU, ⟨ U a ⟩ ⊂ νµ ⟨ U a ⟩ , µ ⟨ U a ⟩ = µνµ ⟨ U a ⟩ . For all (⟨ U a ⟩ , V ) ∈ Σ A ⊕ A R ∞ A µ ⟨ U a ⟩ we have the associated Thom complex T K ⟨ U a ⟩ V ∶ = Σ K S V − f ⟨ U a ⟩ / ( f, ∞ ) ∼ ( g, ∞ ) ∈ Top ∗ , where Σ K S V − f ⟨ U a ⟩ is topologized as a subspace of K × S V , with the equivalenceclass [ f, ∞ ] as base point. We will use the notation ⃗ vf ∶ = [ f, ⃗ v ] ∈ T K ⟨ U a ⟩ V .The twisted half-smash product is defined as L A ⋉ − ∶ Sp ⊕ A R ∞ → Sp ; L A ⋉ Z ∶ = ⟨ Colim K ⊂ cpct L A T K νUU ∧ Z νU ⟩ ,σ UV [[ ⃗ uf , z ] , ⃗ v ] ∶ = [ ( ⃗ u + ⃗ v ) V − fνV f , σ νUνV [ z, f − ( ⃗ u + ⃗ v ) fνV ]] . We define the monad ( L ; η, µ ) on Sp with L Y ∶ = L ⋉ Y ; ηy ∶ = [ ⃗ id , y ] , µ [ ⃗ uf , [ ⃗ vg , y ]] ∶ = [ ⃗ u + f ⃗ vfg , y ] .
29e refer to the L -algebras as L -spectra and for ( Y, y ) ∈ L [ Sp ] we use thenotation ⃗ uf y ∶ = y [ ⃗ uf , y ] .The sphere spectrum S , the Eilenberg-Maclane spectra HG and the Thomspectrum M O in example 4.1, as well as the suspensions Σ ∞ X of L -spaces,deloopings B ∞ X of E ∞ -rings and the spectrifications ̃ Ω Y of L -spectra, are all L -spectra with structural morphisms given respectively by: ⃗ uf y S ⃗ u + f ⃗ vHG g a ⊗ ⃗ u + f ⃗ v a M O [⟨ ι fνUU f g i f − ⟩ , t, ⃗ u + f ⃗ v ] Σ ∞ X [ f y, ⃗ u + f ⃗ v ] B ∞ X [[⟨ f ⋉ α a ⟩ , ⟨ f x e ⟩ , ⃗ u + f ⃗ v ] T , t ] ̃ Ω Y [ f ⃗ v ↦ ⃗ u fνV ⊥ f γ ( f − ⃗ u fνV + ⃗ v )] The A -indexed smash product is ∧ L A ∶ Π A L [ Sp ] → L [ Sp ] ; ∧ L A ⟨ Y a ⟩ ∶ = ⟨ L A ⋉ ∧ A Y aU / ⊗ ⃗ uf [ ⃗ vaga y a ] = ⊗ ⃗ u + fa ⃗ va ⟨ faga ⟩ [ y a ] ⟩ with structural maps induced by the ones for the twisted smash product. Inorder to make explicit the parallel between the smash product of spectra withthe tensor product of abelian groups we will use the notation ⊗ ⃗ uf [ y a ] ∶ = [[ ⃗ uf , [ y a ]]] ∈ ∧ L A ⟨ Y a ⟩ , so that the L structural maps are given by the formula ⃗ uf ⊗ ⃗ vg [ y a ] ∶ = ⊗ ⃗ u + f ⃗ v ⟨ fg a ⟩ [ y a ] . For A = Y ∧ L Y ∶ = ∧ L ⟨ Y , Y ⟩ . Associativity follows from the fact that the maps ∧ L A ( ∧ L B a ⟨ Y ab ⟩) → ∧ L Σ A B a ⟨ Y ab ⟩ ; ⊗ ⃗ uf [ ⊗ ⃗ v a g a [ y ab ]] ↦ ⊗ ⃗ u + f a ⃗ v a ⟨ f a g ab ⟩ [ y ab ] are isomorphisms [5, Theorems I.5.4, I.5.5 and I.5.6]. In particular when the B a are a constant set B we have a natural isomorphismΦ A,B ∶ ∧ L A ( ∧ L B ⟨ Y ab ⟩) → ∧ L B ( ∧ L A ⟨ Y ab ⟩) and we set the notation ⊗ ⃗ vg [ ⊗ ⃗ u b f b [ y ab ]] ∶ = Φ A,B ⊗ ⃗ uf [ ⊗ ⃗ v a g a [ y ab ]] . τ Y ,Y ∶ Y ∧ L Y ≅ − → Y ∧ L Y , τ ⊗ ⃗ u ⟨ f ,f ⟩ [ y , y ] ∶ = ⊗ ⃗ u ⟨ f ,f ⟩ [ y , y ] . For all Z ∈ L [ Sp ] set the notation Σ Z ∶ = − ∧ L Z ∶ L [ Sp ] → L [ Sp ] .This smash product almost has a unit given by the sphere spectrum S , inthat there are natural weak equivalences ρ Y ∶ Σ S Y ∼ − → Y, ⊗ ⃗ uf [ y, ⃗ v ] ↦ σ f νU U [ ⃗ f y, ⃗ u + f ⃗ v ] . Unfortunately ρ is not in general a natural isomorphism, though it is a naturalweak equivalence. The category of S -modules is the full subcategory Mod S ∶ = { Y ∈ L [ Sp ] ∣ ρ Y is an isomorphism } . With the same smash product and unit S the category Mod S is a symmetricmonoidal category.From the nontrivial fact that L A / L A has a single equivalence class [5,Theorems I.8.1 and Section XI.2] the sphere spectrum S , the Eilenberg-Maclanespectra HG and the Thom spectrum M O in 4.1, as well as Σ S Y for Y ∈ L [ Sp ] and spetrifications ̃ Ω Y for Y ∈ Mod S , are all S -modules with inverse maps givenrespectively by: ρ − y S ⊗ ⃗ f [ f − ⃗ u, ⃗ ] HG ⊗ ⃗ f [ g a ⊗ f − ⃗ u a , ⃗ ] M O [⟨ f − g i f ⟩ , t, f − ⃗ u ] , ⃗ ] Σ S Y ⊗ ⃗ u ⟨ f ,g ,g ⟩ [ y, g − f ⃗ v, ⃗ ] ̃ Ω Y [ ⃗ v ↦ ρ − γ ⃗ v ] where in the first three lines f ∈ L U ⊂ f R ∞ ,in the fourth ⟨ g , g ⟩ ∈ L ⟨ f , g , g ⟩ ∈ L f νU ⊂ g R ∞ .The functor Σ S ∶ = − ∧ L S is the right adjoint of the inclusion of Mod S in L [ Sp ] . The functor Σ S is also a left adjoint, with right adjoint induced by aclosed structure on L [ Sp ] given by an L -mappings functor F L . Details of thisconstruction can be found in [5, Section I.7], but we give an overview to establishnotation. The twisted half-smash product L A ⋉ − admits a right adjoint, the twisted function spectrum functor F [ L A, − ) ∶ Sp → Sp ⊕ A R ∞ , F [ L ( A ) , Y ) ∶ = ⟨ Lim K ⊂ cpct L ( A ) Y T K ⟨ Ua ⟩ µ ⟨ Ua ⟩ µ ⟨ U a ⟩ ⟩ ,σ ⟨ V a ⟩⟨ U a ⟩ [ ϕ, ⟨ ⃗ v a ⟩] ∶ = ⟨ ⃗ uf ↦ σ µ ⟨ V a ⟩ µ ⟨ U a ⟩ [ ϕ ( ⃗ u + f a ⃗ v a ) µ ⟨ Ua ⟩ f , ( ⃗ u + f a ⃗ v a ) µ ⟨ U a ⟩ ⊥ ]⟩ . U ∈ A we also have a shift functor − [ U ] ∶ Sp R ∞ ⊕ R ∞ → Sp ; Y [ U ] = ⟨ Y U ,U ⟩ , σ U V [ y, ⃗ v ] ∶ = σ U ,U U ,V [ y, ( ⃗ , ⃗ v )] . If Y ∈ L [ Sp ] then F [ L , Y )[ U ] ∈ L [ Sp ] with structural map ⃗ uf ϕ ∶ = ⟨ ⃗ vg ↦ ϕ ⃗ v + g ⃗ u ⟨ g ,g f ⟩ ⟩ . Finally, we can now define F L ( − , − ) ∶ L [ Sp ] op × L [ Sp ] → L [ Sp ] ; F L ( Z, Y ) ∶ = ⟨⎧⎪⎪⎪⎨⎪⎪⎪⎩ φ ∈ L [ Sp ]( Z, F [ L , Y )[ U ]) »»»»»»»»»»» ⃗ uf ( φz ⃗ vg ) = φz ⃗ u + f ⃗ v ⟨ fg ,fg ⟩ ,φ ( ⃗ uf z ) ⃗ vg = φz ⃗ u + g ⃗ v ⟨ g ,g f ⟩ . ⎫⎪⎪⎪⎬⎪⎪⎪⎭⟩ ,σ U V [ φ, ⃗ v ] ∶ = ⟨ z ↦ σ U ,U V ,U [ φz, ( ⃗ v, ⃗ )]⟩ , ⃗ uf φ ∶ = ⟨[ z, ⃗ vg ] ↦ φz ⃗ v + g ⃗ u ⟨ g f,g ⟩ ⟩ . The functor F S ∶ = F L ( S , − ) ∶ Mod S → L [ Sp ] is right adjoint to Σ S .The monoidal structure of S -modules provides a natural definition of ringspectra, module spectra and algebra spectra. Definition 4.4. A commutative ring spectrum R is a commutative monoid in Mod S , ie an S -module equipped with a multiplication map µ ∶ R ∧ L R → R and a unit map η ∶ S → R satisfying natural associative, unit and commutativelaws. The category of commutative ring spectra is denoted CRingSp .For R ∈ CRingSp an R -module M is a module over R , ie an S -moduleequipped with an action λ ∶ R ∧ L M → M , satisfying natural associative andunit laws. The category of R -modules is denoted as Mod R .The category of R -modules admits a monoidal structure with associativeand symmetric tensor product the coequalizer M ∧ R N ∶ = Coeq ( M ∧ L R ∧ L N ⇉ M ∧ L N ) and unit R . The category of R -modules is denoted Mod R .A commutative R -algebra is a commutative monoid in ( Mod R , ∧ R , R ) , andthe category of commutative R -algebra is denoted CAlg R .The category of commutative algebra spectra is defined as CAlgSp ∶ = Σ CRingSp
CAlg R . As in the classical set theoretical setting there is a natural isomorphism
CAlgSp ≅ CRingSp (cid:1) [5, VII.1]. Alternatively we have a monad ( P (cid:1) ; η, µ ) on L [ Sp ] with P (cid:1) Y ⋆ ∶ = ∫ S ⋆ ∧ L A ⟨ Y c a ⟩ ; η ⋆ y ∶ = [ ⊗ ⃗ id y ] , µ ⋆ [ ⊗ ⃗ uf [ ⊗ ⃗ v a g a [ y a,b ]]] ∶ = [ ⊗ ⃗ u + f a ⃗ v a ⟨ f a g ab ⟩ [ y a,b ]] Mod S . The objects of P (cid:1) [ L [ Sp ] ] behave likealgebra spectra over ring spectra except they have units only up to weak equiv-alence and are refered to as E (cid:1) ∞ -algebra spectra, similarly to how algebras in L [ Sp ] over the nonrelative version P of this monad are called E ∞ -ring spectra.By the same argument as in [5, Prop. II.4.5] we have an isomorphism P (cid:1) [ Mod S ] ≅ CAlgSp . (4.2)For R = (( R d , R c ) ; η, µ ) ∈ CAlgSp and ⊗ ⃗ uf [ r a ] ∈ ∧ L A ⟨ R c a ⟩ we will use thenotation ∏ ⃗ uf [ r a ] ∶ = µ [ ⊗ ⃗ uf [ r a ]] . The sphere spectrum S , the Eilenberg-MacLane spectrum of a commutativering HR , the Thom spectrum M O , suspensions Σ ∞ X of L -spaces, deloopings B ∞ X of E ∞ -rings, the S -module Σ S R associated to an E ∞ -ring spectrum R in P [ L [ Sp ]] and spectrifications ̃ Ω R of ring spectra R are all commutative ringspectra with η ⃗ u ∏ ⃗ uf [ y a ] S ⃗ u ⃗ u + f a ⃗ v a HR R ⊗ ⃗ u ∏ A ⟨ ba ⟩ r ab a ⊗ ⃗ u + f a ⃗ v a,b a M O [ id, ∅ , ⃗ u ] [ ∏ A ⟨ ι f a νU a U f a g a,i f − a ⟩ ⋅ δ a , ⊲ A t a , ⃗ u + f a ⃗ v a ] Σ ∞ X [ X , ⃗ u ] [ f [ x a ] , ⃗ u + f a ⃗ v a ] B ∞ X [[ ∅ , X , ⃗ u ] , ∅ ] [[⟨ f a ⋉ ⟨{ id α a,v ⟩⟩ , ⟨ f a [ x a,e ]⟩ , ⃗ u + f a ⃗ v a ] Π A T a ⋅ δ a , ⊲ A t a ] Σ S R ⊗ ⃗ uf [ Y , ⃗ u ] ⊗ ⃗ vg [ ∏ ⃗ u f [ r a ] , ⃗ u + f a ⃗ v a ] ̃ Ω R [ ⃗ v ↦ σ UV [ η ⃗ u, ⃗ v ]] [ f a ⃗ v a ↦ ∏ ⃗ u fνV ⊥ f [ γ a ( f − a ⃗ u f a νV a + ⃗ v a )]] with the implicit conditions in the fifth line as in the formula 3.11.There is a natural isomorphism CAlg S ≅ CRingSp , which is analogous to theisomorphism between commutative rings and commutative Z -algebras. More-over ( M O, HR ) ∈ CAlgSp with ∏ ⃗ uf [[⟨ g ,i ⟩ , t , ⃗ v ] , r b ⊗ ⃗ v ,b ] ∶ = r b ⊗ ⃗ u + f ◦ m g ,i ⃗ v + f ⃗ v ,b . The stable mixed model structure of
Mod S is right transferred from the one in Sp by the adjunction ( Σ S L ⊣ F S ) ∶ Sp ⇋ Mod S as described in [1, 4, 5], so that weak equivalences and fibrations in Mod S are those maps whose underlying spectrum mapping are q -equivalences and33 -fibrations respectively. The Hurewicz/Strøm factorization systems are con-structed as in Sp with the S -module structures of Γ f and E f defined point-wise.The mixed model structure of CAlgSp is right transferred from the one in
Mod S by the adjunction ( P (cid:1) ⊣ U ) ∶ Mod S ⇋ CAlgSp . The Quillen model structure is transferred due to the fact that
CAlgSp has con-tinuous coequalizers and satisfies the “Cofibration Hypothesis” as in [5, Theo-rem VII.4.7]. The Hurewicz/Strøm model structure is transferred since we candefine an algebra structure on Γ f for f ∈ CAlgSp as η ⃗ u ∶ = ( η X ⃗ u, , r ↦ η Y ⃗ u ) ; ∏ ⃗ uf [( x a , t a , γ a )] ∶ = ( ∏ ⃗ uf x a , max A t a , r ↦ ∏ ⃗ uf γ a r ) As in Sp we have that ( Γ; C t , F ) forms an algebraic weak factorization sys-tem in CAlgSp . On the other hand there doesn’t seem to be any naturalalgebra structure on E f such that the h -cofibration/trivial h -fibration factor-ization ( E ; C, F t ) in Sp induces a factorization in CAlgSp . We do have an h -cofibration/ h -equivalence factorization X (cid:31) (cid:127) in X / / X ∧ P X P ( Γ f ∧ [ , ∞ ] + ) ∧ P Γ f Y ≅ ( f ,F t f † ,id ) / / Y and the fact that C t f has the left lifting property against h -fibrations in Mod S induces the left lifting property against h -fibrations in CAlgSp on in X . The map ( f , F t f † , id ) is an h -equivalence, but it is not necessarily an h -fibration. Applying ( Γ; C t , F ) then gives us the h -cofibration/trivial h -fibration factorization X (cid:31) (cid:127) C t ( f ,F t f † ,id ) in X / / Γ ( f , F t f † , id ) ≅ F ( f ,F t f † ,id ) / / / / Y which determines the Hurewicz/Strøm, and therefore also the mixed, modelstructure on CAlgSp . Let E (cid:1) be an E (cid:1) ∞ -operad equipped with an L (cid:1) -action. The functors F S andΣ S induces objectwise adjoint functors F S (cid:1) and Σ S (cid:1) on the morphism categories.We can then define the functorsΩ ∞ , S ∶ CAlgSp → ( L (cid:1) , E (cid:1) )[ Top ] , Ω ∞ , S R ∶ = Ω ∞ F S (cid:1) η ; f [ φ a ] ∶ = [ ⃗ u ↦ ⟨[ ⃗ u , ⃗ vg ] ↦ ∏ ⃗ f φ a [ ⃗ u , ⃗ u ] ⃗ vg ⟩] and B S , ∞ (cid:1) ∶ ( L (cid:1) , E (cid:1) )[ Top ] → AlgSp ; B S , ∞ (cid:1) X ∶ = Σ S (cid:1) B ∞ (cid:1) X. heorem 4.5. There is an idempotent quasiadjunction ( B S , ∞ (cid:1) ⊣ B , ̃ Ω (cid:1) Ω ∞ , S ) ∶ ( H (cid:1) ∞ , L (cid:1) )[ Top ] ⇋ CAlgSp that induces an equivalence of homotopy categories ( L B S , ∞ (cid:1) ⊣ R Ω ∞ , S ) ∶ H o ( H (cid:1) ∞ , L (cid:1) )[ Top ] alg ⇋ H o CAlgSp con . Proof:
The natural weak equivalences η ′ and ǫ ′ are defined as in the proofof theorem 4.2. The other natural transformations of the unit span and counitcospan are η ⋆ [[ α r , ⟨ α v ⟩ , ⟨ x e ⟩] T ,t ] ∶ = [ ⃗ u ↦ {⟨[ ⃗ u , ⃗ vf ] ↦ ⊗ ⃗ vf [[[⟨ α v ⟩ , ⟨ x e ⟩ , α r, − e ′ ⃗ u ] T ≥ e ′ , t ] , ⃗ u ]⟩ ∞ ] ; ǫ ⋆ ,U ⊗ ⃗ uf [[[⟨ α v ⟩ , ⟨ φ e ⟩ , ⃗ v ] T , t ] , ⃗ v ] ∶ = [ ⃗ w ↦ ◦ T α v ⟨ φ e ⟩[ ⃗ v + ⃗ w f , ⃗ v + ⃗ w f ] ⃗ u + ⃗ w f ⊥ f ] . with the conditions in the first formula as in the proof of theorem 4.2 and withthe domain in the last formula any W ∈ A U + f V with V a common domain ofrepresentatives of the loops φ e .That these maps satisfy the conditions for an idempotent quasiadjunctionfollows from the fact that ( Σ S (cid:1) ⊣ F S (cid:1) ) ∶ P (cid:1) [ L [ Sp ] ] ⇋ CAlgSp is a Quillenequivalence and the same argument as for 4.2 and 4.3. (cid:4)
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