The cube axiom and resolutions in homotopy theory

We show that a version of the cube axiom holds in cosimplicial unstable coalgebras and cosimplicial spaces equipped with a resolution model structure. As an application, classical theorems in unstable homotopy theory are extended to this context.


Introduction
The cube axiom holds in a model category M if in every commutative cube the top square is a homotopy pushout if the bottom square is a homotopy pushout and the vertical faces are homotopy pullbacks.It was proved by Mather that the cube axiom is valid in topological spaces with the Strøm structure [M].In many other model categories it is known to hold.Examples include every stable model category, every ∞-topos and the model category of motivic spaces [Ho,Proposition 3.15].
In the model categorical axiomatic approach to Lusternik-Schnirelmann category, due to Doerane, a variant of the cube axiom is to be assumed [Do], [Do2].If it is satisfied many of the classical topological results on Lusternik-Schnirelmann category extend.To know that the cube axiom holds brings some more benefits.There is a good theory of fiberwise localization [C.S].Moreover, in a recent paper of Devalpurka and Haine it is shown that a form of the James splitting and a weak form of the Hilton-Milnor theorem hold under some assumptions in an ∞-category [D.H].The main assumption made is the cube axiom.
In this paper, we investigate in which form the cube axiom holds in two resolution model categories of cosimplicial objects.The first is the one of cosimplicial spaces cS G and the second the one of cosimplicial unstable coalgebras cCA E for a prime field F. The classes of group objects G and E which define the resolutions are the Eilenberg-MacLane spaces of type F and their homology respectively [Bou].These model categories give home to the constructions of unstable Adams spectral sequences of Bousfield and Kan [Bou2], [BK].For this reason, they also play a central role in the obstruction theory for realizations of unstable coalgebras and the construction of moduli spaces for such realizations [Bl], [BRS].Since cCA E and cS G are not known to be cofibrantly generated, the general results of [Re] and [R.S.] do not apply.In fact, it is the class of fibrations which is defined by cocell attachments.
As an application we see that versions of classical theorems of James, Hilton-Milnor and Ganea hold in cCA E and cS G .For the rest of this paper M stands for CA or S and F for E or G.We show the following: Theorem 1.1.The cube axiom holds in cM F for cubes in which the map B → A is 0-connected.
Remark 1.2.That one has to make assumptions on the maps in the cube is not uncommon.Several examples of categories in which such a restriction is necessary for the cube axiom or its dual to hold can be found in [Do, Appendix].A particular important example is the category of commutative differential graded algebras over the rationals.This serves as one model for rational homotopy theory.
Let Z be a fibrant and cofibrant object in cCA E or cS G and write cCA E Z and cS G Z for the model categories of objects over Z with coaugmentation under Z.These are pointed model categories with 0-object Z.
All suspensions, loop spaces, products, smash and wedge products in cM F Z are taken in the derived sense and over and under the 0-object Z. Combining 1.1 with some of the main theorems in [D.H] we obtain the following application.
Theorem 1.3.Let X, Y be objects in cM F Z .Assume that the structure maps X → Z and Y → Z are 0-connected.Then there are equivalences in cM and there are homotopy fiber sequences Let F → E → X be a fibration sequence and write E ∪ CF for the homotopy cofiber of the inclusion of the homotopy fiber Then there is a homotopy fiber sequence In the usual form of the Hilton-Milnor theorem the right hand side of (3) in 1.3 is further decomposed into an infinite product.We prove such a version in 1.4 below.This needs a convergence property which is implied by some connectivity estimates which are derived from the dual Blakers-Massey theorem in [BRS].
where w runs through all basic words in (x, y) and w(X, Y ) is the iterated smash power of X, Y defined by w.
Remark 1.5.Under the additional assumptions that the homotopy groups π * (X), π * (Y ) are free π 0 (Z)-modules and that π * (Z) is concentrated in degree 0 a (dual) Hilton-Milnor theorem was established by Goerss in the category of simplicial unstable algebras [G].He noted that possibly this restriction may be omitted by restructuring the proof along the lines of Milnor's original argument [Mi], [Wh].That is what we did.Computational applications of his theorem where given in [G2] and [G3].Many of them generalize due to 1.4.In particular, 1.4 helps in the computation of André-Quillen cohomology of coabelian objects in cCA E which shows up in the E 2 -term of the unstable Adams spectral sequence.
Remark 1.6.Many of our arguments extend to resolution model categories defined by the spaces in the Ω spectrum of Morava K-theory.The unstable homology operations are also known due to the determination of the Hopf ring of Morava K-theory by Wilson [W].One ingredient which is missing so far is the homotopy excision theorem.
The paper is organized as follows.In section 2 we give background information on the model categories cCA E and cS G .The proof of 1.1 is given in section 3. Section 4 is devoted to the proof of 1.3.Connectivity estimates for relative infinite products, wedge and smash products are derived in section 5.These are needed in the proof of 1.4 which is the content of section 6.

Conventions
Throughout the paper we fix a prime field F. We will use the following notation.
• S = the category of simplicial sets; • V ec = the category of non-negatively graded F-vector spaces • CA = the category of unstable coalgebras over the Steenrod algebra A over F For any category C we let • cC = the category of cosimplicial objects over C; We use freely notions and standard facts about model categories.Besides the founding [Q] some widely used references are [Hi], [H] and [GJ].In section 5 we use the language of ∞-categories.The needed definitions and results are to be found in [L].

Acknowledgments
I would like to thank Hadrian Heine and Markus Spitzweck for patiently answering my questions on ∞-categorical issues.Finally, I would like to thank the Deutsche Forschungsgemeinschaft for support through the Schwerpunktprogramm 1786 "Homotopy theory and algebraic geometry".

Resolution model categories
The categories S and CA are equipped with the standard Quillen and the discrete model structure respectively.Let K(F, m) denote the Eilenberg-MacLane space of type F. A product of those spaces will be called an The categories cS of cosimplicial spaces and cosimplicial unstable coalgebras cCA carry simplicial resolution model category structures relative to G and E respectively [D.K.S.], [Bou].We explain the basic facts about the model categories cS G and cCA E .As any resolution model category they carry an external simplicial structure.
The weak equivalences are the maps of cosimplicial objects which induce an isomorphism on π * [−, F ] for every F ∈ F. This can be equivalently described as maps which induce an isomorphism on π * H * (−) and π * (−) respectively.For E this equivalence is due to the fact that objects in E are cofree unstable coalgebras.
The cofibrations are the Reedy cofibrations such that the induced homomorphism is a fibration of simplicial groups.A concrete description of the fibrations can be found and will be of importance later on.
is surjective for all F-monic i : A → B.
For D ∈ cM and K ∈ S we write D ⊗ K and D K for the tensor and cotensor.Now we can describe the fibrations in cM F .Recall from [BRS,2.3.].
such that for all s ≥ 0, there are homotopy pullback diagrams in the Reedy model structure of cM cosk s (f ) is called a quasi-cocell attachment of dimension s.The map f is called the coattaching map.
A map in cM F is a fibration if and only if it is a Reedy fibration and a retract of a quasi-cofree map [BRS,corollary 2.3.12].We recall the definition of the external loop functor.
Definition 2.4.The collapsed boundary gives the n-sphere S n := ∆ n /∂∆ n a canonical basepoint.If X is pointed in cM we define the s-th external loop object by as the fiber taken at the basepoint of X of the map induced by the basepoint of ∆ n /∂∆ n .
We will often omit the subscript ext when the reference to the external structure is clear from the context.
The following observation from [BRS] will be useful to us later on.Let E be as in 2.3.Then each cosimplicial degree t there is an isomorphism (2.1.3) As a consequence of homotopy excision the model category cM F is proper [BRS,4.4.2 6.2.5].We recall the notion of cosimplicial connectivity.
A pointed object (C, * ) of cM F is n-connected if and only if C is cosimplicially (n − 1)-connected.Every pointed object is cosimplicially 0connected.
The next result from [BRS] is used to reduce questions in cS G to questions in the simpler category cCA E .
Objects in M satisfy the following Fflatness property with respect to the product.
Lemma 2.7.Let X ∈ M and i : A → B an F-injective map in Ho (M).
Proof: Suppose first that M = CA.In this case a map is E-injective if and only if it is injective.This follows from the fact that the homology of an Eilenberg-MacLane space of type F is cofree.The product in CA is the tensor product which is an exact functor of the underlying graded vector spaces.
In case M = S note that a map i : 3 The cube theorem in cCA E and cS G Definition 3.1.Let N be a model category.We say that the cube axiom holds in N if in any commutative cube in N whose bottom square is a homotopy pushout and whose vertical squares are homotopy pull backs the top square is a homotopy push out.
Proposition 3.2.Let C be a cube as in 3.1 where p : E D → D be a quasi-cocell attachment in cM F with coattaching map f and cocell (cF Proof: We may assume that all the vertical squares are pullbacks along fibrations and that D ∼ = A ⊔ B C is a pushout along two cofibrations in cM F [Do, Theorem A.1].Let us start with the case cM F = cCA E .Recall that pullbacks in CA are given by the cotensor product [BRS,remark p.46] and that colimits are computed in the underlying vector spaces.There is a natural map q : In each cosimplcial degree n there are isomorphisms In each cosimplicial degree n the map q n is the chain of isomorphism We have to show that j n is Einjective that is to say injective for all n.But j n is isomorphic to the map k and a trivial fibration w.We have to see that the induced map from the (homotopy Apply the homology functor to the given cube C. The cube H * C so obtained in cCA E has bottom square a homotopy pushout with 0-connected H * (B) → H * (A) and all vertical squares homotopy pullbacks by 2.6 .The top square is a homotopy pushout, as we just have seen above, and the map Definition 3.3.We say that the restricted cube axiom holds in if the assertion in 3.1 holds for all cubes in which the map B → A is 0-connected.
Theorem 3.4.The restricted cube axiom holds in cM F .
Proof: Using the fact that fibrations are retracts of quasi-cofree maps the assertion follows from the coskeletal tower by means of 3.2.
We record some invariance properties of 0-connected maps.
be a homotopy pullback diagram in cM F .If p is 0-connected so is q.
Proof: The assertion for cCA E is in [BRS,Lemma 4.6.3.].For the proof in cS G , note that a map f in cS G is 0-connected if and only if H * (f ) is 0connected in cCA E and that homotopy pullbacks are preserved under H * . .
Proof: We start with the case cCA E and may assume that i is a cofibration.So the map i is a fibration of simplicial groups for each m.By the cofreenes of H * (K(F, m)), this sequence can be identified with the induced map on the dual in degree Because i * is 0-connected it is a π 0 -surjective Kan fibration and hence, surjective.It follows that i is injective.Then clearly ī is injective as well since colimits are formed in the underlying cosimplicial vector spaces or equivalently in the underlying cochain complex.But then the restriction to the cocycles ī : Z 0 (C) = H 0 (C) → Z 0 (P ) = H 0 (P ) is injective.Suppose that the cube is in cS G .Apply H * to get a cube in cCA E which is a homotopy pushout by 2.6.Since H * (i) is 0-connected H * ( ī) is 0-connected and hence so is ī.

Applications of the cube theorem
Let N be a model category and Z an object of N .Denote the categories of objects over and under Z in N by (N ↓ Z) and (N ↑ Z) respectively.If the category M has an initial objects ∅ then ∅ → Z is initial in (N ↓ X) and 1 Z is terminal.If N has a terminal object 1 then Z → 1 is terminal in (N ↑ Z) and 1 Z is initial.
There are model structures on these categories in which a morphism is defined to be a weak equivalence, cofibration or fibration if its image under the forget functor to N is in the corresponding class.Let N be a category and X ∈ N .The category objects coaugmented under Z is defined as For short we denote it by X.
Lemma 4.2.The restricted cube axiom holds in (cM Proof: This follows from 3.4 and the fact all the relevant structure is defined by the forget functor to cM F . Recall that one says that homotopy pushouts are universal in an ∞category if homotopy pushouts are stable under homotopy pullbacks.The following result which is key for us can be found except for the items ( 4) and ( 6) in [D.H].
Theorem 4.4.Let M be an ∞-category with finite limits, pushouts in which the cube axiom holds.Let X and Y be pointed objects in M * and F → E → X a fibration sequence.Then all the assertions made in 1.3 hold true in M where for (2) and (3) we assume in addition that countable coproducts exist in M.
Remark 4.5.In the context of model categories, (3) and ( 5) were proved by Doerane in [Do2] as we learned from [R.S.] where a version of ( 4) is given.
Now we apply 4.4 to the model category cM F Z .Since it is model category enriched over pointed simplicial sets S * the categories of cofibrant and fibrant objects define pointed ∞-categories in the model of simplicial categories.We denote the underlying pointed ∞-categories by cM F Z .They have all small limits and colimits.The restricted cube axiom holds in (cM F ) Z by 3.4 and 4.2.
Proof: of (1.3).One simply checks that under the connectivity assumptions made the proofs in [D.H] go through in the ∞-category cM F Z since the restricted cube axiom is sufficient in all applications of the cube axiom made.For ( 4) and ( 6) one uses 4.3.

Connectivity estimates
In this section we provide the results on cosimplicial connectivity needed in the proof of the Hilton-Milnor theorem.The proofs flow from three sources.First the fact that colimits in CA E Z are formed in the underlying cosimplicial vector spaces.Second 2.6 which enables us to reduce the proofs in S G Z to the ones in CA E Z .Third the dual Blakers-Massey theorem which replaces the generalized Blakers-Massey theorem of [A.B.F.J.] valid in ∞-topoi.
Recall that finite products in CA E Z are defined by the cotensor product over Z. Infinte products are described as the filtered limit of all finite subproducts.Now filtered limits are given as the colimit of all unstable subcoalgebras of finite dimension in the limit of the underlying vector spaces.So all in all J ( ) J C j where J runs through the finite subsets of I and the colimit over the finite dimensional subcoalgebras D a in the vector space limit.This may be seen as in the absolute case to be found for example in [G.4, 1.1.b].Another way to the construction of infinite products is via the cofree functor right adjoint to the forget functor I: as in [A].
Proposition 5.1.Let I be a discrete category and D i ∈ cM F Z fibrant and D i → Z s-connected for all i.Then I D i → Z is s-connected.
Proof: First note that finite products cCA C are defined by the degreewise cotensor product over C and this functor has the asserted property by the spectral sequence [BRS,Theorem 4.5.1 (b)].Since D i is fibrant there is an isomorphism So the limit under consideration is just an ordinary infinite tensor product.
The structure maps of the system in degree n are given by the projections which are clearly surjective.By [BRS,Proposition 3.4.3]we may assume that E n i = F for n ≤ s.Consequently D n i = Z n for n ≤ s.By [We,p.84]there is a short exact sequence for each r For r − 1 ≤ s the derived functor is trivial since in this case the system is constant.So there are isomorphisms for r ≤ s The actual limits in coalgebras are then defined by the colimits over the filtered system of finite dimensional subcoalgebras.Colimits are computed in the underlying cosimplicial vector spaces or equivalently cochain complexes.
Hence, this colimit commutes with π * (−).This implies the assertion in cCA E C because every unstable coalgebra is the colimit of its finite subcoalgebras .
The claim in cS G may be reduced to the case just proved as follows.By the same argument as above the products which show up are the ordinary products of the form (× i∈I G n i ) × Z n with G i ∈ G.We have to estimate the connectivity of the functor π * H * applied to these products.Now recall the fact that homology commutes with infinite products of Eilenberg-MacLane spaces of type F and combine this with the assertion in cCA E just seen.For finite characteristic this fact is a consequence of [Bou3,4.4.] and in characteristic zero it may be deduced from the Milnor-Moore theorem as in the proof of [BRS,Theorem 4.2.2.].This completes the proof.
Lemma 5.2.Let X n ∈ cM F Z with q n : X n → Z k n -connected such that k n ≥ 0 for all n ∈ N. Then ∨X n → Z is at least m-connected with m = min(k n |n ∈ N).
Proof: We may assume that the maps i n : Z → X n are cofibrations.
Consider the case of cCA E Z first.The maps i n : Z → X n are injective as they are 0-connected cofibrations.Consider the exact sequence of cochain complexes for X 1 , X 2 .
is injective for j ≤ 2 the long exact cohomology sequence of 5.0.1 of decomposes in short exact sequences.Since H k j ≤ (i j ) is an isomorphism it follows that there is an isomorphism By an easy induction the assertion holds for any finite subset of N. Now there is an isomorphism and since filtered colimits are exact the assertion holds in cCA E Z .We turn to the case cS G Z .By [BRS,Proposition 6.1.7]the statement for finite subsets of the natural numbers hols there as well.The infinite coproduct cS G Z is again the colimit of its finite subcoproducts and since singular homology commutes with filtered colimits of cofibrations the assertion holds in cS G Z .
We turn to smash powers now.The next result is a consequence of the homotopy excision theorem in cM F [BRS,Theorems 4.6.5,6.2.6.].
Proof: We consider the case cCA E Z .Note that in the abelian category of coaugmented cosimplicial Z-comodules cComod(Z) Z coproducts and products coincide.Consider where the map on the left is the canonical inclusion and the map on the right is the canonical map of Z-comodules.The composition is the identity and q is k + l + 1-connected by [BRS,4.6.5. b)].It follows that i is k + l + 1connected as well.By the homotopy pushout square -connected as a pointed object which was to be seen.We consider the case cS G .Since H * commutes with homotopy pullbacks [BRS,6.2.3.]there are isomorphisms Since H * commutes with homotopy pushouts along pairs of maps one of which is 0-connected 2.6, there are isomorphisms Apply the same fact again to the homotopy pushout square and find an isomorphism Now the assertion follows from the proved statement in cCA E Z .
Lemma 5.4.Let X ∈ cM F Z and assume that X → Z is k-connected.Then ΩΣ(X) → Z is also k-connected.
Proof: We start as usual by considering cCA E Z .The assertion is true in the abelian category cComod(Z) Z since the loop and suspension functors Ω Comod and Σ Comod = Σ are inverse to each other.By [BRS,Theorem 4.6.5. b)] the natural map ΩΣ(X) → Ω Comod Σ(X) is (2k + 1)-connected.This proves the assertion in cCA E Z .

It holds in cS G
Z by an application of 2.6.

6
The Hilton-Milnor theorem in cM F

Z
We start with a preparatory lemma established by Gray in the category of topological spaces [Gr].As a consequence of the James splitting in 1.3 we get: Lemma 6.4.Let ΣX, ΣY ∈ cM F Z be 1-connected as pointed objects.Then there is an equivalence The proof of the Hilton-Milnor theorem below proceeds from here on essentially as in the classical sources [Mi], [Wh] but amplified by the connectivity estimates proved in section 5.These are needed in order to control the convergence of a series of maps which define the asserted equivalence.
We show that h induces an isomorphism on π r () for each r.Choose N such that X m and R m+1 are r-connected for all m ≥ N .Consider the obvious maps i N : P N → P and h N : As a consequence of 5.1, 5. gives the assertion.Remark 6.7.Porter has extended the classical Hilton-Milnor theorem by decomposing the loop space of the fat wedge of suspension spaces [P].Such an extension also holds in cM F Z .
Hence the maps E B → E A and k are 0-connected.By 2.6 again H * (P ) is a homotopy pushout.It follows that the map H * (P ) → H * (E D ) is a weak equivalence which is what we want.

Lemma 4 . 3 .
Homotopy pushouts along pairs of maps one of which is 0connected are universal in cM F .Proof: The proof of [D.H, Lemma 2.5.]applies word for word.
is a homotopy pushout by 4.3 as is its composition withY / / ΣX × Y * / / ΣX ∨ Σ(X ∧ Y ) by [D.H, Corollary 2.21.1].The pasting law for homotopy pushouts implies that the second diagram is a homotopy pushout as well.Corollary 6.2.Let ΣX, Y ∈ cM FZ be 1-connected as pointed objects.Then there is a homotopy fiber sequenceΣX ∨ Σ(X ∧ ΩY ) → ΣX ∨ Y → YProof: This follows from 6.1 together with (6) in 1.3 .Using this we arrive at.Lemma 6.3.Let ΣX, Y ∈ cM F Z be 1-connected as pointed objects.Then there is an equivalenceΩ(ΣX ∨ Y ) ≃ Ω(Y ) × Ω(ΣX ∨ Σ(X ∧ ΩY ))Proof: The equivalence is induced by the inclusion Y → ΣX ∨ Y and the map ΣX ∨ Σ(X ∧ ΩY ) → ΣX ∨ Y .The loops of these maps can be multiplied in Ω(ΣX ∨ Y ) which is a group object in the homotopy category of cM F Z and this map is the searched for equivalence.