On the James and Hilton–Milnor Splittings, and the Metastable EHP Sequence

This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton–Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results ΣΩΣX ≃ ΣX ∨ (X ∧ ΣΩΣX) and Ω(X ∨ Y ) ≃ ΩX × ΩY × ΩΣ(ΩX ∧ ΩY ) in the maximal generality of an ∞-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather’s Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton–Milnor Splittings. Moreover, working in this generality shows that the James and Hilton–Milnor splittings hold in many new contexts, for example in: elementary ∞-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting results in this last context extend Wickelgren and Williams’ splitting result for motivic spaces over a perfect field. We also give two proofs of the metastable EHP sequence in the setting of ∞-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers–Massey Theorem, while the second reduces to the classical computational proof. 2020 Mathematics Subject Classification: 55P35, 55P40, 55P99, 55Q20, 18N60, 14F42

Abstract. This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton-Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results and Ω(X ∨ Y ) ≃ ΩX × ΩY × ΩΣ(ΩX ∧ ΩY ) in the maximal generality of an ∞-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather's Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton-Milnor Splittings. Moreover, working in this generality shows that the James and Hilton-Milnor splittings hold in many new contexts, for example in: elementary ∞-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting results in this last context extend Wickelgren and Williams' splitting result for motivic spaces over a perfect field. We also give two proofs of the metastable EHP sequence in the setting of ∞-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers-Massey Theorem, while the second reduces to the classical computational proof.

Introduction
A classical result of James shows that given a pointed connected space X, the homotopy type ΣΩΣX given by suspending the loopspace on the suspension of X splits as a wedge sum of suspensions of smash powers of X [8,19]. Hilton and Milnor proved a related splitting result [16;17;28,Theorem 3]: given pointed connected spaces X and Y , they showed that there is a homotopy equivalence In the classical setting, these splitting results follow from combining a connectedness argument using the hypothesis that X and Y are connected with the following more fundamental splittings: given any pointed spaces X and Y , there are natural equivalences and Ω(X ∨ Y ) ≃ ΩX × ΩY × ΩΣ(ΩX ∧ ΩY ) . (1.4) In general, the infinite splitting ΣΩΣX ≃ i≥1 ΣX ∧i need not hold; roughly speaking, the problem is that if X is not connected, then (X ∧n ∧ ΣΩΣX) need not vanish as n → ∞. However, there is always a natural map i≥1 ΣX ∧i → ΣΩΣX.
Theorem 1.6 (Fundamental Hilton-Milnor Splitting; Theorem 3.1). Let X be an ∞-category with finite limits and pushouts, and assume that Mather's Second Cube Lemma holds in X. Then for every pair of pointed objects X, Y ∈ X * there is an natural equivalence Ω(X ∨ Y ) ≃ ΩX × ΩY × ΩΣ(ΩX ∧ ΩY ) .
In § 4.2 we explain in what generality the the infinite James Splitting (1.1) and Hilton-Milnor Splitting (1.2) hold, and how to deduce them from Theorems 1.5 and 1.6. It might seem that knowing that the James and Hilton-Milnor Splittings in this level of generality is of dubious advantage; the settings in which one is most likely to want to apply these splittings are the ∞-category Spc of spaces (where the results are already known), or an ∞-topos (where the results follows immediately from the results for Spc; see § 4.2). However, algebraic geometry provides an example that does not immediately follow from the result for spaces: motivic spaces. The obstruction is that the ∞-category of motivic spaces over a scheme is not an ∞-topos; since motivic localization almost never commutes with taking loops, knowing the James and Hilton-Milnor Splittings in the ∞topos of Nisnevich sheaves does not allow one to deduce that they hold in motivic spaces. Wickelgren and Williams used the James filtration to prove that the infinite James Splitting (1.1) holds for A 1 -connected motivic spaces over a perfect field [43,Theorem 1.5]. The reason for the restriction on the base is because their proof relies on Morel's unstable A 1 -connectivity Theorem [29, Theorems 5.46 and 6.1], which implies that motivic localization commutes with loops [4, Theorem 2.4.1; 29, Theorem 6.46]. However, the unstable A 1 -connectivity property does not hold for higher-dimensional bases [4,Remark 3.3.5;5], so a different method is needed if one wants to prove James and Hilton-Milnor Splittings for motivic spaces over more general bases. This is where our generalization pays off: work of Hoyois [18,Proposition 3.15] shows that, in particular, Mather's Second Cube Lemma holds in motivic spaces over an arbitrary base scheme. Therefore, Theorems 1.5 and 1.6 apply in this setting. We use these splittings to give a description of the motivic space ΣΩP 1 in terms of wedges of motivic spheres S i+1,i (Example 2.14), and also give a new description of ΩΣ(P 1 {0, 1, ∞}) (Example 3.3). Over a perfect field, we give a new decomposition of ΩΣ 2 (P 1 {0, 1, ∞}) in terms of motivic spheres of the form S 2m+1,m (Example 4.25).
The second goal of this note is to give a modern construction of the metastable EHP sequence in an ∞-topos X. For every pointed connected object X ∈ X * , the James Splitting provides Hopf maps h n : ΩΣX → ΩΣ(X ∧n ) .
There is also a James filtration {J m (X)} m≥0 on ΩΣX, and, moreover, the composite is trivial. The sequence (1.7) is not a fiber sequence in general 1 , but is in the metastable range: Theorem 1.8 (metastable EHP sequence; Theorem 5.19). Let X be an ∞topos, k ≥ 0 an integer, and X ∈ X * a pointed k-connected object. Then for each integer n ≥ 1, the morphism We note here that a morphism is m-connected in our terminology if and only if it is (m + 1)-connected in the classical terminology (see Warning 4.9). We provide two proofs of Theorem 1.8. The first proof is new and noncomputational; it only makes use of some basic connectedness estimates involving the James filtration and the Blakers-Massey Theorem. In the second proof we simply note that Theorem 1.8 for a general ∞-topos follows immediately from the claim for the ∞-topos of spaces. In the case of spaces, we provide a computational proof; we include this second proof because we were unable to find the computational proof we were familiar with in the literature.

Linear overview
We have written this note with two audiences in mind: the student interested in seeing proofs of Theorems 1.5, 1.6 and 1.8 in the classical setting of spaces, and the expert homotopy theorist interested in applying these results to more general contexts such as motivic spaces or profinite spaces. The student can always take X to be the ∞-category of spaces, and the expert can safely skip the background sections provided for the student. We also note that this text should still be accessible to the reader familiar with homotopy (co)limits but unfamiliar with higher categories, since all we use in our proofs are basic manipulations of homotopy (co)limits. Section 2 is dedicated to proving Theorem 1.

Notation & background
In this subsection we set the basic notational conventions that we use throughout this note as well as recall a bit of relevant background. Notation 1.9. Let X be an ∞-category. If X has a terminal object, we write * ∈ X for the terminal object and X * for the ∞-category of pointed objects in X. If X * has coproducts and X, Y ∈ X * , we write X ∨ Y for the coproduct of X and Y in X * . If X * has coproducts and products, note that there is a natural comparison morphism X ∨ Y → X × Y induced by the morphisms We also make repeated use of the following easy fact. The unfamiliar reader should consult [6, §2; 31].
Lemma 1.14. Let X be an ∞-category with pushouts and a commutative diagram in X. Then the colimit of the diagram (1.15) exists and is equivalent to both of the following two iterated pushouts: (1.14.1) Form the pushout of the rows of (1.15), then take the pushout of the resulting span (1.14.2) Form the pushout of the columns of (1.15), then take the pushout of the resulting span

The James Splitting
In this section, we present a proof of the James Splitting which holds in any ∞-category with finite limits and pushouts, where pushout squares remain pushouts after basechange along an arbitrary morphism. The argument we give roughly follows the argument Hopkins gave in his course on stable homotopy theory in the setting of spaces [15, Lecture 4, §3]; Hopkins attributes this proof to James [19][20][21] and Ganea [10].

Universal pushouts and Mather's Second Cube Lemma
The key property utilized in the proofs we present of the James and Hilton-Milnor Splittings is that pushout squares are preserved by arbitrary basechange. This implies that, in particular, the James and Hilton-Milnor Splittings hold in any ∞-topos, but also in other situations (such as motivic spaces). In this subsection, we provide the categorical context that we work in for the rest of the paper and give a convenient reformulation of the stability of pullbacks under basechange in terms of Mather's Second Cube Lemma (Lemma 2.6).
Recollection 2.1. Let I be an ∞-category and let X be an ∞-category with pullbacks and all I-shaped colimits. We say that I-shaped colimits in X are universal if I-shaped colimits in X are stable under pullback along any morphism. That is, for every diagram F : I → X and pair of morphisms colim i∈I F (i) → Z and Y → Z in X, the natural morphism is an equivalence.
Example 2.2. Let 0 ≤ n ≤ ∞, and let X be an n-topos. One of the Giraud-Lurie axioms for n-topoi guarentees that all small colimits in X are universal [ In particular, finite colimits are universal in X.
Example 2.4 (motivic spaces). Let S be a scheme. The ∞-category H(S) of motivic spaces over S is defined as the A 1 -localization of the ∞-topos Sh nis (Sm S ) of sheaves of spaces on the category Sm S of smooth schemes of finite type over S equipped with the Nisnevich topology. Concretely, H(S) is the full subcategory of Sh nis (Sm S ) spanned by those Nisnevich sheaves F on Sm S with the property that for each smooth S-scheme X, the projection pr 1 : X × S A 1 S → X induces an equivalence Example 2.5 (profinite spaces). We say that a space X is π-finite if X is truncated, has finitely many connected components, and π i (X, x) is finite for every integer i ≥ 1 and point x ∈ X. Write Spc π ⊂ Spc for the full subcategory spanned by the π-finite spaces and Pro(Spc π ) for the ∞-category of profinite spaces. Infinite coproducts in Pro(Spc π ) are not universal [24, Warning E.6.0.9], however, finite colimits and geometric realizations of simplicial objects are universal in Pro(Spc π ) [24, Theorem E. 6 Lemma 2.6. Let X be an ∞-category with pullbacks and pushouts. The following conditions are equivalent: (2.6.1) Pushouts in X are universal.
(2.6.2) Mather's Second Cube Lemma holds in X: Given a commutative cube Documenta Mathematica 26 (2021)  in X where the bottom horizontal face is a pushout square and all vertical faces are pullback squares, then the top horizontal square is a pushout square.
Proof. The implication (2.6.1) ⇒ (2.6.2) is immediate. To see that (2.6.2) ⇒ (2.6.1), suppose that we are given a pushout square in X and morphisms f : are pullbacks. Since the bottom horizontal square of the cube in (2.8) is a pushout, (2.6.2) implies that the top horizontal square is also a pushout. Thus the pushout square (2.7) remains a pushout after base change along an arbitrary morphism, as desired.
Since the main results of this note are about pointed objects, we make the following mildly abusive convention: Convention 2.9. We say that an ∞-category X has universal pushouts if X has finite limits and pushouts, and pushouts in X are universal.

Statement of the James Splitting & Consequences
The James Splitting, originally proven in [19], provides a splitting of the space ΩΣX after a single suspension. The goal of this subsection is to provide a proof of the James Splitting that only relies on the universality of pushouts and a few elementary computations involving the interaction between forming suspensions, loop objects, and smash products.
Using the fact that Σ(X ∧ ΩΣX) ≃ X ∧ ΣΩΣX (Lemma 2.26) and iterating the equivalence of Theorem 2.10, we see: Corollary 2.11 (Fundamental James Splitting, redux). Let X be an ∞category with universal pushouts. For each pointed object X ∈ X * and integer n ≥ 1, there is a natural equivalence Notation 2.12. Let X be an ∞-category with universal pushouts and X ∈ X * a pointed object. Assume that X * has countable coproducts. Passing to the colimit as n → ∞, the coproduct insertions 13. The comparison morphism c X need not be an equivalence. For example, if X = Spc and X = S 0 is the 0-sphere, then the map is not an equivalence. Even though both the source and target of c S 0 are countable wedges of copies of S 1 , the map c S 0 is the summand inclusion induced by the inclusion Z ≥1 ⊂ Z. We analyze when the comparison morphism c X is an equivalence in § 4.2.
Example 2.14. Let S be a scheme. Since colimits are universal in the ∞category H(S) of motivic spaces over S (Example 2.4), Theorem 2.10 implies that for any pointed motivic space X ∈ H(S) * and integer n ≥ 1, we have S 1 -James Splittings Write G m for the multiplicative group scheme over S. Since ΣG m ≃ P 1 S , setting X = G m we see that Using the grading convention S a,b := G ∧b m ∧ (S 1 ) ∧(a−b) for motivic spheres, we can rewrite the equivalence (2.15) as Remark 2.16. There is another suspension in motivic homotopy theory, given by smashing with the multiplicative group scheme G m . One would like an analogue of the James Splitting in H(S) * for G m -suspensions. For S = Spec(R), Betti realization defines a functor H(Spec(R)) → Spc C2 to C 2 -spaces which sends G m to the sign representation circle S σ and S 1 to the circle with trivial C 2 -action. Even though Betti realization is not an equivalence, it closely ties Rmotivic homotopy theory with C 2 -equivariant homotopy theory. In [14], Hill studies the signed James construction in C 2 -equivariant unstable homotopy theory, and shows that an analogue of the James Splitting holds for Ω σ Σ σ X after suspending by the regular representation sphere S ρ = S 1 ∧ S σ . This might lead one to hope that there is an analogue of Hill's result in motivic homotopy theory which proves the James Splitting for Ω Gm Σ Gm X after P 1 -suspension; at the moment, we are not aware of such a result.

Proof of the James Splitting
Before we prove Theorem 2.10, we need a few preliminary results. First, we give a convenient expression for ΣΩΣX as the cofiber of the projection pr 2 : X × ΩΣX → ΩΣX. This expression for ΣΩΣX is an immediate consequence of the following: Lemma 2.17. Let X be an ∞-category with universal pushouts. For every pointed object X ∈ X * , there exists a natural morphism a X : X × ΩΣX → ΩΣX and a pushout square for the pushout square defining the suspension ΣX. The definition of ΣX provides an equivalence between the points i 1 , i 2 : * → ΣX, hence there are natural pullback squares The claim now follows from Mather's Second Cube Lemma applied to the cube Warning 2.18. The morphism a X : X × ΩΣX → ΩΣX in Lemma 2.17 cannot generally be identified with the second projection pr 2 : X × ΩΣX → ΩΣX. Indeed, since X is assumed to have universal pushouts, there is a natural pushout square Moreover, the object ΣX × ΩΣX is not generally terminal in X.
Next, we give a convenient expression for the term Σ(X ∧ ΩΣX) in the James Splitting as the pushout of the span Our proof of this appeals to the following fact, which follows immediately from the definitions.
Lemma 2.20. Let X be an ∞-category with pushouts and a terminal object, and let X, Y ∈ X * be pointed objects of X. Then the square is a pushout square.
Proposition 2.21. Let X be an ∞-category with finite limits and pushouts. Then for every pair of pointed objects X, Y ∈ X, there is a pushout square We apply Lemma 1.14 to the commutative diagram * * * Appealing to Lemma 2.20, taking pushouts of the rows of (2.22) results in the span which has pushout C. Alternatively, since the smash product X ∧ Y is the cofiber of the comparison morphism X ∨ Y → X × Y , taking pushouts of the columns of (2.22) results in the span * X ∧ Y * . (2.23) By definition, the pushout of the span (2.23) is the suspension Σ(X ∧ Y ), so Lemma 1.14 shows that C ≃ Σ(X ∧ Y ) . To conclude the proof, note that it follows from the definitions that the induced morphisms Proposition 2.21 also provides a general formula for the cofiber cofib(pr 2 : X × Y → Y ) that allows us to relate the expressions for ΣΩΣX and Σ(X ∧ ΩΣX) from Corollary 2.19 and Proposition 2.21, respectively.
Corollary 2.24. Let X be an ∞-category with finite limits and pushouts. Then, for every pair of pointed objects X, Y ∈ X * : The splitting of Corollary 2.11 is immediate from Theorem 2.10 combined with the following elementary fact: Lemma 2.26. Let X be an ∞-category with universal pushouts. For every pair of pointed objects X, Y ∈ X * , there is a natural equivalence Proof. Since pushouts in X are universal and colimits commute, the squares are both pushouts in X * . By the definition of the smash product and the facts that colimits commute and X ∧ * ≃ * , we see that

Ganea's Lemma
Since the method of proof is similar to the arguments in this section, we close with the following lemma of Ganea [9, Theorem 1.1]. This will not be used in the sequel.
Lemma 2.27. Let X be an ∞-category with universal pushouts. Let f : X → Y be a morphism in X * , and write i : fib(f ) → X for the induced morphism from the fiber of f . Then there is a natural equivalence Proof. By Proposition 2.21, it suffices to show that the square and note that each vertical square is a pullback square. The bottom horizontal square in (2.28) is a pushout square by definition, so the assumption that pushouts in X are universal implies that the top horizontal square is a pushout as well.

The Hilton-Milnor Splitting
The main result of this section is the following: Hilton-Milnor Splitting). Let X be an ∞-category with universal pushouts and X, Y ∈ X * . Then there is a natural equivalence Before giving the proof of Theorem 3.1, we discuss some applications.
Applying the James Splitting of Example 2.14 we see that for each integer n ≥ 1, we can also express ΩΣ(P 1 S {0, 1, ∞}) as We now turn to the proof of the Hilton-Milnor Splitting. We first show that there is a fiber sequence We then show that the sequence (3.4) splits after taking loops. To do this, we construct a section and use the fact that a fiber sequence of group objects with a section splits on the level of underlying objects. After proving that (3.4) is a fiber sequence we give a quick review of group objects and deduce Theorem 3.1 from the Splitting Lemma (Lemma 3.12).
We start with the following observation: Lemma 3.5. Let X be an ∞-category with finite limits and X, Y ∈ X * . Then there is a natural equivalence Next, we prove the existence of the fiber sequence (3.4).
Lemma 3.6. Let X be an ∞-category with universal pushouts and X, Y ∈ X * . Then there is a natural equivalence   .7) is a pushout square by definition, so the assumption that pushouts in X are universal implies that the top horizontal square is a pushout as well.

Reminder on group objects & the Splitting Lemma
In order to split the fiber sequence (3.4) after taking loops, we need a few basic facts about group objects (also called E 1 -groups or grouplike E 1 -algebras) in ∞-categories, which we now review. We begin with a little motivation for the definition of group objects as deloopings. For the genesis of these ideas, we refer the reader to [1,34,35]. Notation 3.8. We write ∆ for the category of nonempty linearly ordered finite sets. As usual, given a simplicial object X : ∆ op → X, we write X n := X([n]) for the n-simplices of X.
Recall that the bar construction is a fully faithful functor from the category of monoids to the category of simplicial sets. The essential image of the bar construction consists of those simplicial sets X : ∆ op → Set satisfying the following conditions: (1) We have X 0 ≃ * .
(2) Segal condition: For each n > 0 and t ∈ [n], the square is a pullback square.
The face map d 1 : X 1 × X 1 ≃ X 2 → X 1 provides a multiplication on X 1 with unit given by the degeneracy map s 0 : * ≃ X 0 → X 1 .
Since groups form a full subcategory of the category of monoids, the bar construction also identifies the category of groups with a full subcategory of the category of simplicial sets. For this it is better to use an alternative characterization of the existence of inverses: a monoid M is a group if and only if the shear maps are bijections. Translating this into simplicial sets one sees that the category of groups is equivalent to the full subcategory of Fun(∆ op , Set) spanned by the simplicial sets X satisfying (1), (2), and: (3) The induced squares are pullback squares.
We emphasize that condition (3) is not implied by the Segal condition (2). The following is the correct generalization of a group object in an arbitrary ∞category. The point is to replace the Segal condition with a stronger condition that also encompasses condition ( Definition 3.9. Let X be an ∞-category. A group object in X is a simplicial object G : ∆ op → X such that: (3.9.1) The object G 0 is a terminal object of X.
is a pullback square in X.
In this case, we call G 1 ∈ X the underlying object of G. We often identify a group object by its underlying object. We write Grp(X) ⊂ Fun(∆ op , X) for the full subcategory spanned by the group objects.
The key example of a group object is loops on a pointed object. As a simplicial object, ΩX can be written as the Čech nerve of the basepoint * → X; since we use Čech nerves in § 4.1, we recall the definition here.
Recollection 3.10. Let X be an ∞-category with pullbacks, and let e : W → X be a morphism in X. The Čech nerveČ(e) of e is the simplicial object HereČ(e) n is the (n+1)-fold fiber product of W over X, each degeneracy map is a diagonal morphism, and each face map is a projection. Note that the morphism e : W → X defines a natural augmentationČ(e) → X.
Lemma 3.11. Let X be an ∞-category with finite limits and X ∈ X * . Then ΩX naturally admits the structure of a group object of X.
Proof. Let U(X) denote the Čech nerve of the basepoint * → X. Since U(X) 0 ≃ * , [22, Proposition 6.1.2.11] shows that the Čech nerve U(X) is a group object of X. Since U(X) 1 ≃ ΩX, it follows that the loop functor Ω : X * → X * factors as the composite X * Grp(X) X * U of the functor given by the assignment X → U(X) followed by the forgetful functor Grp(X) → X * .
We leave the following Splitting Lemma as an amusing exercise for the reader. is an equivalence in X * .
We now prove the Fundamental Hilton-Milnor Splitting.
Proof of Theorem 3.1. By Lemmas 3.6 and 3.11, there is a fiber sequence of group objects of X. Note that the map Ω(X ∨ Y ) → ΩX × ΩY has a section defined by the composite where i 1 : X → X ∨ Y and i 2 : Y → X ∨ Y are the coproduct insertions, and m is the multiplication coming from the group structure on Ω(X ∨ Y ). By Lemma 3.12 the fiber sequence (3.13) splits, so applying Lemma 3.6 we see that there are equivalences ≃ ΩX × ΩY × ΩΣ(ΩX ∧ ΩY ) .

Connectedness & the James Splitting
The purpose of this section is explain how to use a connectedness argument to show that if X is a pointed connected object of an ∞-topos, then the comparison morphism is an equivalence. To do this, we start by reviewing the basics of connectedness in an ∞-topos ( § 4.1). We also prove some basic connectedness results that we need in our proof of the metastable EHP sequence in Section 5. Subsection 4.2 proves the infinite James and Hilton-Milnor Splittings for connected objects and explains how to deduce Wickelgren and William's James Splitting in motivic spaces over a perfect field from these results.

Connectedness and the Blakers-Massey Theorem
In this subsection, we review the basic properties of k-truncated and kconnected morphisms in an ∞-topos. We also recall the Blakers-Massey Definition 4.1. Let X be an ∞-topos. For each integer k ≥ −2, define ktruncatedness for morphisms in X recursively as follows.
An object X ∈ X is k-truncated if the unique morphism X → * is k-truncated. Write X ≤k ⊂ X for the full subcategory spanned by the k-truncated objects.
The inclusion X ≤k ⊂ X admits a left adjoint which we denote by τ ≤k : X → X ≤k .
Example 4.2. Let C be a small ∞-category equipped with a Grothendieck topology τ , and let k ≥ −2 be an integer. Then a sheaf F ∈ Sh τ (C) of spaces on C with respect to τ is k-truncated if and only if F(c) is a k-truncated space for every c ∈ C. That is, F is k-truncated if and only if F is a sheaf of k-truncated spaces.
Remark 4.3. If X is an ∞-topos, then the full subcategory X ≤0 spanned by the 0-truncated objects is an ordinary topos, i.e., a category of sheaves of sets on a Grothendieck site.  Definition 4.6. Let X be an ∞-topos. For each integer k ≥ −2, define kconnectedness for morphisms in X recursively as follows.
An object X ∈ X is k-connected if the unique morphism X → * is k-connected.       (4.10.7) An object X ∈ X is k-connected if and only if the k-truncation τ ≤k (X) of X is terminal in X.
Since the k-truncation functor τ ≤k : X → X preserves filtered colimits, from (4.10.7) we deduce: Corollary 4.11. Let X be an ∞-topos and k ≥ −2 be an integer. Then the class of k-connected objects of X is stable under filtered colimits.
In the ∞-topos of spaces, the following connectedness estimates are usually done by appealing to cell structures. Such arguments are unavailable in an arbitrary ∞-topos, so we deduce these connectedness estimates from Proposition 4.10.
Proof. For (4.12.1) note that since k-connected morphisms are stable under pushout (4.10.2), the definition of the suspension ΣX as a pushout shows that the basepoint * → ΣX is k-connected. Hence ΣX is (k + 1)-connected Finally, (4.12.5) follows from (4.12.4) by induction. Now we record a convenient fact about the interaction between connectedness and pullbacks that we need in out proof of the metastable EHP sequence.
Proposition 4.13. Let X be an ∞-topos, ℓ ≥ −2 be an integer, and be a commutative diagram in X. If a and b are ℓ-connected and c is (ℓ + 1)connected, then the induced morphism on pullbacks a × c b : Proof. Since ℓ-connected morphisms are stable under composition, by factoring the induced morphism A × C B → A ′ × C ′ B ′ as a composite of induced morphisms it suffices to prove the claim in the special case B = B ′ and the morphism b : B → B ′ is the identity. To prove the claim when b is the identity, first write

Documenta Mathematica 26 (2021) 1423-1464
Consider the following commutative diagram of pullback squares In particular, Proposition 4.13 shows that the class of ℓ-connected morphisms in an ∞-topos is closed under finite products. Setting B = B ′ = * in Proposition 4.13 we deduce: Corollary 4.14. Let X be an ∞-topos, ℓ ≥ −2 be an integer, and be a commutative square in X. If a is ℓ-connected and c is (ℓ + 1)-connected, then for every point x : * → C, the induced morphism fib x (f ) → fib cx (f ′ ) on fibers is ℓ-connected.
We conclude this subsection by recalling the Blakers-Massey and Freudenthal Suspension Theorems in the setting of ∞-topoi. be a pushout square in X. If f is k-connected and g is ℓ-connected, then the induced morphism A → B × D C is (k + ℓ)-connected.
As in the classical setting, applying the Blakers-Massey Theorem to the pushout defining the suspension immediately implies the Freudenthal Suspension Theorem.
Corollary 4.16 (Freudenthal Suspension Theorem). Let X be an ∞-topos, and X ∈ X * a pointed k-connected object. Then the unit morphism X → ΩΣX is 2k-connected.

The infinite James and Hilton-Milnor Splittings
We now explain how to use a connectedness argument to show that if X is a pointed 0-connected object of an ∞-topos, then the comparison morphism introduced in Notation 2.12 is an equivalence (Proposition 4.18). The James Splitting gives us access to generalized Hopf invariants in this very general setting, and implies the stable Snaith Splitting for ΩΣX [36]. We also prove the infinite version of the Hilton-Milnor Splitting (Example 4.22). Using Morel's unstable A 1 -connectivity Theorem, we explain why these infinite splittings hold in motivic spaces over a perfect field (Corollary 4.24). As an application, we give a new description of ΩΣ 2 (P 1 {0, 1, ∞}) (Example 4.25).
The key tool to prove all of these results is the following lemma about how infinite James Splittings interact with localizations.
Lemma 4.17. Let Y be an ∞-category, j : X ֒→ Y a full subcategory, and assume that the inclusion j admits a left adjoint L : Y → X. Assume that: (4.17.1) The ∞-categories X and Y have universal pushouts and the ∞-category Y * has countable coproducts. (4.17. 2) The functor L : Y → X commutes with finite products and the formation of loop objects (e.g. L is left exact).
If X ∈ X * is a pointed object with the property that the comparison morphism is an equivalence in Y * , then the comparison morphism is an equivalence in X * .
Proof. Since the localization L : Y → X preserves the terminal object, the functor L descends to the level of pointed objects. Moreover, since L preserves finite products, the functor L : Y * → X * commutes with smash products. Since the functor L : Y * → X * also commutes with the formation of loop objects, we see that the morphism c X is equivalent to L(c j(X) ).
The assumption that the morphism c j(X) is an equivalence completes the proof.
Proposition 4.18 (James Splitting). Let X be an ∞-topos. Then for each 0-connected pointed object X ∈ X * , the natural comparison morphism is an equivalence in X * .
Proof. Since every ∞-topos is a left exact localization of a presheaf ∞-topos and presheaf ∞-topoi are hypercomplete (see Definition 4.8), by Lemma 4.17 we are reduced to the case that X is hypercomplete. That is, it suffices to show that the morphism c X is ∞-connected.

Documenta Mathematica 26 (2021) 1423-1464
is n-connected. The commutativity of the triangle (4.19) combined with (4.10.5) show that the morphism Since this is true for each integer n ≥ 1, we have shown that morphism c X is ∞-connected, as desired.
Remark 4.20. Of course, in the proof of Proposition 4.18 we can further reduce to the case X = Spc and the claim follows from the classical James Splitting. The purpose of our proof is to provide an explaination that does not appeal to the classical result but, rather, only uses basic manipulations available in an ∞-topos.
Proposition 4.21 (Hilton-Milnor Splitting, general version). Let X be an ∞category with universal pushouts and countable coproducts, and let X, Y ∈ X * be pointed objects.
The claim now follows from the natural natural equivalences Similarly, (4.21.2) follows from Lemma 2.26, (4.21.1), and the assumption that c X : i≥1 ΣX ∧i → ΣΩΣX is an equivalence Example 4.22 (Hilton-Milnor Splitting). Let X be an ∞-topos and let X, Y ∈ X * be pointed objects. By Proposition 4.18, the hypotheses of (4.21.1) are satisfied if Y is 0-connected, and the hypotheses of (4.21.2) are satisfied if both X and Y are 0-connected.
The next application is that the infinite James and Hilton-Milnor Splittings hold for A 1 -0-connected motivic spaces over a perfect field. The infinite James Splitting was first proven by Wickelgren and Williams using the James filtration We complete this section by constructing the Hopf maps that appear in the metastable EHP sequence.

The James filtration & metastable EHP sequence
In classical algebraic topology, the metastable EHP sequence is the statement that the composite h2 is a fiber sequence in a range depending on the connectedness of X, known as the metastable range. Here the first map X → ΩΣX is the unit and h 2 is the Hopf map (Construction 4.26). For the higher Hopf maps h n : ΩΣX → ΩΣX ∧n , there is an analogous fiber sequence in a range J n−1 (X) ΩΣX ΩΣX ∧n , hn where J n−1 (X) is the (n − 1) st piece of the James filtration on ΩΣX. This section is dedicated to a non-computational proof of the metastable EHP sequence in an ∞-topos that only makes use of the Blakers-Massey Theorem and some basic connectedness results (see Theorem 5.19). In § 5.1 we review the James filtration. In § 5.2 we refine the James Splitting to a splitting In § 5.3, we give our non-computational proof of the metastable EHP sequence via the Blakers-Massey Theorem, and also record a computational proof for posterity.

The James filtration
Classically, the James filtration {J n (X)} n≥0 provides a multiplicative filtration on the free monoid J(X) on a pointed space X, in the homotopical sense. At the point-set level, J(X) can be presented as the free topological monoid on X, and J n (X) can be identified the subspace of words of length at most n in J(X). Concatenation of words then supplies {J n (X)} n≥0 with the structure of a filtered monoid. Since the trivial monoid and trivial group coincide, if X is connected, then the free monoid J(X) on X coincides with the free group ΩΣX on X.
In a general ∞-category, we can define the James filtration as follows. This definition is provided in [7,Section 3] in the context of homotopy type theory; the arguments made in [7,Section 3] are formal and valid in any ∞-topos.
Construction 5.1 (James filtration). Let X be an ∞-category with finite products and pushouts, and let X ∈ X * be a pointed object. For each integer n ≥ 0 we define a pointed object J n (X) ∈ X * as well as morphisms i n : J n (X) → J n+1 (X) and α n : X × J n (X) → J n+1 (X) in X * recursively as follows.
(5.1.2) For n ≥ 2, we define J n (X), i n−1 , and α n−1 by the pushout square The following is a straightforward application of Lemma 1.14.
Lemma 5.12. Let X be an ∞-category with pushouts and a terminal object and let A B C D be a commutative square in X * . Then there is a natural equivalence We are now ready to show that cofib(i n ) ≃ X ∧n+1 .
Proposition 5.13. Let X be an ∞-category with universal pushouts, and let X ∈ X * . Then for each integer n ≥ 0, there is a natural equivalence cofib(i n : J n (X) → J n+1 (X)) ≃ X ∧n+1 .
Moreover, the composite an is equivalent to the canonical map X ×n+1 → X ∧n+1 .
Proof. We prove the claim by induction on n. For the base case, note that since the morphism i 0 is the basepoint * → X, the cofiber of i 0 is X. For the inductive step we assume that cofib(i n ) ≃ X ∧n+1 and show that cofib(i n+1 ) ≃ X ∧n+2 . From the defining pushout square (5.2), we see that Applying Lemma 5.12 shows that where the map of cofibers is induced by the map i n : J n (X) → J n+1 (X). By Lemma 5.10, there is a cofiber sequence X → cofib(J n (X) → X × J n (X)) → X ∧ J n (X) ; moreover, the map i n : J n (X) → J n+1 (X) induces a map of cofiber sequences Since the leftmost vertical map is the identity, taking vertical cofibers in the map of cofiber sequences (5.14) produces an equivalence between the vertical cofibers of the middle and right vertical maps. Since the cofiber of the middle vertical map is cofib(i n+1 ), we find that Since pushouts in X are universal we have a natural equivalence cofib(id X ∧i n : X ∧ J n (X) → X ∧ J n+1 (X)) ≃ X ∧ cofib(i n : J n (X) → J n+1 (X)) By the inductive hypothesis, cofib(i n ) ≃ X ∧n+1 , so cofib(i n+1 ) ≃ X ∧n+2 , as desired.
Next we split the term in the pushout square (5.2) defining ΣJ n+1 (X) and prove Proposition 5.8.

(5.16)
Under the equivalence Σ(X × J n−1 (X)) ≃ Σ(X ∧ J n−1 (X)) ∨ ΣX ∨ ΣJ n−1 (X) of Corollary 2.24, the left vertical map in (5.16) is the coproduct insertion. Hence on pushouts we see that Proof of Proposition 5.8. We prove the claim by induction on n. The base case where n = 1 is obvious. For the inductive step, assume that n ≥ 1 and ΣJ n (X) ≃ n i=1 ΣX ∧n . By Proposition 5.13 we have a cofiber sequence in so the inductive hypothesis and the duals of Lemmas 3.11 and 3.12, it suffices to define a retraction r n : ΣJ n+1 (X) → ΣJ n (X) of the map Σi n . We construct the retractions r n : ΣJ n+1 (X) → ΣJ n (X) inductively. For the base case, the retraction r 0 : ΣX → * of Σi 0 is the unique morphism. For the inductive step, assume that n ≥ 1 and we have constructed a retraction r n−1 : ΣJ n (X) → ΣJ n−1 (X) of Σi n−1 ; we use this to construct a retraction r n of Σi n . Since suspension preserves pushouts, suspending the defining pushout square (5.2) yields a pushout square In order to define a retraction of Σi n , it suffices to define a retraction of the left vertical map in (5.17), i.e., it suffices to define a retraction By Corollary 2.24 and Lemma 5.15, we have equivalences Σ(X × J n (X)) ≃ Σ(X ∧ J n (X)) ∨ ΣX ∨ ΣJ n (X) and Σ X × J n−1 (X) Moreover, the left vertical map in (5.17) is induced by the suspensions of the identity on X, identity on J n (X), and the map i n : J n−1 (X) → J n (X). Under the identifications Σ(X ∧ J n−1 (X)) ≃ X ∧ ΣJ n−1 (X) and Σ(X ∧ J n (X)) ≃ X ∧ ΣJ n (X) of Lemma 2.26, we see that the map is a retraction of Σ(id X ∧i n−1 ). Hence the map supplies the desired retraction of the left vertical map in (5.17).

Proofs of the metastable EHP sequence
In this subsection, we present two proofs of the metastable EHP sequence in the setting of ∞-topoi. Before making a precise statement of the main result, we need the following easy lemma.
Lemma 5.18. Let X be an ∞-topos, X ∈ X * be a pointed 0-connected object, and n ≥ 1 an integer. Then the composite Proof. It suffices to prove the corresponding statement on adjoints: in other words, we need to show that the composite ΣJ n−1 (X) ΣΩΣX ΣX ∧n Σun is null. This is an immediate consequence of Proposition 5.8.
We can now state the metastable EHP sequence.
Remark 5.20. The metastable EHP sequence for ∞-topoi of hypersheaves on an ordinary site with enough points proven by Asok-Wickelgren-Williams [4, Proposition 3.1.4] is a special case of Theorem 5.19.
The first proof of Theorem 5.19 we present is internal to ∞-topoi, and only uses basic facts about connectedness and the James construction, as well as the Blakers-Massey Theorem. The second reduces to the ∞-topos Spc of spaces, then uses the homology Whitehead Theorem and Serre spectral sequence to give a calculational proof of the metastable EHP sequence in the classical setting. Both perspectives are valuable, and we present the second here in part because the calculational proof of the metastable EHP sequence does not seem to be easy to locate in the literature.
(1) If the conclusion of Theorem 5.19 holds for the ∞-topos Spc, then it holds for any presheaf ∞-topos.
Remark 5.23. In the case that n = 2 and X = Spc, the computational proof of the metastable EHP sequence given here reduces to the proof presented in [4, Proposition 3.1.2].