aa r X i v : . [ m a t h . A T ] J u l FLATNESS AND SHIPLEY’S ALGEBRAICIZATION THEOREM
JORDAN WILLIAMSON
Abstract.
We provide an enhancement of Shipley’s algebraicization theorem which behavesbetter in the context of commutative algebras. This involves defining flat model structures as inShipley and Pavlov-Scholbach, and showing that the functors still provide Quillen equivalencesin this refined context. The use of flat model structures allows one to identify the algebraiccounterparts of change of groups functors, as demonstrated in forthcoming work of the author.
Contents
1. Introduction 12. Model categorical preliminaries 43. Symmetric spectra in general model categories 64. Flat model structures 75. Shipley’s algebraicization theorem in the flat setting 126. Extension to commutative algebras 147. A symmetric monoidal equivalence for modules 19References 221.
Introduction
Many concepts and constructions in algebra can be understood in a homotopy invariant sense,and the derived category of a ring is the universal category in which to study these. In turn,these homotopy invariant algebraic notions can be translated into stable homotopy theory [13]and this translation to spectral algebra has led to a powerful new point of view on many areassuch as modular representation theory [12, 14]. Robinson [31] showed that the category ofspectra contains ‘extraordinary’ derived categories generalizing the derived category of a ring.Shipley [37] gave a more precise and general version of Robinson’s result in terms of a zig-zagof Quillen equivalences. This paper is a contribution to the understanding of the relationshipbetween spectral and homological algebra.Passing between the worlds of spectral algebra and homological algebra is a valuable technique.It allows the reduction of topological questions to algebraic questions, and conversely, allowsthe importation of algebraic methods to the realm of spectra. Associated to any ring R thereis an Eilenberg-MacLane spectrum HR , and the homological algebra of R is equivalent to thespectral algebra of HR . This relation is particularly striking in the case that R = Q , as therational sphere spectrum is equivalent to H Q .Let R be a commutative ring. It was shown by Shipley [37] that there is a zig-zag of Quillenequivalences between HR -module spectra and chain complexes of R -modules. Moreover, thisis a zig-zag of symmetric monoidal Quillen equivalences, so that it gives a zig-zag of Quillenequivalences between HR -algebra spectra and differential graded R -algebras. Shipley’s alge-braicization theorem shows that spectral algebra is a vast generalization of homological algebra. oreover, it provides a bridge between the worlds of topology and algebra. This bridge hasbeen widely used in the construction of algebraic models for rational equivariant cohomologytheories by Barnes, Greenlees, Kędziorek and Shipley, see [1, 2, 5, 15, 16, 17, 23, 29, 34].By Shipley’s algebraicization theorem, an HR -algebra X corresponds to a differential graded R -algebra Θ X and there is a Quillen equivalence Mod X ≃ Q Mod Θ X . However, if X is in additiona commutative HR -algebra, it does not correspond to a commutative differential graded R -algebra, but rather to a differential graded E ∞ - R -algebra, see [30].When R = Q , more is true. A commutative H Q -algebra X does correspond to a commutativedifferential graded Q -algebra by [37, 1.2]. More precisely, there is zig-zag of natural weakequivalences Θ X ≃ Θ ′ X where Θ ′ X is a commutative DGA. However, despite the fact that thecategories of modules have symmetric monoidal structures, the Quillen equivalence Mod X ≃ Q Mod Θ ′ X is not a symmetric monoidal Quillen equivalence. This is because the upgrading of aQuillen equivalence to the categories of modules involves cofibrant replacement of monoids [33,3.12(1)] which will destroy commutativity and hence the symmetric monoidal structure.The stable model structure on spectra does not behave well with respect to commutative alge-bras, in the sense that for a commutative ring spectrum S , cofibrant commutative S -algebrasare not cofibrant as S -modules in general. Shipley [35] constructed the flat model structure(also called the S -model structure) on symmetric spectra, which does satisfy the property thatcofibrant commutative algebras are cofibrant as modules. Pavlov-Scholbach [28] extended thisto the case of symmetric spectra in general model categories. This extra compatibility betweencommutative algebras and modules provides several useful tools that would otherwise not bevalid. For a concrete example of where this compatibility can be useful, see the next section ofthe introduction.In light of these considerations, the goal of this paper is threefold. Firstly, we show that thezig-zag of Quillen equivalences in Shipley’s algebraicization theorem still holds in flat modelstructures which satisfy the extra compatibility between commutative algebras and modulesdiscussed above. See Section 5 for a precise statement of the zig-zag of symmetric monoidalQuillen equivalences in this first theorem. Theorem 1.1.
There is a zig-zag of symmetric monoidal Quillen equivalences
Mod flat H Q ≃ Q Ch Q where the intermediate categories have the flat model structure. In fact we show that the flat model structures on the intermediate categories are the same asthe stable model structures used by Shipley [37], see Corollary 4.13.Secondly, we use this theorem to give a new proof of the following theorem, which appears inthe body of the paper as Theorem 6.6. In particular, our approach does not pass through thecategory of E ∞ -algebras as in the proof given by Richter-Shipley [30]. Theorem 1.2.
There is a zig-zag of Quillen equivalences between the category of commutative H Q -algebras and the category of commutative rational DGAs. Finally, we prove the following theorem which appears as Theorem 7.2 in the main body of thepaper.
Theorem 1.3.
For a commutative H Q -algebra X there is a zig-zag of weak symmetric monoidalQuillen equivalences Mod X ≃ Q Mod Θ X where Θ X is a commutative DGA. Motivation and related work.
The author’s main motivation comes from the study of al-gebraic models for rational equivariant cohomology theories. A key step in the construction ofalgebraic models is the passage from modules over a commutative H Q -algebra to modules overa commutative DGA via Shipley’s algebraicization theorem. Therefore, a deep understanding f Shipley’s algebraicization theorem provides key insights into the understanding of algebraicmodels for rational equivariant cohomology theories.Working in the flat model structure provides valuable techniques which are not valid in thestable model structure. In forthcoming work [42], the author considers the correspondence ofthe change of groups functors in rational equivariant stable homotopy theory with functors be-tween the algebraic models. In particular, this includes studying how the extension-restriction-coextension of scalars adjoint triple along a map of commutative H Q -algebras θ : S → R behaveswith respect to the Quillen equivalences in Shipley’s algebraicization theorem.The restriction of scalars functor along a map of commutative monoids θ : S → R in a symmetricmonoidal model category is always right Quillen in the model structure right lifted from theunderlying category, but it is not left Quillen in general. If the monoidal unit of the underlyingcategory is cofibrant, then restriction of scalars is left Quillen if and only if R is cofibrantas an S -module. Since a key step in the proof of algebraic models is a formality argumentbased on the fact that polynomial rings are formal as commutative DGAs, one needs to beable to replace R in such a way that it is still a commutative S -algebra, and is cofibrant asan S -module. This replacement is possible in the flat model structure, but not in the stablemodel structure on spectra. Therefore, Theorem 1.1 provides the necessary setup in which toattack the correspondence of functors along the bridge which Shipley’s algebraicization theoremprovides between topology and algebra.The use of the flat model structure allows the extension of the result to commutative algebraobjects, so that we prove a Quillen equivalence between the category of commutative H Q -algebras and the category of commutative rational DGAs. Richter-Shipley [30] prove thatthe category of commutative HR -algebras is Quillen equivalent to the category of differentialgraded E ∞ - R -algebras for any commutative ring R . Since E ∞ -algebras in chain complexes of Q -modules can be rectified to strictly commutative objects, see for example [25, §7.1.4], as acorollary [30, 8.4] of Richter and Shipley’s result one obtains that the category of commuta-tive H Q -algebras is Quillen equivalent to the category of commutative rational DGAs. Wegive a concrete zig-zag of Quillen equivalences which lands naturally in commutative DGAs,bypassing the need for the rectification step. We expect that this direct approach will enablea better understanding of algebraic models for naive-commutative rational G -spectra as stud-ied by Barnes-Greenlees-Kędziorek [3, 4]. White-Yau [41] give an alternative approach to thiszig-zag of Quillen equivalences by using the stable model structure and their theory of liftingQuillen equivalences to categories of coloured operads. The generality of their theory leads tomore stringent hypotheses than our approach, see for example [41, 3.27]. Our approach exploitsthe fact that in the flat model structure, cofibrant commutative algebras forget to cofibrantmodules.Finally we give a concrete zig-zag of symmetric monoidal Quillen equivalences between thecategory of modules over a commutative H Q -algebra and the category of modules over a com-mutative DGA. The result is assumed without proof in the literature, see for example [6, 3.4.4].Due to the importance of this result in the construction of algebraic models, we believe it isvaluable to make the proof explicit. Shipley proved that there is a Quillen equivalence [37, 2.15]between modules over an HR -algebra X and modules over a DGA Θ X for any ring R . In thecase that R = Q , Shipley furthermore proves that Θ X is naturally weakly equivalent to a com-mutative DGA Θ ′ X [37, 1.2]. A dual of a result of Schwede-Shipley [33, 3.12(2)] allows one toconclude moreover that there is a commutative DGA Θ X and a zig-zag of symmetric monoidal Quillen equivalences Mod X ≃ Q Mod Θ X . The fact that this is a symmetric monoidal Quillenequivalence has been a vital ingredient in the construction of symmetric monoidal algebraicmodels, see [6, 3.4.4] and [29, 9.6].
Outline of the paper.
We recall the key background on model categories in Section 2, andon symmetric spectra in general model categories in Section 3. In Section 4, we recall results rom Pavlov-Scholbach [28] which enable the construction of flat model structures on symmetricspectra in general model categories, and apply these results to our cases of interest. Section5 is dedicated to the proof of Theorem 1.1. In Section 6 we extend our results to show thatthe category of commutative H Q -algebras is Quillen equivalent to the category of commutativerational DGAs. Finally, in Section 7 we consider the extension to modules over commutative H Q -algebras. Conventions.
We write the left adjoint above the right adjoint in an adjoint pair displayedhorizontally, and on the left in an adjoint pair displayed vertically.
Acknowledgements.
I am grateful to John Greenlees and Luca Pol for their comments onthis paper and many helpful discussions. I would also like to thank Brooke Shipley and SarahWhitehouse for many useful conversations and suggestions. I am also grateful to the referee fortheir helpful comments and suggestions on the preliminary version of this paper.2.
Model categorical preliminaries
In this section we recall the necessary background on model categories which we require for thepaper.2.1.
Bousfield localization.
Firstly we recall the definitions and key properties of left Bous-field localizations from [18].
Definition 2.1.
Let C be a model category and let S be a collection of maps in C . • An object W in C is S -local if it is fibrant in C and for every s : A → B in S , the naturalmap map( B, W ) → map( A, W ) is a weak equivalence of homotopy function complexes. • A map f : X → Y in C is an S -local equivalence if for every S -local object W , the naturalmap map( Y, W ) → map( X, W ) is a weak equivalence of homotopy function complexes.The left Bousfield localization of C at S (if it exists), denoted L S C , is the model structure on C in which the weak equivalences are the S -local equivalences and the cofibrations are the sameas in C . The fibrant objects are the S -local objects. We call the fibrations the S -local fibrations.The left Bousfield localization of C at S exists if S is a set of maps and C is left proper andcellular [18, 4.1.1], or if S is a set of maps and C is left proper and combinatorial [7, 4.7]. Anyweak equivalence in C is an S -local equivalence, so it follows that the identity functors give aQuillen adjunction C ⇄ L S C . Proposition 2.2 ([18, 3.3.16], [22, 7.21]) . Let C be a model category and S a set of maps in C .(1) If f is an S -local equivalence between S -local objects, then f is a weak equivalence in C .(2) If f is a fibration between S -local objects, then f is an S -local fibration. We now recall a result of Dugger [11, A.2], which when used in conjunction with Proposition 2.2simplifies the process of proving a Quillen adjunction between left Bousfield localizations.
Proposition 2.3.
Let F : C ⇄ D : G be an adjunction, where C and D are model categories.Then G is right Quillen if and only if G preserves fibrations between fibrant objects and allacyclic fibrations. Algebras and modules.
We next recall the theory of (commutative) monoids, (commuta-tive) algebras and modules in symmetric monoidal model categories due to Schwede-Shipley [32]and White [40].Recall that a model category is said to be symmetric monoidal if it has a closed symmetricmonoidal structure and satisfies the following two conditions: pushout-product axiom : if f : A → B and g : X → Y are cofibrations, then the pushout-product map f (cid:3) g : A ⊗ Y [ A ⊗ X B ⊗ X → B ⊗ Y is a cofibration, which is acyclic if either f or g is acyclic;(2) unit axiom : for c → a cofibrant replacement of the unit, the natural map c ⊗ X → ⊗ X ∼ = X is a weak equivalence for all cofibrant X . Definition 2.4.
Suppose that F : C ⇄ D : U is a Quillen adjunction between symmetricmonoidal model categories. We say that ( F, U ) is a weak symmetric monoidal Quillen adjunction if the right adjoint U is lax symmetric monoidal (which gives the left adjoint F an oplaxsymmetric monoidal structure) and the following conditions hold:(1) for cofibrant A and B in C , the oplax monoidal structure map ϕ : F ( A ⊗ B ) → F A ⊗ F B is a weak equivalence in D ;(2) for a cofibrant replacement c C of the unit in C , the map F ( c C ) → D is a weakequivalence in D .If the oplax monoidal structure maps are isomorphisms, then we say that ( F, U ) is a strongsymmetric monoidal Quillen adjunction . We say that (
F, U ) is a weak (resp. strong) symmetricmonoidal Quillen equivalence if (
F, U ) is a weak (resp. strong) symmetric monoidal Quillenadjunction which is also a Quillen equivalence. Note that if F is strong monoidal and the unitof C is cofibrant, then the Quillen pair ( F, U ) is a strong symmetric monoidal Quillen pair.In this paper, we will be particularly interested in the interaction of model structures andQuillen functors with categories of modules and (commutative) algebras. Let ( C , ⊗ , ) be asymmetric monoidal model category. For a monoid S in C , we denote the category of (left) S -modules by Mod S ( C ). If the underlying category is clear, we will instead write Mod S .The categories of modules and algebras often inherit a model structure from the underlyingcategory in the following way. Let F : C ⇄ D : U be an adjunction in which C is a modelcategory and D is a bicomplete category. Kan’s lifting theorem [18, 11.3.2] provides conditionsunder which D inherits a model structure in which a map f in D is a weak equivalence (resp.fibration) if and only if U f is a weak equivalence (resp. fibration) in C . We call such a modelstructure right lifted .Under mild hypotheses, the categories of modules and (commutative) algebras obtain rightlifted model structures. We refer the reader to [32, 2.4] for the precise smallness condition in thefollowing theorem, and instead note that it is satisfied if C is locally presentable. Similarly, werefer the reader to [32, 3.3] and [40, 3.1] for the definitions of the monoid axiom and commutativemonoid axiom respectively. Theorem 2.5 ([32, 4.1], [40, 3.2]) . Let C be a cofibrantly generated, symmetric monoidal modelcategory (with some smallness condition) and let S be a commutative monoid in C .(1) If C satisfies the monoid axiom then the categories of S -modules and S -algebras haveright lifted model structures in which a map is a weak equivalence (resp. fibration) ifand only if it is a weak equivalence (resp. fibration) in C .(2) If C satisfies the commutative monoid axiom and the monoid axiom, then the categoryof commutative S -algebras has a right lifted model structure in which a map is a weakequivalence (resp. fibration) if and only if it is a weak equivalence (resp. fibration) in C . We say that a symmetric monoidal model category C satisfies Quillen invariance of modules if for any weak equivalence θ : S → R of monoids in C , the extension-restriction of scalarsadjunction Mod S Mod RR ⊗ S − θ ∗ s a Quillen equivalence, see [32, 4.3]. Throughout we write θ ∗ = R ⊗ S − for the left adjoint ofthe restriction of scalars functor θ ∗ .2.3. Cofibrations of modules and (commutative) algebras.
In general there is not anexplicit description of the cofibrations in a right lifted model structure, but in many situationsthey have desirable properties.
Theorem 2.6 ([32, 4.1]) . Let C be a symmetric monoidal model category and let S be a commu-tative monoid in C . Every cofibration of S -algebras whose source is cofibrant as an S -module isalso a cofibration of S -modules. In particular, if the unit of C is cofibrant, then every cofibrant S -algebra is a cofibrant S -module. The case of commutative algebras is more subtle. White [40, 3.5, 3.6] has given an answerto this question in general, but it requires stronger assumptions that just the existence of themodel structure on commutative algebras. We recall some relevant examples.
Example 2.7. If S a commutative DGA over a field of characteristic zero and R is a cofi-brant commutative S -algebra, then R is cofibrant (i.e., dg-projective) as an S -module, see forinstance [40, §5.1]. Note that it fails in non-zero characteristic since Maschke’s theorem doesnot apply. Example 2.8.
In categories of spectra the situation is more complicated. It is well known byLewis’ obstruction [24] that the stable model structure on (symmetric) spectra cannot be rightlifted to a model structure on commutative algebra spectra as the sphere spectrum is cofibrant.Instead, one must consider the positive stable model structure in which the sphere spectrum isnot cofibrant. This model structure can be right lifted to give a model structure on commutativealgebras, however, a cofibrant commutative algebra in the positive stable model structure onspectra need not be cofibrant as a module. Nonetheless there is a model structure on spectracalled the flat model structure, for which this property is true, see Corollary 4.11.3.
Symmetric spectra in general model categories
In this section we recall the definition of the category of symmetric spectra in general modelcategories and its properties and stable model structure as in [20]; see also [30, §2].Let ( C , ⊗ , ) be a bicomplete, closed symmetric monoidal category and K ∈ C . Let Σ be thecategory whose objects are the finite sets n = { , . . . , n } for n ≥ ∅ , and whosemorphisms are the bijections of n . The category of symmetric sequences in C is the functorcategory C Σ . The category C Σ inherits a closed symmetric monoidal structure from C via theDay convolution, with tensor product given by( A ⊙ B )( n ) = a p + q = n Σ n × Σ p × Σ q A ( p ) ⊗ B ( q ) . The category of symmetric spectra Sp Σ ( C , K ) is the category of modules over Sym( K ) in C Σ ,where Sym( K ) = ( , K, K ⊗ , · · · ) is the free commutative monoid on K . Therefore, Sp Σ ( C , K )inherits a closed symmetric monoidal structure with tensor product defined by the coequalizer X ∧ Y = coeq ( X ⊙ Sym( K ) ⊙ Y ⇒ X ⊙ Y )of the actions of Sym( K ) on X and Y . More explicitly, an object X of Sp Σ ( C , K ) is a collectionof Σ n -objects X ( n ) ∈ C with Σ n -equivariant maps K ⊗ X ( n ) → X ( n + 1)for all n ≥
0, such that the composite K ⊗ m ⊗ X ( n ) → X ( n + m ) s Σ m × Σ n -equivariant for all m, n ≥
0. Note that taking C = sSet ∗ and K = S recovers theusual notion of symmetric spectra as defined and studied by Hovey-Shipley-Smith [21].We now sketch the construction of the stable model structure on Sp Σ ( C , K ) due to Hovey [20].If C is a left proper and cellular model category, one can equip Sp Σ ( C , K ) with a level modelstructure in which the weak equivalences and fibrations are levelwise weak equivalences andlevelwise fibrations in C respectively [20, 8.2]. One can then left Bousfield localize this levelmodel structure to obtain the stable model structure [20, 8.7]. We call the weak equivalencesin this model structure the stable equivalences and the fibrations the stable fibrations.There is also a positive stable model structure, which allows the construction of right liftedmodel structures on commutative algebras, see for instance [26, §14]. However, these modelstructures do not have the property that cofibrant commutative algebras are cofibrant modules.In order to rectify this, we turn to the flat model structure in the next section. Notation 3.1.
We set notation for the categories of symmetric spectra of interest. • We write Sp Σ = Sp Σ (sSet ∗ , S ) for the category of symmetric spectra in simplicial sets. • We write Sp Σ (s Q -mod) for the category Sp Σ (s Q -mod , e Q S ) where s Q -mod is the cat-egory of simplicial Q -modules and e Q : sSet ∗ → s Q -mod is the functor which takes thelevelwise free Q -module on the non-basepoint simplices. • We write Sp Σ (Ch + Q ) for the category Sp Σ (Ch + Q , Q [1]) where Ch + Q is the category of non-negatively graded chain complexes of Q -modules and Q [1] is the chain complex whichcontains a single copy of Q concentrated in degree 1.4. Flat model structures
In this section we show that the categories used in Shipley’s algebraicization theorem supporta flat model structure. Recall from Example 2.8 that a cofibrant commutative algebra need notbe a cofibrant module in the stable model structure on spectra. To rectify this, Shipley [35]constructs a flat (and a positive flat) model structure on symmetric spectra in simplicial setsin which this property holds. Pavlov-Scholbach [28] extended these flat model structures tosymmetric spectra in general model categories. The flat model structure has the same weakequivalences as the stable model structure on spectra (i.e., the stable equivalences), but hasmore cofibrations. In particular, the identity functor from the stable model structure to the flatmodel structure is a left Quillen equivalence.4.1.
Equivariant model structures.
The stable model structure on symmetric spectra dis-regards the actions of the symmetric groups on each level. Instead, the flat model structureproceeds by remembering this equivariance and building it into the model structure. There aretwo extreme cases: the naive case is where no equivariance is recorded and the genuine case iswhen all equivariance is recorded. The flat model structure on Sp Σ ( C , K ) (when it exists) isbuilt from the blended model structure on G -objects in C which is intermediate between thenaive and genuine structures. Note that some authors refer to this model structure as the mixedmodel structure, but we do not since it is not mixed in the sense of Cole mixing [10].From now on, we assume that C is a pretty small model category [27, 2.1]. We note that thiscondition is satisfied for simplicial sets, simplicial Q -modules and non-negatively graded chaincomplexes of Q -modules.We now recall the conditions needed for the genuine and blended model structures to exist, seefor instance [38]. Let G be a finite group. We write G C for the category of G -objects in C ;that is, the functor category [ BG, C ] where BG is the one-object category whose morphisms areelements of G . Definition 4.1.
We say that C satisfies the weak cellularity conditions for G if the followingare true for all subgroups H, K ≤ G :
1) ( − ) H preserves directed colimits of diagrams in G C where each underlying arrow in C is a cofibration,(2) ( − ) H preserves pushouts of diagrams where one leg is of the form G/K ⊗ f for f acofibration in C ,(3) ( G/K ⊗ − ) H takes generating cofibrations to cofibrations and generating acyclic cofi-brations to acyclic cofibrations.We say that it satisfies the strong cellularity conditions for G if (1) and (2) from above hold,and for any H, K ≤ G and any X ∈ C ,( G/H ⊗ X ) K ∼ = ( G/H ) K ⊗ X. Definition 4.2.
We say that a map f : X → Y in G C is: • a naive weak equivalence if the underlying morphism is a weak equivalence in C ; • a naive fibration if the underlying morphism is a fibration in C ; • a naive cofibration if it has the left lifting property with respect to the naive acyclicfibrations; • a genuine weak equivalence if for every subgroup H of G , the map f H : X H → Y H is aweak equivalence in C ; • a genuine fibration if for every subgroup H of G , the map f H : X H → Y H is a fibrationin C ; • a genuine cofibration if it has the left lifting property with respect to all genuine acyclicfibrations. • a blended fibration if it has the right lifting property with respect to maps which areboth naive weak equivalences and genuine cofibrations.The cellularity conditions control when the genuine model structure on G C exists. Proposition 4.3.
If the weak cellularity conditions hold for C then the genuine weak equiva-lences, genuine cofibrations and genuine fibrations give a cofibrantly generated, model structureon G C called the genuine model structure. Furthermore, if C is proper, then so is the genuinemodel structure on G C , and if C is a monoidal model category with cofibrant unit, then so is thegenuine model structure on G C .Proof. The claim that the genuine model structure exists and is cofibrantly generated is due toStephan [38, 2.6]. The generating cofibrations and generating acyclic cofibrations are given by ∪ H ≤ G { G/H ⊗ i | i ∈ I } and ∪ H ≤ G { G/H ⊗ j | j ∈ J } respectively, where I and J are the setsof generating cofibrations and acyclic cofibrations for C respectively.We now prove that the genuine model structure is left proper. It suffices to prove that in adiagram of pushouts of the form G/H ⊗ A C XG/H ⊗ B D Y p ∼ p where A → B is a generating cofibration for C , the map D → Y is a genuine weak equivalence.This is because as C is pretty small, weak equivalences are closed under transfinite composi-tion [27, 2.2], and therefore the class of maps for which pushing out along them preserves weakequivalences is closed under retracts, pushouts and transfinite compositions. By the secondcellularity condition, after taking K fixed points, the left hand square and the outer rectangleare still pushouts. It follows that the right hand square is also still a pushout.By the third cellularity condition, the left most vertical map is still a cofibration after taking K fixed points. Since cofibrations are stable under pushout, the map C K → D K is a cofibration, nd since C is left proper, we have that D K → Y K is a weak equivalence for all K . Hence thegenuine model structure is left proper. The fact that the model structure is right proper followsimmediately from the fact that fixed points determine fibrations and weak equivalences.We now prove that the genuine model structure is monoidal. Firstly we must show that thepushout-product of two genuine cofibrations is a genuine cofibration. We use the description ofthe generating cofibrations ∪ H ≤ G { G/H ⊗ i | i ∈ I } for the genuine model structure where I isthe set of generating cofibrations for C . Take generating cofibrations G/H ⊗ i : G/H ⊗ A → G/H ⊗ B and G/K ⊗ i ′ : G/K ⊗ X → G/K ⊗ Y for the genuine model structure. Since G/H ⊗ − and
G/K ⊗ − are left adjoints, the pushoutproduct map (
G/H ⊗ i ) (cid:3) ( G/K ⊗ i ′ ) can be identified with the map ( G/H ⊗ G/K ) ⊗ ( i (cid:3) i ′ ),which in turn can be identified with a x ∈ [ H \ G/K ] G/ ( H ∩ xKx − ) ⊗ ( i (cid:3) i ′ )by the double coset formula. Since C is monoidal, i (cid:3) i ′ is a cofibration in C and hence thepushout product map ( G/H ⊗ i ) (cid:3) ( G/K ⊗ i ′ ) is a genuine cofibration as required. It followsby a similar argument that the pushout product of a genuine cofibration with a genuine acycliccofibration is a genuine acyclic cofibration.For the unit axiom, note that the monoidal unit in G C is the unit of C equipped with the trivial G -action. The functor which equips an object with the trivial G -action is left adjoint to the G -fixed points functor, and hence is left Quillen. It then follows that since the unit of C iscofibrant, the unit in G C is genuine cofibrant. (cid:3) We can then localize the genuine model structure to give the blended model structure.
Theorem 4.4.
Let C be a simplicial, proper model category which satisfies the weak cellularityconditions. Then the naive weak equivalences, genuine cofibrations and blended fibrations give aproper, cofibrantly generated model structure on G C which we call the blended model structure.Proof. We apply Bousfield-Friedlander localization [9, 9.3] to the genuine model structure on G C ,with QX = map( EG, b f X ) where b f is a genuine fibrant replacement functor and map denotesthe simplicial cotensor. We must verify that the conditions (A1), (A2) and (A3) from [9, 9.3]are satisfied. Note that a map f : X → Y in G C is a naive weak equivalence if and only if Qf : QX → QY is a genuine weak equivalence. To see this, if f : X → Y is a naive weakequivalence, then map( G, f ) is a genuine weak equivalence. Therefore, map(
Z, f ) is a genuineweak equivalence if Z is built from free cells. Conversely, since EG → ∗ is a naive weakequivalence, if map( EG, f ) is a genuine weak equivalence then f is a naive weak equivalence.The conditions (A1) and (A2) follow from this observation. Since Q preserves fibrations andpullbacks, condition (A3) follows from the right properness of the genuine model structure on G C . (cid:3) Note that [9, 9.3] also gives an explicit description of the blended fibrations as those maps X → Y which are genuine fibrations and have the property that( ⋆ ) X map( EG, b f X ) Y map( EG, b f Y )is a homotopy pullback square. The genuine fibrant replacement ensures that this is equivalentto the square being a homotopy pullback after taking H -fixed points for all H ≤ G . Proposition 4.5.
The blended model structure exists on G C for C = sSet ∗ , s Q - mod and Ch + Q . roof. The categories of based simplicial sets and simplicial Q -modules satisfy the strong cellu-larity conditions by [38, 2.14]. The category of non-negatively graded rational chain complexessatisfies the weak cellularity conditions by [38, 2.19]. Therefore the result follows from Theo-rem 4.4. (cid:3) Finally we note that in these cases, the blended model structure can be identified with theinjective model structure in which the weak equivalences and cofibrations are both underlying.
Proposition 4.6. (i)
A map f in G - sSet ∗ is an underlying cofibration if and only if it is a genuine cofibration. (ii) For C = s Q - mod and Ch + Q , a map f in G C is an underlying cofibration if and only if itis a naive cofibration if and only if it is a genuine cofibration.Proof. Part (i) is well known; see for example [35, 1.2] or [38, 2.16].For part (ii), let C = Ch + Q or s Q -mod and note that we can give the same style of proof sincethey are both Q -additive. From the description of the generating cofibrations of the genuinemodel structure given in the proof of Proposition 4.3, it is clear that any genuine cofibrationis an underlying cofibration. Since any naive cofibration is also a genuine cofibration it followsthat any naive cofibration is an underlying cofibration.We now turn to proving the forward implication. Since any naive cofibration is a genuinecofibration, it suffices to show that if f is an underlying cofibration then it is a naive cofibration.Let f : X → Y be an underlying cofibration in G C . In order to prove that f is a naive cofibrationwe must show that it has the left lifting property with respect to the naive acyclic fibrations.Consider a commutative square X AY B αf hβ in G C , in which h is a naive acyclic fibration. Since f is an underlying cofibration, there is alift θ : Y → A making the diagram commute, but this need not be an equivariant map. Define ϕ : Y → A by ϕ ( y ) = 1 | G | X g ∈ G gθ ( g − y ) . This is an equivariant map, so it remains to check that it is indeed a lift.Since f and α are equivariant maps, ϕ ( f ( x )) = 1 | G | X g ∈ G gθ ( f ( g − x )) = 1 | G | X g ∈ G gα ( g − x ) = α ( x ) . In a similar way, one can show that hϕ = β . Therefore ϕ is a lift, and the map f is a naivecofibration and hence also a genuine cofibration. (cid:3) Corollary 4.7.
For C = s Q - mod and Ch + Q , the blended model structure, injective model struc-ture and the naive model structure on G C are the same.Proof. The weak equivalences in all three model structures are the naive weak equivalences.The cofibrations in each coincide by Proposition 4.6. (cid:3)
We emphasize that in the case of C = sSet ∗ , the blended model structure is the same as theinjective model structure on G C , but is not the same as the naive model structure. Corollary 4.8.
For C = sSet ∗ , s Q - mod and Ch + Q , the blended model structure on G C ismonoidal. roof. Note that in each case, C is monoidal and has cofibrant unit. Since the blended modelstructure is the same as the injective model structure by Proposition 4.6, it is immediate that thepushout-product axiom holds. The unit axiom holds by the same argument as in Proposition 4.3. (cid:3) The flat model structure.
We can equip Sp Σ , Sp Σ (s Q -mod) and Sp Σ (Ch + Q ) with the level flat model structure , in which the weak equivalences (resp. fibrations) are the levelwisenaive weak equivalences (resp. levelwise blended fibrations) [28, 3.1.3]. The cofibrations in thelevel flat model structure are the flat cofibrations ; that is, the maps which have the left liftingproperty with respect to maps which are both levelwise naive weak equivalences and levelwiseblended fibrations. In a similar manner, Sp Σ , Sp Σ (s Q -mod) and Sp Σ (Ch + Q ) can be given the positive level flat model structure in which the weak equivalences (resp. fibrations) are the mapswhich are naive weak equivalences (resp. blended fibrations) for each level n > flat model structure . Theweak equivalences in the flat model structure are the stable equivalences, and the cofibrationsare the flat cofibrations. We call the fibrations in the flat model structure the flat fibrations .Similarly, a left Bousfield localization of the positive level flat model structure gives the positiveflat model structure in which the weak equivalences are also the stable equivalences. Theorem 4.9.
The flat and positive flat model structures on Sp Σ , Sp Σ (s Q - mod) and Sp Σ (Ch + Q ) (and on modules over monoids in these categories) exist. Furthermore, they satisfy Quilleninvariance of modules, and are stable, left proper, symmetric monoidal and combinatorial modelstructures.Proof. Since the genuine cofibrations are the same as the underlying cofibrations by Proposi-tion 4.6, the blended model structure coincides with the injective model structure. The injectivemodel structure is strongly admissible by [28, 2.3.7] and therefore the flat model structure ex-ists by [28, 3.2.1]. Quillen invariance holds by [28, 3.3.9], monoidality follows as it is definedto be a monoidal left Bousfield localization, stability by [28, 3.4.1] and left properness andcombinatoriality follows from [28, 3.4.2]. (cid:3)
We now record some key properties of the flat model structure which we will use throughoutthis paper.
Proposition 4.10. (i)
A map is an acyclic flat fibration if and only if it is a levelwise acyclic flat fibration. (ii)
A map between flat fibrant objects is a flat fibration if and only if it is a levelwise flatfibration. (iii)
The identity functor is a left Quillen equivalence from the stable model structure to theflat model structure.Proof.
Part (i) follows from the fact that left Bousfield localization does not change the acyclicfibrations and part (ii) follows from Proposition 2.2. For part (iii), since the stable modelstructure and the flat model structure have the same weak equivalences, it suffices to show thatany stable cofibration is a flat cofibration. A map is a stable cofibration if and only if it hasthe left lifting property with respect to maps which are levelwise naive acyclic fibrations, and amap is a flat cofibration if and only if it has the left lifting property with respect to maps whichare both levelwise naive weak equivalences and blended fibrations. Any blended fibration is anaive fibration, and therefore a stable cofibration is also a flat cofibration. (cid:3)
Corollary 4.11.
Let S be a commutative monoid in Sp Σ , Sp Σ (s Q - mod) or Sp Σ (Ch + Q ) . Thepositive flat model structure can be right lifted to give a model structure on commutative S -algebras. Moreover, a positively flat cofibrant commutative S -algebra is also flat cofibrant as an S -module. roof. Since the blended model structure coincides with the injective model structure in thesecases by Proposition 4.6, and the injective model structure is strongly admissible [28, 2.3.7],this is a consequence of [28, 4.1, 4.4]. (cid:3)
The flat model structure is a left Bousfield localization of the level flat model structure whereweak equivalences and fibrations are determined levelwise in the blended model structure. Wecan give a characterization of the fibrant objects in the flat model structure.
Proposition 4.12 ([28, 3.2.1]) . An object X of Sp Σ ( C , K ) is flat fibrant if and only if X islevel flat fibrant and X n → Hom(
K, X n +1 ) is a naive weak equivalence where Hom( K, − ) is theright adjoint to K ⊗ − . The following corollary shows that stable model structures on Sp Σ (s Q -mod) and Sp Σ (Ch + Q )satisfy extra compatibility between commutative algebras and modules, unlike the stable modelstructure on Sp Σ . Corollary 4.13.
The flat (resp. positive flat) model structure on Sp Σ (s Q - mod) and Sp Σ (Ch + Q ) is the same as the stable (resp. positive stable) model structure.Proof. The weak equivalences in both the flat and stable model structure are the stable equiv-alences. Therefore it suffices to show that they have the same acyclic fibrations. A map isan acyclic fibration in the stable model structure if and only if it is a levelwise naive acyclicfibration. By Corollary 4.7, this is the case if and only if it is a levelwise acyclic fibration in theblended model structure, i.e., an acyclic flat fibration. (cid:3)
In light of the previous corollary, we could call the model structure which we use on Sp Σ (s Q -mod)and Sp Σ (Ch + Q ) either flat or stable. However, we will often refer to it as the flat model structureto remind the reader of the extra compatibility between commutative algebras and modulesgiven by Corollary 4.11, which we will use throughout the paper.5. Shipley’s algebraicization theorem in the flat setting
In this section we show that the chain of Quillen equivalences given by Shipley [37] for the stablemodel structure are still Quillen equivalences in the flat model structure, for the rational case.The identity functor from the stable model structure to the flat model structure is a left Quillenequivalence by Proposition 4.10. Therefore by the 2-out-of-3 property of Quillen equivalences, itis sufficient to check that we get Quillen adjunctions in the flat model structure. In fact, since thestable and flat model structure are the same on Sp Σ (s Q -mod) and Sp Σ (Ch + Q ) by Corollary 4.13,this reduces to just checking that the first adjunction is a Quillen adjunction. The followingdiagram summarises all of the adjunctions between H Q -modules and chain complexes of Q -modules.( † ) Mod flat H Q Sp Σ (s Q -mod) flat Sp Σ (Ch + Q ) flat Ch Q Mod stable H Q Sp Σ (s Q -mod) stable Sp Σ (Ch + Q ) stable Ch Q Z ≃ Q ϕ ∗ NU = = L D = RZ ≃ QU ϕ ∗ N ≃ Q L D ≃ Q R The functors will be defined throughout the rest of the section.The model structure on simplicial Q -modules is right lifted from simplicial sets along theforgetful functor s Q -mod → sSet ∗ . Applying this functor levelwise gives a forgetful functor e U : Sp Σ (s Q -mod) → Sp Σ . Note that e U Sym( e Q S ) = ( Q , e Q S , e Q S , ... ) which is H Q [21, 1.2.5].Therefore the forgetful functor e U can be viewed as a functor U : Sp Σ (s Q -mod) → Mod H Q . irstly, we show that the forgetful functor e U is right Quillen when Sp Σ is equipped with theflat model structure. Even though the flat model structure and the stable model structureon Sp Σ (s Q -mod) are the same by Corollary 4.13, in order to prove the following it is actuallyconvenient to work with the description of the acyclic fibrations in the flat model structure. Lemma 5.1.
The forgetful functor e U : Sp Σ (s Q - mod) flat → Sp Σflat preserves fibrations, and preserves and detects weak equivalences.Proof.
The forgetful functor preserves and detects weak equivalences by [37, Proof of 4.1]. Wenow show that it preserves the fibrations. By Proposition 2.3 it is sufficient to show that e U preserves the acyclic flat fibrations and the flat fibrations between flat fibrant objects.A map is an acyclic flat fibration if and only if it is a levelwise acyclic flat fibration, so itsuffices to show that the forgetful functor s Q -mod → sSet ∗ preserves naive weak equivalencesand blended fibrations. Since the model structure on s Q -mod is right lifted from sSet ∗ , theforgetful functor preserves naive weak equivalences and genuine fibrations. It remains to checkthe homotopy pullback condition ( ⋆ ), which is an immediate consequence of the fact that theforgetful functor preserves homotopy pullbacks.A flat fibration between flat fibrant objects is a levelwise flat fibration and hence e U sends it to alevelwise flat fibration by the previous paragraph. Therefore, it remains to show that e U preservesflat fibrant objects. Let X be a flat fibrant object in Sp Σ (s Q -mod) flat . By Proposition 4.12, X is level flat fibrant and X n → Hom( e Q S , X n +1 ) is a naive weak equivalence. It followsthat e U X is level flat fibrant, and since e U preserves naive weak equivalences, we also have that e U X n → e U Hom( e Q S , X n +1 ) is a naive weak equivalence. By the e Q ⊣ U adjunction, it followsthat e U X n → Hom( S , e U X n +1 )is a naive weak equivalence, and hence by Proposition 4.12, e U X is flat fibrant. Therefore, e U preserves flat fibrant objects. (cid:3) Corollary 5.2.
The forgetful functor U : Sp Σ (s Q - mod) flat → Mod flat H Q preserves fibrations, and preserves and detects weak equivalences. Recall from [37, 4.3] that the forgetful functor U : Sp Σ (s Q -mod) → Mod H Q has a left adjoint Z defined by Z ( X ) = H Q ⊗ e Q H Q e Q X where H Q is viewed as a e Q H Q -module via the ring map β : e Q H Q → H Q given by the monadstructure on e Q . Proposition 5.3.
The adjunction
Mod flat H Q Sp Σ (s Q - mod) flat ZU is a strong symmetric monoidal Quillen equivalence with the respect to the flat model structures.Proof. The forgetful functor U preserves weak equivalences and fibrations in the flat modelstructure by Corollary 5.2. Therefore, Z ⊣ U is a Quillen adjunction and hence by the 2-out-of-3 property of Quillen equivalences, is a Quillen equivalence; see Diagram ( † ). It is a strongsymmetric monoidal Quillen equivalence as Z is strong symmetric monoidal and the unit H Q is a cofibrant H Q -module. (cid:3) pplying the normalization functor N : s Q -mod → Ch + Q levelwise yields a functor N : Sp Σ (s Q -mod) → Mod N (cid:16) (Ch + Q ) Σ (cid:17) where N = N (Sym( e Q S )). There is a ring map ϕ : Sym( Q [1]) → N induced levelwise by thelax symmetric monoidal structure on N , and therefore composing N and ϕ ∗ gives a functor ϕ ∗ N : Sp Σ (s Q -mod) → Sp Σ (Ch + Q ). This functor has a left adjoint denoted L by [33, §3.3]. It isimportant to note that the left adjoint is not just the composite of the left adjoints of N and ϕ ∗ . Shipley [37, 4.4] shows thatSp Σ (s Q -mod) Sp Σ (Ch + Q ) ϕ ∗ NL is a weak symmetric monoidal Quillen equivalence.The final step is the passage from symmetric spectra in non-negatively graded chain complexesto unbounded chain complexes. The inclusion Ch + Q → Ch Q of non-negatively graded chaincomplexes into unbounded complexes has a right adjoint C called the connective cover. Thisis defined by ( C X ) n = X n for n ≥ C X ) = cycles( X ). Using the connective cover,one defines a functor R : Ch Q → Sp Σ (Ch + Q ) by ( RY ) n = C ( Y ⊗ Q [ n ]). Recall from [37] thatthis functor has a left adjoint D . Moreover, D is strong symmetric monoidal as proved byStrickland [39]. Note that this fact has been subject to some confusion, see [36]. Shipley [37,4.7] shows that Sp Σ (Ch + Q ) Ch Q DR is a strong symmetric monoidal Quillen equivalence where Ch Q is equipped with the projectivemodel structure.Combining the results of this section gives a proof of Theorem 1.1.6. Extension to commutative algebras
Let F : C ⇄ D : G be a weak symmetric monoidal Quillen pair. As G is lax symmetric monoidal,it preserves commutative monoids and therefore gives rise to a functor G : CMon( D ) → CMon( C ).If the Quillen pair is a strong symmetric monoidal Quillen pair, then F also lifts to a functoron commutative monoids. However, when F is only oplax symmetric monoidal, it will notnecessarily preserve commutative monoids.We always equip the category of commutative monoids with the model structure right liftedalong the forgetful functor, see Theorem 2.5. The forgetful functor U : CMon( C ) → C has a leftadjoint given by P C ( X ) = _ n ≥ X ∧ n / Σ n . The adjoint lifting theorem [8, 4.5.6] implies that the lift of G to the categories of commutativemonoids has a left adjoint e F defined by the coequalizer diagram P D F P C X P D F X e F X.
One of the maps is obtained from the counit of the P C ⊣ U adjunction, and the other map isadjunct to the natural map F P C X ∼ = _ n ≥ F ( X ∧ n ) / Σ n → _ n ≥ ( F X ) ∧ n / Σ n ∼ = P D F X btained from the oplax structure on F . Since G preserves commutative monoids, there is anatural isomorphism U G ∼ = GU and by adjunction there is a natural isomorphism P D F ∼ = e F P C . Before we can state a theorem about lifting weak symmetric monoidal Quillen equivalences toQuillen equivalences on commutative monoids, we need to impose a hypothesis.
Hypothesis 6.1.
Let F : C ⇄ D : G be a weak symmetric monoidal Quillen equivalence. Forany cofibrant object X of C , the natural map F ( X ∧ n ) / Σ n → ( F X ) ∧ n / Σ n is a weak equivalence in D . Lemma 6.2.
Let F : C ⇄ D : G be a weak symmetric monoidal Quillen equivalence. Thissatisfies Hypothesis 6.1 if either of the following conditions hold: (i) F : C ⇄ D : G is a strong symmetric monoidal Quillen equivalence; (ii) underlying cofibrant objects in Σ n D are naive cofibrant.Proof. The first part follows immediately from the definition. For the second part, let X becofibrant in C . By definition of a weak symmetric monoidal Quillen pair, the natural map F ( X ∧ n ) → ( F X ) ∧ n is a weak equivalence between cofibrant objects in the injective model struc-ture on Σ n D . By hypothesis, this is moreover a naive weak equivalence between naive cofibrantobjects. From the description of the generating (acyclic) cofibrations given in Proposition 4.3one can see that the orbits functor ( − ) / Σ n : Σ n D → D is left Quillen when Σ n D is equippedwith the genuine model structure. Since the identity is a left Quillen functor from the naivemodel structure on Σ n D to the genuine model structure, it follows that ( − ) / Σ n : Σ n D → D isleft Quillen when Σ n D is equipped with the naive model structure. By Ken Brown’s lemma, itthen follows that F ( X ∧ n ) / Σ n → ( F X ) ∧ n / Σ n is a weak equivalence in D . (cid:3) We now state when weak symmetric monoidal Quillen equivalences lift to Quillen equivalencesbetween the categories of commutative monoids. This result is closely related to work ofSchwede-Shipley, White and White-Yau. Schwede-Shipley [33, 3.12(3)] consider the relatedquestion on associative monoids without the commutativity assumption, White [40, 4.19] pro-vides hypotheses under which strong monoidal Quillen equivalences lift to the categories ofcommutative monoids and White-Yau [41, 5.8] provide hypotheses under which weak monoidalQuillen equivalences lift. The most general of the statements is that of White-Yau where theresult follows from a more general result about lifting Quillen equivalences to categories ofcoloured operads.For orientation in the following statement and proof, the reader might like to consider C beingthe positive flat model structure on spectra and e C being the flat model structure on spectra.The hypotheses are designed in such a way that this example fits into the framework. We notethat we write left adjoint functors on the left in an adjoint pair displayed vertically. Theorem 6.3.
Let F : C ⇄ D : G be a weak symmetric monoidal Quillen equivalence betweencofibrantly generated model categories which satisfy the commutative monoid axiom and themonoid axiom. Suppose that the underlying categories of C and D support other model structuresdenoted e C and e D respectively, with the same weak equivalences, such that C D e C e D F GF G re all Quillen adjunctions. Suppose that cofibrant commutative monoids in C (resp. D ) arecofibrant in e C (resp. e D ), the generating cofibrations I of C have cofibrant source (and hencetarget), the monoidal unit C of C is cofibrant and that Hypothesis 6.1 is satisfied. Then thereis a Quillen equivalence e F : CMon( C ) ⇄ CMon( D ) : G. Proof.
Since the model structures are right lifted, G preserves fibrations and acyclic fibrationsand therefore is right Quillen as a functor CMon( D ) → CMon( C ). Let A be a cofibrant com-mutative monoid in C and B be a fibrant commutative monoid in D . We must show that themap A → GB is a weak equivalence in C if and only if e F A → B is a weak equivalence in D .The adjunction unit of e F ⊣ G gives rise to a map U A → U G e F A ∼ = GU e F A and hence byadjunction there is a natural map
F A → e F A where we neglect to write the forgetful functors.The composite
F A → e F A → B is adjunct to the map A → GB in C .Let X ∈ D . Write f X for a fibrant replacement of X in D and e f X for a fibrant replacement of X in e D . Consider the square X e f Xf X ∗ in which the left vertical arrow is an acyclic cofibration in D , and the right vertical is a fibrationin e D and hence in D . By lifting properties, we obtain a map f X → e f X which is a weakequivalence.We must show that the map A → GB is a weak equivalence in C if and only if e F A → B is aweak equivalence in D , where A is cofibrant in CMon( C ) and B is fibrant in CMon( D ). By theprevious paragraph, we have a weak equivalence B → e f B where e f B is fibrant in e D and hencein D . By Ken Brown’s lemma, GB → G e f B is a weak equivalence, and therefore A → GB is aweak equivalence if and only if A → G e f B is a weak equivalence.Note that since C and e C have the same weak equivalences, the identity functor C → e C is aleft Quillen equivalence, and similarly for D . Therefore, by the 2-out-of-3 property of Quillenequivalences, F : e C ⇄ e D : G is a Quillen equivalence. Since A is cofibrant in CMon( C ) andhence in e C , and e f B is fibrant in e D , A → G e f B is a weak equivalence if and only if F A → e f B is a weak equivalence. Since B → e f B is a weak equivalence, F A → e f B is a weak equivalenceif and only if F A → B is a weak equivalence. Since the composite F A → e F A → B is adjunctto the map A → GB in C , it follows that it is enough to show that λ A : F A → e F A is a weakequivalence.As C is cofibrantly generated, A is a retract of a P C ( I )-cell complex where I is the set ofgenerating cofibrations for C , i.e., ∅ → A is a retract of a transfinite composition of pushouts ofmaps in P C ( I ). We proceed by transfinite induction on the transfinite composition which definesa cofibrant object. The base case is the claim that F ( C ) → e F ( C ) is a weak equivalence. Theleft adjoint e F takes the initial object C of CMon( C ) to the initial object D of CMon( D ). Since C is cofibrant, F ( C ) → D is a weak equivalence by the unit axiom of the weak monoidalQuillen adjunction F ⊣ G . Therefore the base case holds.Write P n X = X ∧ n / Σ n , so that P X = ∨ n ≥ P n X. By Hypothesis 6.1, if X is a cofibrant objectof C , F ( P n C X ) = F ( X ∧ n ) / Σ n → ( F X ) ∧ n / Σ n = P n D ( F X ) is a weak equivalence. Since X iscofibrant in C , P n C X is cofibrant in e C and therefore F ( P n C X ) is cofibrant in e D . In a similarway, one sees that P n D ( F X ) is cofibrant in e D . Therefore F ( X ∧ n ) / Σ n → ( F X ) ∧ n / Σ n is aweak equivalence between cofibrant objects in e D and Ken Brown’s lemma shows that taking oproducts preserves this weak equivalence. Therefore, F P C X = F _ n ≥ X ∧ n / Σ n ∼ = _ n ≥ F ( X ∧ n ) / Σ n ∼ −→ _ n ≥ ( F X ) ∧ n / Σ n = P D F X is a weak equivalence. Hence using the isomorphism P D F X ∼ = e F P C X , if X is cofibrant in C ,one sees that both F P C X → e F P C X and F P n C X → e F P n C X are weak equivalences.We now prove that if P C X Y P C X ′ P f is a pushout square in CMon( C ), and F Y → e F Y is a weak equivalence where Y is cofibrant,then F P → e F P is a weak equivalence. Since I consists of cofibrations with cofibrant source,we may assume that X and X ′ are cofibrant in C . By [40, B.2], f : Y → P has a filtration Y = P → P → · · · where P n − → P n is defined by the pushout Y ∧ Q n ( f ) / Σ n P n − Y ∧ P n C X ′ P n in C . For our purposes, it is not important precisely what Q n ( f ) is, apart from the fact that itis a colimit of a punctured n -dimensional cube whose vertices are given by tensor products of X , X ′ and Y . It follows from the commutative monoid axiom that Q n ( f ) / Σ n is cofibrant in C ,see [40, Proof of 4.17] for details.Since F sends pushouts in C to pushouts in D , F ( Y ∧ Q n ( f ) / Σ n ) F P n − F ( Y ∧ P n C X ′ ) F P n is a pushout in D . Since e F preserves pushouts of commutative monoids, P D F X e F Y P D F X ′ e F P e F f is a pushout in CMon( D ) using the isomorphism P D F ∼ = e F P C . Applying [40, B.2] again, weobtain a filtration e F Y = R → R → · · · of e F f : e F Y → e F P where R n − → R n is defined bythe pushout e F Y ∧ Q n ( e F f ) / Σ n R n − e F Y ∧ P n D F X ′ R n in D . This filtration is compatible with the filtration of Y → P and therefore λ Y sends F P n to R n . By applying F to the pushout square shown Diagram 1 in [40, Proof of A.1] and using that F ⊣ G is a weak monoidal Quillen pair, one argues by induction that there is a natural weakequivalence F Q n ( f ) ∼ −→ Q n ( e F f ) . Similarly, since taking orbits commutes with taking pushouts, here is a natural weak equivalence F Q n ( f ) / Σ n ∼ −→ Q n ( e F f ) / Σ n by an inductive argument onDiagram 6 in [40, Proof of A.3].We now show by induction that λ P n : F P n → R n is a weak equivalence. The base case holdssince λ P = λ Y which was a weak equivalence by assumption. Suppose that λ P n − is a weakequivalence. Consider the diagram F ( Y ∧ Q n ( f ) / Σ n ) e F Y ∧ Q n ( e F f ) / Σ n F P n − R n − F ( Y ∧ P n C X ′ ) e F Y ∧ P n D F X ′ F P n R n in which the leftmost face and the rightmost face are pushouts in D . The horizontal map F P n − → R n − is a weak equivalence by the inductive hypothesis.The horizontal map F ( Y ∧ Q n ( f ) / Σ n ) → e F Y ∧ Q n ( e F f ) / Σ n factors as the composite F ( Y ∧ Q n ( f ) / Σ n ) → F Y ∧ F Q n ( f ) / Σ n → e F Y ∧ F Q n ( f ) / Σ n → e F Y ∧ Q n ( e F f ) / Σ n where the first map is a weak equivalence since F ⊣ G is a weak monoidal Quillen pair. The map F Y → e F Y is a weak equivalence between cofibrant objects in e D , and the map F Q n ( f ) / Σ n ∼ −→ Q n ( e F f ) / Σ n is a weak equivalence between cofibrant objects in D . Since cofibrant objects in D are also cofibrant in e D , both of these maps are weak equivalences between cofibrant objectsin e D . By Ken Brown’s lemma, tensoring with cofibrant objects preserves weak equivalencesbetween cofibrant objects, and hence the second and third map are weak equivalences.The horizontal map F ( Y ∧ P n C X ′ ) → e F Y ∧ P n D F X ′ is a weak equivalence since it factors as thecomposite F ( Y ∧ P n C X ′ ) → F Y ∧ F P n C X ′ → e F Y ∧ e F P n C X ′ ∼ = e F Y ∧ P n D F X ′ . Therefore, the map
F P n → R n is a weak equivalence by [19, 5.2.6]. Each filtration map is acofibration between cofibrant objects and hence by Ken Brown’s lemma and [19, 5.1.5], the map F P → e F P is a weak equivalence.It remains to show that the property is preserved under the transfinite compositions used tobuild relative cell complexes which again follows from [19, 5.1.5]. Therefore for any cofibrantcommutative monoid object A of C , we have that the map F A → e F A is a weak equivalencewhich concludes the proof. (cid:3)
Remark 6.4.
The hypothesis that C and D satisfy the commutative monoid axiom and themonoid axiom ensures that the categories of commutative monoids inherit a right lifted modelstructure [40, 3.2]. Remark 6.5.
In some cases such as rational chain complexes, cofibrant commutative algebrasare cofibrant as modules, see Example 2.7. In such examples, one can take C = e C in the previoustheorem. Theorem 6.6.
There is a zig-zag of Quillen equivalences between the category of commutative H Q -algebras and the category of commutative rational DGAs.Proof. Consider the adjunctionsMod pf H Q Sp Σ (s Q -mod) pf Sp Σ (Ch + Q ) pf Ch Q Z ϕ ∗ NU L DR here pf denotes the positive flat model structure. Recall from Corollary 4.13 that on Sp Σ (s Q -mod)and Sp Σ (Ch + Q ) the positive stable and positive flat model structures are the same.Firstly, we must justify that these are Quillen adjunctions. The adjunction D ⊣ R is Quillensince it can be viewed as the composite of Quillen adjunctionsSp Σ (Ch + Q ) pf Sp Σ (Ch + Q ) Ch Q DR where the second adjunction was proved to be Quillen in [37, 4.7].For the adjunction Z ⊣ U , by Proposition 2.3 it is sufficient to check that the right adjoint U preserves acyclic positive flat fibrations and positive flat fibrations between positive flat fibrants.Recall that a map is an acyclic positive flat fibration if and only if it is a levelwise acyclic blendedfibration for levels n >
0, and that a map is a positive flat fibration between positive flat fibrantsif and only if it is a levelwise blended fibration for levels n > X of Sp Σ ( C , K ) is positively flat fibrant if and only if X islevelwise blended fibrant for levels n > X n → Hom(
K, X n +1 ) is a naive weak equivalencefor all n ≥ K, − ) is the right adjoint to K ⊗ − . One notes that all the conditionsthat must be checked, except for the last condition, are all levelwise. Therefore, applying thearguments given in Lemma 5.1 and to levels n > X n → Hom(
K, X n +1 ) is a naive weak equivalence is unchangedbetween the flat model structure and the positive flat model structure. This condition was alsoverified in Lemma 5.1. For the L ⊣ ϕ ∗ N adjunction one can argue similarly, using [37, 4.4].We now apply Theorem 6.3. For each of the categories of symmetric spectra, we take C to bethe version equipped with the positive flat model structure, and e C to be equipped with the flatmodel structure. For the category of chain complexes, we take C = e C . In each case, cofibrantcommutative algebras forget to flat cofibrant modules by Corollary 4.11. Hypothesis 6.1 holdsfor the first and last adjunctions since they are strong symmetric monoidal Quillen equivalencesand therefore they give Quillen equivalences on the commutative monoids by Theorem 6.3.For the L ⊣ ϕ ∗ N adjunction, we argue that condition (ii) in Lemma 6.2 holds. We show that fora finite group G , if f : X → Y is an underlying cofibration in G -Sp Σ (s Q -mod) pf then f is a naivecofibration in G -Sp Σ (s Q -mod) pf . A G -object X in Sp Σ (s Q -mod) consists of G × Σ n -objects X ( n )in s Q -mod with G × Σ n -equivariant structure maps. Similarly, a map ϕ : X → Y between objectsin G -Sp Σ (s Q -mod) consists of a collection of G × Σ n -equivariant maps ϕ ( n ) : X ( n ) → Y ( n )making the evident diagrams commute.Write U for the forgetful functor G -Sp Σ (s Q -mod) → Sp Σ (s Q -mod). Suppose that f : X → Y is an underlying cofibration in G -Sp Σ (s Q -mod) pf , i.e., U f : U X → U Y is a positive flatcofibration, and that p : A → B is a naive acyclic fibration in G -Sp Σ (s Q -mod) pf , i.e., U p isan acyclic positive flat fibration. Therefore
U f has the left lifting property with respect to
U p . It remains to argue that the lift θ : U Y → U A can be made into an equivariant map ϕ : Y → A . The lift θ : U Y → U A is a collection θ ( n ) : Y ( n ) → A ( n ) of Σ n -equivariant maps.Since the maps are determined levelwise, one can apply the averaging method as in the proofof Proposition 4.6 to construct G × Σ n -equivariant maps ϕ ( n ) : Y ( n ) → A ( n ) and it followsthat ϕ is a map in G -Sp Σ (s Q -mod) which is also a lift. Therefore, f is a naive cofibrationin G -Sp Σ (s Q -mod). Hence by Theorem 6.3, the middle Quillen equivalence also lifts to thecommutative monoids. (cid:3) A symmetric monoidal equivalence for modules
In this section, we give a symmetric monoidal Quillen equivalence between the categories ofmodules over a commutative H Q -algebra and a commutative DGA. We note that this resulthas been assumed without proof in the literature; for more details see the introduction. We rstly explain why this result is not an immediate corollary of the zig-zag of Quillen equivalencesMod H Q ≃ Q Ch Q .Let F : C ⇄ D : G be a strong symmetric monoidal Quillen equivalence and suppose that theunit objects of C and D are cofibrant. If S is a cofibrant monoid in C , Schwede-Shipley [33,3.12] show that F : Mod S ( C ) ⇄ Mod
F S ( D ) : G is a Quillen equivalence. Now suppose that S is a commutative monoid in C , which is not cofibrant as a monoid. Since S is commutative,the category Mod S ( C ) of modules is symmetric monoidal, with tensor product defined by thecoequalizer of the two maps M ⊗ S N = coeq( M ⊗ S ⊗ N M ⊗ N )defined by the action of S on M and N .However, a cofibrant replacement q : cS ∼ −→ S as a monoid will no longer be commutative, andhence the zig-zag of Quillen equivalencesMod S ( C ) Mod cS ( C ) Mod F cS ( D ) q ∗ −⊗ cS S FG cannot be symmetric monoidal. We explain how to rectify this.Before we can prove the desired symmetric monoidal Quillen equivalence, we require an abstractlemma about lifting symmetric monoidal Quillen equivalences to the categories of modules. Wenote that this first statement is a counterpart to [33, 3.12(2)]. The proof is effectively the same.
Lemma 7.1.
Let
C D FG be a strong symmetric monoidal Quillen equivalence and let S be a commutative monoid in C . Suppose that C and D satisfy the monoid axiom. If F preserves all weak equivalences andQuillen invariance holds in C and D , then Mod S Mod
F SFG is a strong symmetric monoidal Quillen equivalence.Proof.
Let q : cS → S be a cofibrant replacement of S as a monoid in C . As F preserves allweak equivalences F q : F cS → F S is a weak equivalence. Consider the diagram of left Quillenfunctors Mod S Mod cS Mod
F S
Mod
F cSF S ⊗ cS − FF S ⊗ F cS − which is commutative since F is strong monoidal. By [33, 3.12(1)] the right hand vertical is aQuillen equivalence, and by Quillen invariance the horizontals are Quillen equivalences. Henceby 2-out-of-3 the left vertical is a Quillen equivalence as required. As a functor between themodule categories, F is strong symmetric monoidal since the tensor product in the modulecategory Mod S is defined by a coequalizer which F preserves. ThereforeMod S Mod
F SFG is a strong symmetric monoidal Quillen equivalence. (cid:3) e recall from Shipley [37, 1.2] the zig-zag of natural weak equivalences between Zc and α ∗ e Q where α is the ring map H Q → e Q H Q induced by the unit of the monad structure on e Q . Let β : e Q H Q → H Q be the ring map induced by the multiplication map of the monad structure.We have Zc = β ∗ e Q c ∼ = α ∗ β ∗ β ∗ e Q c since βα = 1. There is then a natural map α ∗ e Q c → α ∗ β ∗ β ∗ e Q c arising from the unit of the β ∗ ⊣ β ∗ adjunction. This is a weak equivalence since e Q preservescofibrant objects, the β ∗ ⊣ β ∗ adjunction is a Quillen equivalence and α ∗ preserves all weakequivalences. Finally there is a natural map α ∗ e Q c → α ∗ e Q which is a weak equivalence as α ∗ and e Q preserve all weak equivalences. We can now apply the previous lemma to obtain thedesired statement. Theorem 7.2.
Let A be a commutative H Q -algebra. There are zig-zags of weak symmetricmonoidal Quillen equivalences Mod stable A ≃ Q Mod Θ A and Mod flat A ≃ Q Mod Θ A where Θ A = Dϕ ∗ N α ∗ e Q A is a commutative DGA.Proof. The proof for each part of the theorem follows the same method. Namely, we apply [33,3.12(2)] together with Lemma 7.1 to the underlying Quillen equivalences given by Shipley [37]in the stable case, and given by Theorem 1.1 in the flat case. Since the weak equivalences inboth the stable model structure and the flat model structures are the same, the following proofapplies in both cases.The first step is the adjunctionMod A (Mod H Q ) Mod e Q A (cid:16) Mod e Q H Q (cid:17) . e Q U Since e Q preserves all weak equivalences, this is a strong symmetric monoidal Quillen adjunctionby Lemma 7.1.Recall that there is a ring map α : H Q → e Q H Q . Since α ∗ is lax symmetric monoidal it givesrise to a functor Mod e Q A (cid:16) Mod e Q H Q (cid:17) α ∗ −→ Mod α ∗ e Q A (Sp Σ (s Q -mod)) . It follows from [33, §3.3] that the left adjoint to α ∗ at the level of modules, is given by α e Q A ∗ ( M ) = e Q A ⊗ α ∗ α ∗ e Q A α ∗ M . We claim that α e Q A ∗ is strong monoidal. As α ∗ preserves colimits and is strongmonoidal, we have α ∗ ( M ⊗ α ∗ e Q A N ) = α ∗ coeq( M ⊗ α ∗ e Q A ⊗ N ⇒ M ⊗ N ) ∼ = coeq( α ∗ M ⊗ α ∗ α ∗ e Q A ⊗ α ∗ N ⇒ α ∗ M ⊗ α ∗ N )= α ∗ M ⊗ α ∗ α ∗ e Q A α ∗ N. From this, one sees that α e Q A ∗ ( M ⊗ α ∗ e Q A N ) ∼ = α e Q A ∗ ( M ) ⊗ e Q A α e Q A ∗ ( N )and hence α e Q A ∗ is strong symmetric monoidal. Since α ∗ preserves all weak equivalences, itfollows from [33, 3.12(2)] thatMod e Q A (cid:16) Mod e Q H Q (cid:17) Mod α ∗ e Q A (Sp Σ (s Q -mod)) α ∗ α e Q A ∗ is a strong symmetric monoidal Quillen equivalence. he next step is the passage along the Dold-Kan type equivalence. Recall that applying thenormalization functor levelwise gives a lax monoidal functorSp Σ (s Q -mod) → Mod N (cid:16) (Ch + Q ) Σ (cid:17) where N = N (Sym( e Q S ), and that there is a ring map ϕ : Sym( Q [1]) → N . The composite ϕ ∗ N : Sp Σ (s Q -mod) → Sp Σ (Ch + Q ) is lax monoidal.Let S be a commutative monoid in Sp Σ (s Q -mod). We now show that the induced functor ϕ ∗ N : Mod S (Sp Σ (s Q -mod)) → Mod ϕ ∗ NS (Sp Σ (Ch + Q ))on the categories of modules is lax symmetric monoidal. Recall that colimits in categories ofmodules are calculated in the underlying category of symmetric spectra where they are computedlevelwise. Therefore, N preserves colimits as it is an equivalence of categories s Q -mod → Ch + Q .The restriction of scalars ϕ ∗ also preserves colimits since it is left adjoint to the coextension ofscalars functor. Therefore we have the following map ϕ ∗ N A ⊗ ϕ ∗ NS ϕ ∗ N B = coeq( ϕ ∗ N A ⊗ ϕ ∗ N S ⊗ ϕ ∗ N B ⇒ ϕ ∗ N A ⊗ ϕ ∗ N B ) → coeq( ϕ ∗ N ( A ⊗ S ⊗ B ) ⇒ ϕ ∗ N ( A ⊗ B )) ∼ = ϕ ∗ N (coeq( A ⊗ S ⊗ B ⇒ A ⊗ B )= ϕ ∗ N ( A ⊗ S B )giving ϕ ∗ N a lax symmetric monoidal structure as a functor between the categories of modules.We now must show that L S ⊣ ϕ ∗ N is a weak monoidal Quillen pair, where L S denotes the leftadjoint of ϕ ∗ N . We use the criteria [33, 3.17]. Since the monoidal unit ϕ ∗ N S is cofibrant inMod ϕ ∗ NS the first condition is that L S ϕ ∗ N S → S is a weak equivalence. Since ϕ ∗ N preservesall weak equivalences [37, 4.4] and ϕ ∗ N S is cofibrant, this map is the derived counit of theQuillen equivalence L S ⊣ ϕ ∗ N and as such is a weak equivalence. The second condition holdssince ϕ ∗ N S is a generator for the homotopy category of Mod ϕ ∗ NS . By taking S = α ∗ e Q A in the previous discussion, the adjunctionMod α ∗ e Q A (Sp Σ (s Q -mod)) Mod ϕ ∗ Nα ∗ e Q A (Sp Σ (Ch + Q )) ϕ ∗ NL α ∗ e Q A is a weak symmetric monoidal Quillen adjunction. Since ϕ ∗ N preserves all weak equiva-lences [37, 4.4], it follows from [33, 3.12(2)] that this is moreover a weak symmetric monoidalQuillen equivalence.The final step in the zig-zag is the adjunctionMod ϕ ∗ Nα ∗ e Q A (Sp Σ (Ch + Q )) Mod Θ A (Ch Q ) DR which is a strong symmetric monoidal Quillen equivalence by Lemma 7.1, since D preserves allweak equivalences rationally [37, 4.8]. (cid:3) References [1] D. Barnes. Classifying rational G -spectra for finite G . Homology Homotopy Appl. , 11(1):141–170, 2009.[2] D. Barnes. Rational O (2)-equivariant spectra. Homology Homotopy Appl. , 19(1):225–252, 2017.[3] D. Barnes, J. P. C. Greenlees, and M. Kędziorek. An algebraic model for rational naive-commutative ring SO (2)-spectra and equivariant elliptic cohomology. arXiv e-prints , Oct 2018. arXiv:1810.03632.[4] D. Barnes, J. P. C. Greenlees, and M. Kędziorek. An algebraic model for rational naïve-commutative G -equivariant ring spectra for finite G . Homology Homotopy Appl. , 21(1):73–93, 2019.[5] D. Barnes, J. P. C. Greenlees, and M. Kędziorek. An algebraic model for rational toral G–spectra.
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