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A constructive approach to higher homotopy operations

In this paper we provide an explicit general construction of higher homotopy operations in model categories, which include classical examples such as (long) Toda brackets and (iterated) Massey products, but also cover unpointed operations not usually considered in this context. We show how such operations, thought of as obstructions to rectifying a homotopy-commutative diagram, can be defined in terms of a double induction, yielding intermediate obstructions as well.

Toda's construction of what we now call Toda brackets in [T1] (cf. [T2,Ch. I]) was the first example of a secondary homotopy operation stricto sensu, although Adem's secondary cohomology operations and Massey's triple products in cohomology appeared at about the same time (see [Ade, Ms]).
In [Ada,Ch. 3], Adams first tried to give a general definition of secondary stable cohomology operations (see also [Ha]). Kristensen gave a description of such operations in terms of chain complexes (cf. [Kr, KK]), which was extended by Maunder and others to n-th order cohomology operations (see [Mau,Hol,K1,K2]).
Higher operations have also figured over the years in rational homotopy theory, where they are more accessible to computation (see, e.g., [Ald, Bu, Re, Ta]). In more recent years there has been a certain revival of interest in the subject, notably in algebraic contexts (see for example, [Bk, Ga, S, E, CF, HW]).
In [Sp2], Spanier gave a general theory of higher order homotopy operations (extending the definition of secondary operations given in [Sp1]). Special cases of higher order homotopy operations appeared in [Wa, K, Mo, BBG], and other general definitions may be found in [BM,BJT2].
The last two approaches cited present higher order operations as the (last) obstruction to rectifying certain homotopy-commutative diagrams (in spaces or other model categories). In particular, they highlight the special role played by null maps in almost all examples occurring in practice. Implicitly, they both assume an inductive approach to rectifying such diagrams. However, in earlier work no attempt was made to describe a useable inductive procedure, which should (inter alia) explain precisely which lower-order operations are required to vanish in order for a higher order operation to be even defined.
The goal of the present note is to make explicit the inductive process underlying our earlier definitions of higher order operations, in as general a framework as possible. We hope the explicit nature of this approach will help in future work both to clarify the question of indeterminacy of the higher operations, and possibly to produce an "algebra of higher operations," in the spirit of Toda's original "juggling lemmas" (see [T2,Ch. I]).
An important feature of the current approach is that we assume that our indexing category is directed, and we consistently proceed in one direction in rectifying the given homotopy-commutative diagram (say, from right to left, in the "right justified" version). As a result, when we come to define the operation associated to an indexing category of length n, we use as initial data a specific choice of rectification for the right segment of length n − 1. This sequence of earlier choices will appear only implicitly in our description and general notation for higher operations, but will be made explicit for our (long) Toda brackets (see §1.7-4.9).
Since our higher operations appear as obstructions to rectification, they fit into the usual framework of obstruction theory: when they do not vanish, one must go back along the thread of earlier choices until reaching a point from which one can proceed along a new branch. From the point of view of the obstruction theory, the important fact is their vanishing or non-vanishing (see Remark 4.9 for the relation to coherent vanishing). Nevertheless, since our higher operations are always described as a certain set of homotopy classes of maps into a suitable pullback, at least in some cases it is possible to describe the indeterminacy more explicitly. However, this would only be a part of the total indeterminacy, since the most general obstruction to rectification consists of the union of these sets, taken over all possible choices of initial data of length n − 1.
After a brief discussion of the classical Toda bracket from our point of view in Section 1, in Section 2.A we describe the basic constructions we need, associated to the type of Reedy indexing categories for the diagrams we consider. The changes needed for pointed diagrams are discussed in Section 2.B. We give our general definition of higher order operations in Section 3: it is hard to relate this construction to more familiar examples, because it is intended to cover a number of different situations, and in particular the less common unpointed version. In all cases the "total higher operation" serves as an obstruction to extending a partial rectification of a homotopycommutative diagram one further stage in the induction.
In Section 4 we provide a refinement of this obstruction to a sequence of intermediate steps (in an inner induction), culminating in the total operation for the given stage in the induction. Section 5 is devoted to a commonly occurring problem: rigidifying a (reduced) simplicial object in a model category, for which the simplicial identities hold only up to homotopy. This serves to illustrate how the general (unpointed) theory works in low dimensions.
In Section 6 we define pointed higher operations, which arise when the indexing category has designated null maps, and we want to rectify our diagram while simultaneously sending these to the strict zero map in the model category. This involves certain simplifications of the general definition, as illustrated in the motivating examples of (long) Toda brackets and Massey products, described in Section 7.
Finally, in Section 8 we make a tentative first step towards a possible "algebra of higher operations," by showing how we can decompose our pointed higher operations into ordinary (long) Toda brackets for a certain class of fully reduced diagrams.
In Appendix A we review some basic facts in model categories needed in the paper; Appendix B contains some preliminary remarks on the indeterminacy of the operations. 0.1. Acknowledgements. We wish to thank the referee and editor for their detailed and pertinent comments. The research of the first author was supported by Israel Science Foundation grants 74/11 and 770/16, and the third author by National Science Foundation grant DMS-1207746.

The classical Toda Bracket
We start with a review of the classical Toda bracket, the primary example of a pointed secondary homotopy operation. In keeping with tradition we give a left justified description, in terms of pushouts, although for technical reasons our general approach will be right justified, in terms of pullbacks.

Left Justified Toda Brackets.
A classical Toda diagram in any pointed model category consists of three composable maps: with each adjacent composite left null-homotopic. We shall assume that all objects in (1.2), and the analogous diagrams throughout the paper, are both fibrant and cofibrant, so we may disregard the distinction between left and right homotopy classes).
To define the associated Toda bracket, we first change h into a cofibration (to avoid excessive notation, we do not change the names of h or its target). By Lemma A.11 we can alter g within its homotopy class to a g ′ to produce a factorization: so g ′ • h is the zero map (not just null-homotopic). We use i : Y(2) ֒→ C Y(2) (an inclusion into a reduced cone) to extend (1.3) to the solid diagram: where all squares (and thus all rectangles) are pushouts, with cofibrations as indicated.
In particular, Σ ′ Y(3) is a model for the reduced suspension of Y(3), M g ′ is a mapping cone on g ′ , and φ is a nullhomotopy for f • g ′ . Note that any choice of such a nullhomotopy φ induces maps ψ φ : Σ ′ Y(3) → Y(0) and κ : M g ′ → Y(0), with κ • j = ψ φ . Suppose that for some choice of φ, the map ψ φ is null-homotopic, so κ • j = ψ φ ∼ 0. Then by Lemma A.11, we could alter κ within its homotopy class to κ ′ such that κ ′ • j = 0, whence the pushout property for the lower right square would induce the dotted map cof(g 2 ) → Y(0). As a consequence, choosing f ′ = κ ′ • i ∼ κ • i = f provides a replacement for f in the same homotopy class satisfying f ′ • g ′ = κ ′ • i • g ′ = 0, rather than only agreeing up to homotopy.
1.4. Definition. Given (1.2), the subset of the homotopy classes of maps [Σ ′ Y(3), Y(0)] consisting of all classes ψ φ (for all choices of φ and g 2 as above) forms the Toda bracket f, g, h . Each such ψ φ is called a value of f, g, h , and we say that the Toda bracket vanishes (at ψ φ : Σ ′ Y(3) → Y(0) as above) if ψ φ ∼ * -that is, if f, g, h includes the null map.

1.5.
Remark. By what we have shown, f, g, h vanishes if and only if we can vary the spaces Y(0), . . . , Y(3) and the maps f, g, h within their homotopy classes so as to make the adjacent composites in (1.2) (strictly) zero, rather than just null-homotopic.
In fact, by considering the cofiber sequence In fact, the choices for homotopy classes of a nullhomotopy for any fixed pointed map ϕ : A → B are in one-to-one correspondence with classes [ΣA, B] (see [Sp1,§1]), and thus the contribution of the choices for φ and g 2 respectively to the value of f, g, h are given by ( The two subgroups ] are referred to as the indeterminacy of f, g, h ; when Y(3) is a homotopy cogroup object or Y(1) is a homotopy group object, the sum of (1.6) is a subgroup of the abelian group [Σ ′ Y(3), Y(0)].
In any case, vanishing means precisely that the (well-defined) class of f, g, h in the double quotient is the trivial element in the quotient set. 1.7. Remark. The 'right justified' definition of our ordinary Toda bracket is given in Step (c) of Section 7.A below. This will depend on a specific initial choice of maps f and g with f • g = * (rather than f • g ∼ * ), and will be denoted by f, g, h , so where the union is indexed over those pairs with f and g in the specified homotopy classes.
The reader is advised to refer to that section for examples of all constructions in Sections 3-4 below, since the example of our long Toda bracket f, g, h, k in Section 7 was the template for our more general setup.

Graded Reedy Matching Spaces
Our goal is now to extend the notions recalled in Section 1 -of Toda diagrams, and Toda brackets as obstructions to their (pointed) realization -to more general diagrams Y : J → E, where E is some complete category (eventually, a pointed model category).

2.2.
Remark. Note that a weak lattice J has no directed loops or non-trivial endomorphisms, and x ∈ Obj J has only Id x mapping out of it if and only if |x| = 0. Moreover, each object is the source of only finitely many morphisms, although there may be elements of arbitrarily large degree.
2.3. Notation. For a weak lattice J as above: (a) We denote by J k the full subcategory of J consisting of the objects of degree ≤ k, with I k : J k → J the inclusion. (b) For any x ∈ Obj J in a positive degree, J x will denote the full subcategory of J whose objects are those t ∈ J with J (x, t) non-empty. Thus x ∈ J x and J x ∩ J 0 = ∅ (by §2.1). (c) We denote by J x k the full subcategory of J x containing x and all objects (under x) of degree at most k, with I x k : J x k → J x the inclusion. We implicitly assume that |x| > k when we use this notation. Similarly, ∂J x k is the full subcategory of J x k containing all objects other than x. (d) Given |x| ≥ k > 0 and a functor Y : → v by projection of the left hand product onto the copy of Y(v) indexed by the composite x f g −→ v (followed by Id Y(v) ).
2.5. Example. Consider the following weak lattice J : where all subdiagrams commute, and the degrees are as indicated. Then is the discrete category with objects {s, t}. Furthermore we have: 2.6. Definition. For a weak lattice J as above and any x ∈ J of degree > k: (a) The comma category (x ↓ J k ) = (x ↓ ∂J x k ) has as objects the morphisms in J from x to objects in J k , with maps in (x ↓ J k ) given by commutative triangles in J of the form where Y (f : x → s) = Y (s) (see [Mc,X.3]). We often write M x k for M x k (Y ) when Y is clear from the context. (c) For any slightly larger diagram Y : J x k → E, there is a canonical map in E defined using the universal property of the limit, m and σ x <k+1 is the composite of m x k with the forgetful map (inclusion) from the limit to the product, so it is closely related to the Reedy matching map when k = |x| − 1.
Note that M x 0 is simply a product of entries of degree zero, indexed by the set of maps from x to the discrete category J x 0 , and m is a fibration; the special case k = |x| − 1 is the standard Reedy matching construction (cf. [Hir,Defn. 15.2.3 (2)]).
Recall that there is an adjoint pair given by forgetting and the right Kan extension over I k x . The fact that I k x is fully faithful implies that the right Kan extension restricts back to the original functor (hence the term extension). Moreover, M x k (Y ) is the formula for the value of the right Kan extension, Ran restricting to the identity away from x. It is thus completely determined by the entry Y (x) → M x k (Y ). Embedding the limit M x k (Y ) as usual into J (x,u),|u|≤k Y(u), we see that there are two kinds of conditions needed for an element in this product to be in the limit (when E is a concrete category): (a) Those not involving Y(s) with |s| = k, yielding M x k−1 (Y ) in the lower left corner of (2.9); (b) Those which do involve Y(s) with |s| = k, where the compatibility conditions necessarily involve objects in degree < k, since all maps in J lower degree. This implies: 2.8. Lemma. If J is a weak lattice and |x| > k > 0, a functor Y : ∂J x k → E induces a pullback square: Proof. Note that the existence of Y suffices to define each component of the diagram. In particular, Y(f ) is defined for each morphism f in ∂J x k , and even forms part of the definition of the factors of the right vertical, but such maps are not defined for any g : x → v with |v| ≥ k.
Denote the pullback of the lower right part of the diagram by R x k . We first show that R x k induces a cone on (x ↓ J x k ), thus inducing a map R x k → M x k by the universal property of the limit: projecting off to the right for targets of degree k, or projecting after moving down followed by the forgetful map for targets of lower degree, yields maps Y (g) : . If the codomain of g has degree less than k, the upper right corner is not involved, and commutativity follows from the fact that the map from R x k factors through M x k−1 (Y ) in the lower left. On the other hand, if the codomain of g has degree exactly k, then projecting off at the chosen pair (g, f ) in the assumed (commutative) pullback diagram, we see that commutes by the definition of the generalized diagonal Ψ, which establishes the cone condition. Thus, the universal property of the limit yields a unique map R x k → M x k . On the other hand, the forgetful map forget : can be split into factors with |t| = k, and the factors with |t| < k, thereby defining maps to the two corners of the pullback which will make the outer diagram commute, by inspection. Thus, there is also a map M x k → R x k and the induced cone, as above, is the standard one, so the composite is the identity on M x k . Finally, starting from R x k , building the cone as above and then projecting as just discussed recovers the same maps Y (h) as entries, so this composite is the identity on R x k as well.

2.B. Pointed Graded Matching Objects
Higher homotopy operations have traditionally appeared as obstructions to vanishing in a pointed context, so we shall need a pointed version of the constructions above.
2.11. Definition. When E is any category with limits (such as a model category), a pointed object in E is one equipped with a map from the final object (or empty limit), denoted by * . The most commonly occurring case is where * is a zero object (both initial and final in E). Similarly, a pointed map in E is one under * . This defines the pointed category E * (which inherits any model category structure on E -cf. [Hov,1.1.8]). Note that there is a canonical zero map, also denoted by * , between any two objects in E * .
2.12. Definition. We say that a small category J as in §2.1 is a pointed indexing category if the set of morphisms has a partition Mor(J ) = J ⊔ J (and thus J (x, t) = J(x, t) ⊔ J (x, t) for each x, t ∈ Obj J ) such that: (a) J (x, x) contains Id X if and only if x is a zero object in J . (b) The subsets J (x, t) are absorbing under composition -that is, if f and g are composable and either of f or g lies in J , then so does their composite. Thus J behaves like a (2-sided) ideal and J like the corresponding cosets.
Given E * and a pointed weak lattice J -that is, a pointed indexing category which is also a weak lattice -a pointed diagram in E * is a functor Y : J → E * such that Y(g) = * whenever g ∈ J (x, t).
2.13. Example. We can make the decreasing poset category pointed by setting J (t, s) := J (t, s) whenever t − s > 1, so only indecomposable maps lie in J. A pointed diagram J → E * is then simply a chain complex in E * .
2.14. Remark. Making a diagram commute while also forcing certain maps to be zero is more restrictive than simply making it commute. Thus, we would like to construct an analog of M x k tailored to the pointed case. Note that in a pointed category E * there is a canonical map * → any t, hence a section (2.15) Θ : of the projection map.
2.16. Definition. Given any diagram Y : J → E * , where J is a pointed weak lattice, define its reduced matching space (for x and k) as the object of E defined by the pullback: which also determines the maps ι x k and forget. In effect, we have replaced any factor indexed on a map in J by * , like reducing modulo the ideal J , precisely as one would expect for a pointed diagram.
We then have the following analogues of Lemmas 2.7 and 2.8: 2.17. Lemma. Given a pointed functor Y : ∂J x k → E * , a pointed extension to Y : J x k → E * is (uniquely) equivalent to a choice of an object Y (x), together with a morphism in 2.18. Lemma. If |x| > k > 0, a pointed functor Y : ∂J x k → E * (for J and E * as above) induces a pullback square: where σ s <k is as in §2.3, and Ψ = Ψ Proof. Follow the proof of Lemma 2.8, with J replacing J . The absence of factors indexed in J implies that the structure maps Y (h) from the pullback of (2.19) to the copy of Y(s) indexed by h : X → s is the zero map whenever h ∈ J , so the result follows from the absorbing property of J .
From the two lemmas we have: for each |x| > k > 0.
Proof. Let k = |x| − 1, and consider a lifting square for m x k with respect to an acyclic cofibration α; extend the diagram to include m x k : Note that a lift in the outer, distorted square will serve as a lift for the inner square, since ι x k is a base change of another monomorphism, so is itself monic. To show that M x k (Y ) is fibrant in E * whenever Y is pointed Reedy fibrant, we adapt the argument of Lemma 15.3.9(2) through Corollary 15.3.12(2) of [Hir], as follows: Given a lifting diagram in E * , we construct the dotted lift by induction on 0 ≤ k < n. For a pointed Reedy fibrant object, we assume the zero entries are each fibrant, so their product M x 0 will also be fibrant. For the induction step, suppose we have a lift in the diagram (2.24) Note that the structure for any f : x → s with |s| = k induces a commutative diagram combining the previous two, the lift h f exists because Y was assumed to be pointed Reedy fibrant. All of these maps together define h 0 : D → Y(s).
Compatibility with lower degree pieces then implies that h 0 factors through the limit defining M Proof. In the following commuting diagram: factor the top horizontal composite as an acyclic cofibration will allow us to construct inductively a weak equivalence between the new diagram Y and the standard Reedy fibrant replacement Y for Z.

General Definition of higher order operations
From now on E will be a model category, and we assume given a "homotopy commutative diagram" in E -that is, a functor Y : J → ho (E), with J as in §2.1. Our higher homotopy operations will serve as obstructions to rectification of such a Y -that is, lifting it to Y : J → E.
We may assume for simplicity that each Y (s) is both cofibrant and fibrant, which can always be arranged without altering any homotopy types (see §3.2).
3.1. The double induction. We attempt to construct the rectification Y by a double induction: I. In the outer induction, we assume we have succeeded in finding a functor Y n : J n → E (Y n is assumed to be Reedy fibrant), realizing Y | J n . In fact, for our induction step it suffices to assume only the existence of Y n+1 : J n+1 → ho(E) extending Y n . II. By the Reedy conditions, lifting Y n+1 to Y n+1 : J n+1 → E extending Y n is equivalent to extending the latter to a point-wise extension Y x n : J x n → E for each x ∈ Obj J of degree n + 1 separately.
Given such an x, the restriction of Y n produces a diagram Y k : ∂J x k → E for each k ≤ n and the restriction of Y n+1 produces a diagram Y x k : J x k → ho (E), with the two remaining compatible. Thus, for our inner induction hypothesis, assume a pointwise extension of Y k−1 at x (agreeing with appropriate restrictions of both of these) has been chosen, so Y x k−1 : and Y k , with the final case of the inner induction being k = n. Notice, our inner induction step is equivalent to making coherent choices for each homotopy class of maps out of x to an object of degree k, leaving all maps not involving x (so those from Y k ) or maps into objects of lower degree (so those from Y x k−1 ) unchanged. By Lemma 3.3 below, we may start the inner induction with Y x 0 defined by the values on objects of Y x 0 . The assumption that Y n is Reedy fibrant implies that Y 1 is Reedy fibrant, too, which will allow us to use the homotopy pullback property to extend Y x 0 to Y x 1 . The general step in the inner induction will use Lemma 2.8: By assumption, we have a map into the lower left corner of (2.9), which we want to extend to a map into the upper left corner still representing the appropriate class required by Y x k . 3.2. Remark. Our induction assumption that the diagram Y n is Reedy fibrant implies that Y n (t) is fibrant in E for each t ∈ Obj J n , and the same will hold for the pullbacks that we consider below (see, e.g., §3.13). We will assume in addition that in the inner induction, for each Together this will ensure that the left and right homotopy classes, appearing in various results from the Appendix, coincide (cf. [Hov,1.2.6]), and the distinction can thus be disregarded.
Theorem 4.24 then yields an obstruction theory for this step in the inner induction.
3.3. Lemma. In the setup described in §3.1 given x ∈ Obj J with |x| > 0: (a) Any choice of representatives for a homotopy commutative has no non-trivial compositions by definition.
where the right vertical is a fibration (being a product of fibrations by the Reedy fibrancy assumption). This is a special case of (2.9) where the forgetful (inclusion) map on the lower left is the identity, since ∂J x 0 is discrete. Note that the outer diagram commutes up to homotopy, since it simply compares composites representing maps in Y x 1 in a somewhat unusual presentation. By Lemma A.5, we can then alter the dashed map σ Y 1 within its homotopy class to obtain the dotted map m x 1 into M x 1 . Equivalently, by Lemma 2.7 one can find a representative of Y x 1 extending to J x 1 without altering the restriction to ∂J x 1 (although this may not be the original Y x 1 , since we might have altered σ Y 1 within its homotopy class when applying Lemma A.5).
3.5. Remark. Using Lemma 3.3, we shall henceforth assume that in the inner induction we may start with k ≥ 1. In order to ensure Reedy fibrancy for k = 1, we factor m We must verify that Y (x) and m x 1 may be chosen in such a way that the maps to the other objects Y (s) (with |s| > 1) have the correct homotopy type. However, by assumption all such objects Y (s) are fibrant, so we can use the left lifting property for to ensure that α and α have the same homotopy class.
In the inner induction on k, we build up the diagram under the fixed x ∈ Obj J by extending Y x k−1 to objects in degree k, using: at the target of f g. Moreover, given a morphism showing that we have a cone, and thus a map ρ(g) to the limit.
3.8. Definition. A pullback grid is a commutative diagram tiled by squares where each square, hence each rectangle in the diagram, is a pullback.
Next, we embed the maps ρ k−1 and m x k−1 in a pullback grid, in order to apply Lemma 2.8: 3.9. Lemma. Assuming |x| > n ≥ k ≥ 2, any functor Y x k−1 : J x k−1 → E induces a pullback grid defined by the lower horizontal and right vertical maps, with the natural (dashed) maps into the pullbacks: Proof. To verify commutativity of the outer diagram, note that for each composable pair We now set the stage for our obstruction theory by combining all of these pieces in a single diagram: 3.11. Proposition. Assuming |x| > n ≥ k ≥ 2, any functor Y k : ∂J x k → E as in §3.1 induces maps into a pullback grid: Here Proof. The outer pullback is M x k by Lemma 2.8 and the fact that m s k−1 followed by the inclusion "forget" is σ s <k (cf. §2.3). Note that the lower half of the grid involves only objects of J in degrees < k, so the fact that Y k agrees with Y x k−1 : J x k−1 → E implies that β k−1 and η k−1 exist, by Lemma 3.9.
The outer diagram commutes up to homotopy because ( and lifts Y x k , which is homotopy commutative. Since the upper left square is a pullback, producing a lift of β k−1 : The fact that we only alter σ x k within its homotopy class ensures that r k •α k ∼ σ x k , with the left hand side serving as the replacement for the right hand side. 3.13. Remark. The problem here is that even though the two maps from Y(x) into the lower right corner of (3.12)) agree up to homotopy, this need not hold for the two maps into J (x,s) M s k−1 , the middle term on the right. Thus we cannot simply apply Lemma A.5 to work with just the upper half of (3.12).
In connection with Remark 3.2, one should note that all three of the objects along the right vertical edge of (3.12) are fibrant in E. The top and bottom objects are products of entries we assumed were fibrant. However, the middle object is a product of the usual Reedy matching spaces for the factors in the product above, so by [Hir,Cor. 15.3.12 (2)], our assumption of Reedy fibrancy implies these factors are also fibrant.
Lemma 2.22 implies that this holds in the pointed case, too.
3.14. The Total Higher Homotopy Operation. Following our inner induction hypothesis as in §3.1(II), assume given Y (If we want a canonical choice of F 1 , we will use the product of free path spaces for the non-zero factors appearing in the target and the reduced path space for each zero factor (see §2.B), with ι defined by the constant paths for non-zero factors.) We then pull back the right vertical maps of (3.12) to produce the following pullback grid, with fibrations indicated as usual by → → : where the outermost diagram commutes up to homotopy (and the map η k−1 exists by Lemma 3.9). In order to construct a lift Y x k : J x k → E, by Proposition 3.11, we need to produce the dotted map α k with p k−1 • α k = η k−1 and r k The problem is that the large square is a strict pullback, but not a homotopy pullback, so the outermost diagram commuting up to homotopy is not enough.
However, the top left square is a pullback over a fibration, so by Lemma A.5 producing α k is equivalent to finding a map κ with µ • κ ∼ γ • η k−1 and r ′ k • κ ∼ σ x k . Moreover, Lemma A.5 applies to the right vertical rectangle, which implies that choosing κ is equivalent to finding a map ϕ in the same homotopy class as the composite ι • σ x <k , making the outer diagram commute. Thus, the only question is whether the two composites Y(x) → F 2 agree: that is, given ϕ, with the map κ induced by ϕ (for which necessarily r Definition. We define the total higher homotopy operation for x to be the set We may apply Corollary A.10 to the right vertical rectangle with horizontal fibrations, since by assumption the outer diagram commutes up to homotopy. This implies that the subset Y x k−1 of Definition 3.16 is non-empty: i.e., some such ϕ and so some κ and in turn some θ, exist. Thus the total higher homotopy operation is defined at this point. The total higher homotopy operation vanishes if there is such a κ with µ This somewhat incongruous terminology of "vanishing" is explained by the following. 3.18. Proposition. Assume given Y : J → ho (E) with J a weak lattice, and Proof. Note that forget is a monomorphism, since the class of monomorphisms is closed under categorical products and the inclusion of a limit into the underlying product is always a monomorphism. Thus, the last statement in Corollary A.10 After possibly altering κ (and so µ • κ, ϕ, and r ′ k κ) within their homotopy classes, by Lemma A.5 applied to the upper left square in (3.15) we then have a dotted map we still have the same homotopy commutative diagram since κ ′ ∼ κ. Moreover, if we disregard the dashed arrows κ and ϕ, the remaining solid diagram commutes on the nose, since q 3.19. Remark. As a consequence of Proposition 3.18, our total higher homotopy operations are the obstructions to extending a certain choice of representative of a (k)-truncation of a homotopy commutative diagram in order to produce a (k + 1)truncated representative. As in any obstruction theory, if the obstruction does not vanish at a certain stage, we must backtrack and reconsider earlier choices, to see whether by altering them we can make the new obstruction vanish at the stage in question.
It is natural to ask more generally whether there is any (k + 1)-truncated (strict) representative of the given homotopy commutative diagram. Rephrasing this in our context, we ask whether for any choice of a (k)-truncated representative our obstruction sets contain the particular class which constitutes "vanishing". In those cases where one can identify the ambient collections of homotopy classes of maps with one another, a positive answer to the more general question is equivalent to that particular class lying in the union of our obstruction subsets.

Separating Total Operations
At this level of generality, we cannot expect Proposition 3.18 to be of much help in practice: its purpose is to codify an obstruction theory for rectifying certain homotopy-commutative diagrams, using the double induction described in §3.1.
We now explain how to factor the right vertical map of (3.12) or (3.15) as a composite of (mostly) fibrations with a view to decomposing the obstruction Y x k−1 into more tractable pieces. A key tool will be the following 4.1. The Separation Lemma.
Assume given a solid commutative diagram as follows: in which: • all rectangles are pullbacks, • the indicated maps are fibrations, • the objects F 0,1 x,k and F j,k 0,k are fibrant, and • the vertical map z is a monomorphism.
Note that as a consequence, all objects in the diagram, other than possibly P x,k by Φ j , so Φ k−1 = µ k−1 , and also define ϕ k to be the identity on Q x k−1 with q k = γ k . In addition, let β j denote the vertical composite F j,k+1 x,k → F j,j+2 x,k . Now assume that we also have a map κ 0 : Then by Lemma A.5 applied to the right vertical rectangle (with horizontal fibrations) there exists κ 1 with u 1 • κ 1 = κ 0 and r 1 • Φ 1 • κ 1 ∼ ϕ 1 • η k−1 . We are interested in decomposing the question of whether s • κ 1 ∼ Γ k−1 • η k−1 into a series of smaller questions. This question will become important once we demonstrate it to be an instance of asking for a total higher homotopy operation to vanish.
If it is true that q 2 • ϕ 2 • η k−1 ∼ Φ 1 • κ 1 , then Lemma A.5 for the next vertical rectangle imply the existence of the dashed map κ 2 , such that u 2 • κ 2 = κ 1 and r 2 • Φ 2 • κ 2 ∼ ϕ 2 • η k−1 . Proceeding in this manner, and assuming the maps into the indicated "staircase terms" remain homotopic, even though we are only certain they agree up to homotopy after applying the relevant r j , one produces κ k−1 such that and if so, it follows by composing with most of the rectangle across the top of the diagram that s • κ 1 ∼ Γ k−1 • η k−1 . In fact, we will be able to characterize when this procedure is possible in terms of obstructions, which we will view as "separated" versions of the total higher homotopy operation corresponding to the original question.

Separation Lemma. Given the pullback grid as indicated above along with a choice of κ
The reader should note that with our conventions, in the final case j = k − 1, the conclusion is that γ k • η k−1 ∼ µ k−1 • κ k−1 .

4.4.
Corollary. If either of the two equivalent conditions of Lemma 4.2 holds, then by changing κ 1 : Y(x) → F 1,k+1 x,k within its homotopy class, (and so using its image under u 1 to replace κ 0 within its homotopy class as well) but without altering Proof of Corollary 4.4. This follows from Lemma A.5, since the long horizontal rectangle across the top of the diagram is a pullback over a vertical fibration. 4.5. Remark. In the case we have in mind, F 0,1 x,k will be a product of objects Y(s), as will F 0,k+1 x,k , this time with |s| = k, and F 0,k x,k will be the corresponding product of matching objects M s k−1 , which will be fibrant by [Hir,Cor. 15.3.12 (2)]. Later, we will also have a pointed version, instead relying on pointed Reedy fibrancy and Lemma 2.22. Note that the second vertical map in each column of the grid is not required to be a fibration, but instead a monomorphism. Recall that monomorphisms are closed under base change and forgetting from a limit to the underlying product is always a monomorphism, so its first factor in any factorization must also be a monomorphism, hence these conditions will arise naturally in our cases of interest.
Proof of Lemma 4.2. We will repeatedly apply Lemma A.5 using a vertical rectangle with horizontal fibrations, with κ j−1 as p and ϕ j • η k−1 as f , showing κ j exists and satisfies (4.6) Since κ 1 exists by the assumption on κ 0 , which is really (4.7) for j = 1, we begin the induction by assuming κ 1 satisfies (4.7) for j = 2, in which case κ 2 exists and satisfies (4.6) for j = 2. Now assuming the stricter condition (4.7) for j = 3 implies the existence of κ 3 satisfying (4.6) for j = 3, and so on. When our induction constructs κ k−1 satisfying (4.6) for j = k − 1, we assume the stricter condition (4.7) for j = k, which, as noted above, is the statement that γ k •η k−1 ∼ µ k−1 •κ k−1 . However, then composing with the horizontal rectangle across the top of the diagram from µ k−1 to s implies the constraint Γ k−1 • η k−1 ∼ s • κ 1 .
On the other hand, if κ 1 satisfies the constraint Γ k−1 • η k−1 ∼ s • κ 1 , then we proceed by applying Lemma A.5 inductively to each square along the top of the diagram using κ j−1 for p and γ k • η k−1 followed by the composite F k−1,k x,k → F j,k x,k for f , exploiting the horizontal fibrations in the rectangle. This yields κ j satisfying more than (4.7), since the homotopy relation is satisfied up in F j,k x,k , and this also implies (4.6) by construction.
, assume that we can refine diagram (3.15) (used to define Y x k−1 , the total higher homotopy operation for x) to a pullback grid as in Lemma 4.2. Then F 0,k+1 , in conformity with Remark 4.5, while one of the two equivalent conditions in Lemma 4.2 is the vanishing of the total higher homotopy operation. Recall the vertical composite F j,k+1 4.8. Definition. If we can produce a pullback grid as in Lemma 4.2 refining diagram (3.15), then for each 1 ≤ j < k, the associated separated higher homotopy operation for x of order j + 1, denoted by Y x k−1 j+1 , is the set of homotopy classes of maps θ : Y(x) → F j,j+1 x,k such that: (using the notation of the top two rows of vertical arrows in §4.1).

We say that
x,k as above if θ ∼ q j+1 • ϕ j+1 • η k−1 (in the notation of the Lemma), and we say it vanishes if it vanishes at some value.
can then be defined and by Corollary A.10 each θ j+1 will satisfy θ j+1 ∼ Φ j • κ j . Thus, the vanishing of some value θ j+1 becomes equivalent to assuming q j+1 • ϕ j+1 • η k−1 ∼ Φ j • κ j . In other words, the vanishing of Y x k−1 j+1 (scilicet at some map θ j+1 ) is a necessary and sufficient condition for Y x k−1 j+2 to be defined. (For comments on coherent vanishing, see Remark 4.9).

4.9.
Remark. Those familiar with other definitions of higher homotopy operations may have expected a stricter, coherent vanishing condition in order for a subsequent operation to be defined. However, this need not be made explicit in our framework, as it is a consequence of compatibility with previous choices.
For example, our version of the ordinary Toda bracket, denoted by f, g, h , is the obstruction to having a given (2)-truncated commuting diagram, satisfying just f • g = * , extending to a (3)-truncated diagram simply by altering h within its homotopy class to satisfy g • h = * , without altering g or f . Each choice of (2)truncation (of which there is at least one, by Lemma 3.3) has an obstruction which is a subset of the homotopy classes of maps [Y(3), Ω ′ Y(0)]. The usual Toda bracket is the union of these subsets: f, g, h = ∪ f, g, h . Thus, the more general existence question has a positive answer (i.e., a vanishing Toda bracket) exactly when, for some choice of (2)-truncation, the obstruction vanishes in our sense.
When defining our long Toda brackets, say f, g, h, k , we will begin by building the (3)-truncation only if the "front" bracket f, g, h vanishes for some choice of (2)-truncation, and we make an appropriate choice of h. At that point, we only consider values of the "back" bracket g, h, k which use the previously chosen maps g and h. Thus asking that our obstruction vanish is automatically a kind of coherent vanishing. If it does not vanish, we must alter our choice of (3)-truncation until we obtain a coherently vanishing "back" bracket. Once again, one interpretation of the traditional long Toda bracket would then be a union ∪ f, g, h, k , this time indexed over all possible strict rectifications of f, g, h , so all such 3-truncations.

Applying the Separation Lemma.
By Proposition 3.18, a necessary and sufficient condition for the inner induction step in §3.1 is the vanishing of the total higher homotopy operation Y x k−1 -that is, by Lemma 2.7, the existence of a suitable map m x k in (3.12). According to Proposition 3.11, this in turn is equivalent to having a map κ in (3.15) satisfying a certain homotopy-commutativity requirement.
In order to apply Lemma 4.2, we need to break up the lower right square of (3.15) into a pullback grid (which then induces a horizontal decomposition of the upper right square). This will be done by decomposing the lower right vertical map, which is a product (over J (x, s), with |s| = k) of the forgetful maps M s k−1 → J (s,v) Y(v) (with |v| ≤ k − 1). The target of this forgetful map can be further broken up as in (2.9) to a product over |v| = k − 1 and one over |v| < k − 1. 4.11. Example. When |s| = 3, we factor the top horizontal arrow in (2.9) as a weak equivalence followed by a fibration: Similarly, we can factor the map in (3.10) from N s 1 to the product of lower degree copies of Y(t) to produce a factorization for the lower degree forgetful map in (2.9). Together these yield a factorization of the full forgetful map: (4.14) with the second map a fibration and the first necessarily a monomorphism, since the composite is a monomorphism as the inclusion of a limit into the underlying product. Precomposing with structure maps Y(s) → → M s 2 (which are fibrations, because we assumed our diagram Y was Reedy fibrant) yields This is a refinement of the right column in (3.15), in which all but the second map is a fibration, and that second map is a monomorphism. Taking (4.15) as the right column in the diagram of Lemma 4.2, we pull it back along the bottom row of (3.15) to get the two right columns of the intended diagram, as shown in (4.16).
For the next column, note that the two maps out of Q x 2 in (3.15) induce a map Q x 2 → F 1,2 x,3 , in the notation of (4.16). Factoring this as an acyclic cofibration followed by a fibration: x,3 → → F 1,2 x,3 and taking pullbacks yields the required pullback grid: Note that F 1,1 x,3 is the F 1 of Definition 3.16, while F 1,3 x,3 is F 2 -that is, the target of our total higher operation θ. Separation Lemma 4.2 tells us that this operation vanishes precisely when the following two "separated" operations vanish: (a) The first, landing in F 1,2 x,3 , being defined by the two composite maps from Y(x); (b) The vanishing of the first yields a second map into F 2,3 x,3 , where this second map defines the values of the second of the "separated" operations, and the formally defined first map defines the possible vanishing of such operations.
This example is indicative of the general pattern, described by: we have a pullback grid as in Lemma 4.2, these induce a pullback grid: to P x k . Note that the two top right slots in (4.18) are consistent with Remark 4.5.
Proof. We prove the Lemma by induction on k, beginning with (4.16) for k = 3. We start with a decomposition of the top map in (2.9), where all but the first map are fibrations; this first map is a monomorphism since the composite is such, being the inclusion of a limit into the underlying product. This is generated using Step k − 1 in the induction, by precomposing the top row in (4.18) for k − 1 with the map M s k−1 → P s k−1 of (3.12).
For N s k−1 → Q s k−1 → |s|=k M s k−1 (the middle row of (3.12)), we pull back the right column of (4.18) for k − 1 along the generalized diagonal Ψ of (2.4) to obtain a sequence of pullbacks (4.20) for each 1 ≤ j ≤ k − 3, where the right vertical map is a fibration by the induction assumption.
For j = k − 2, we instead factor the composite of the top row in: into an acyclic cofibration i followed by a fibration r, as shown (where the top maps are those of (3.12) and (4.18) for k − 1, respectively). Precomposing this with the map M s k−1 → N x k−2 of (3.12) and then taking products as in Example 4.11 yields the desired factorization of the forgetful map: Now factor the next generalized diagonal Ψ x k−1 as an acyclic cofibration followed by a fibration p 1,1 : F 1,1 x,k → → J (s,v) |v|<k Y(v). Pulling back the tower (4.22) along p 1,1 yields the second column on the right in our new grid (4.18). The total higher operation will then land in the twice-boxed pullback object F 1,k x,k . To construct the j-th column from the right (j ≥ 2), with entries F j+1,• x,k , factor the previously defined map Q x k−1 → F j,j+1 x,k as an acyclic cofibration Q x k−1 ∼ → F j+1,j+1 x,k followed by a fibration p : F j+1,j+1 x,k → → F j,j+1 x,k . We then pull back the (j − 1)-st column along p to form the j-th column of (4.18).
Note that upon completion of this process, the map Q x k−1 → F k−1,k x,k need not be a fibration, but the vertical maps in the upper left square are fibrations, by successive base-change from the product of maps Y(s) → → M s k−1 , each of which is a fibration by Reedy fibrancy of Y k . 4.23. Definition. The diagram of Lemma 4.2, when constructed inductively as in Lemma 4.17, will be called a separation grid for Y k .
Combining Lemma 4.17 with the Separation Lemma 4.2 and Corollary 4.4 yields the following refinement of Proposition 3.18: 4.24. Theorem. Assume given Y x k : J x k → ho (E), Y x k−1 : J x k−1 → E and a Reedy fibrant Y k : ∂J x k → E as in §3.1(II) for |x| > n ≥ k ≥ 2. Then our total higher homotopy operation separates into a sequence of k − 1 obstructions and the following are equivalent: (1 ≤ j < k) of separated higher homotopy operations of §4.8 vanish (so in particular each in turn is defined).

4.25.
Remark. The machinery of the separated higher homotopy operations has been formulated to agree with (long) Toda brackets in pointed cases. We shall deal with these in Section 7, after a more detailed study of the special issues involving pointed diagrams. In particular, the role of Q x k−1 will be played by a point, so the weak equivalence followed by a fibration factorizations out of it will be provided by taking reduced path objects on the target. However, we first present a simple example of the (less familiar) general unpointed situation before focusing on the details for the pointed situation.

Rigidifying Simplicial Diagrams up to Homotopy
A commonly occurring instance of a homotopy-commutative diagram which needs to be rectified are restricted (co)simplicial objects, also known as ∆-simplicial objects (i.e., without (co)degeneracies). Examples appear in [BJT1,§6], [BJT3, §4.1], [B2, §5], and implicitly in [May, Se, Pr], and more. We now show how the double inductive approach described in §3.1 applies to such diagrams. We denote the objects of the simplicial indexing category ∆ by 0, 1, . . . , n, . . . , with the value of Y : ∆ → E at n thus denoted by Y(n) instead of the usual Y n .

1-Truncated ∆-Simplicial Objects.
We start the outer induction with n = 0. Our 1-truncated diagram in ho(E) then consists of a pair of parallel arrows, so we have only the stage k = 0 in the inner induction: this means choosing representatives for each of the two face maps d 0 , d 1 : Y(1) → Y(0). Making this Reedy fibrant means changing the combined map (d 0 , d 1 ) : Y(1) → Y(0) d 0 × Y(0) d 1 into a fibration (i.e., factoring this as Y(1)
To define Y 2 0 : J 2 0 → E at stage k = 0 in the inner induction, pick representatives for each of the full length composites: in this case, the three maps Y(2) → Y(0) denoted by d 0 d 1 , d 0 d 2 , and d 1 d 2 in canonical form. This means M 2 0 is the product of three copies of Y(0) indexed by d i d j (0 ≤ i < j ≤ 2), and our choice of representatives yields a single map m 2 0 into the product. At stage k = 1, we must first choose representatives for the components of σ 2 1 ( Y 2 1 ) -that is, for the maps d 0 , d 1 , and d 2 : Y(2) → Y(1), which are all the maps 2 → 1 in J ). The generalized diagonal map Ψ = Ψ 2 1 of (2.4) takes Y(0) d i d j (i < j) to the product Y(0) d i d j × Y(0) d j−1 d i , in accordance with the simplicial identities. Note that the target of σ 2 1 is 0≤j≤2 Y(1) d j .
Thus we have a pair of maps into a pullback diagram: where the outer diagram commutes up to homotopy (for any choice of representatives for d 0 , d 1 , and d 2 ). The dotted map exists by Lemma A.5 (after possibly altering the dashed map within its homotopy class), yielding a full 2-truncated ∆-simplicial object (which rectifies Y 2 1 ) by Lemma 2.7. Changing m 2 1 into a fibration provides us with a Reedy fibrant replacement Y 2 : ∂J 2 → E.

3-Truncated ∆-Simplicial
Objects. At stage n = 2 (with x = 3), for the first time we are in the situation of §3.16, somewhat simplified by the fact that we have a single object n in each grading n of J = ∆. In particular, we will have no separated operations yet.
In the inner induction, for k = 0, we choose representatives for each full length map in Y 3 2 to obtain Y 3 0 ; the full length composites are the four maps d i d j d ℓ with 0 ≤ i < j < ℓ ≤ 3, so M 3 0 is a product of four copies of Y(0) indexed by these maps, and the generalized diagonal of (2.4) takes each copy of We make an initial choice (to be modified below) of σ 2 1 ( Y 2 1 ) (i.e., of each composite d j d ℓ : 3 → 1 for 0 ≤ j < ℓ ≤ 3 within its homotopy class). Again this yields a pair of maps into a pullback diagram: where the right vertical is a product of fibrations Y(1) → M 1 0 = i≤1 Y(0) d i (by Reedy fibrancy of Y 2 ).
Since Y 3 2 is homotopy commutative, by Lemma A.5 we obtain a dotted map m 3 1 (after altering the dashed map -that is, our choice for each d j d ℓ -within its homotopy class). By Lemma 2.7 this yields Y 3 1 , still representing Y 3 2 .
It is at stage k = 2 in the inner induction that we first encounter a possible obstruction: we must now choose representatives for d ℓ : 3 → 2 (0 ≤ ℓ ≤ 3) in the homotopy class given by Y 3 2 . As in (2.9), we know that the target of the forgetful map from M 3 1 is the product of the lower left and upper right corners of (5.5). Thus Ψ = Ψ 3 2 is a product of two maps: the first taking each factor Y(1) As in §3.16, we now factor Ψ as a trivial cofibration to F 1 followed by a fibration Ψ ′ , and pull back the product of the forgetful maps as in (5.3), indexed by the first face maps d ℓ : 3 → 2 (0 ≤ ℓ ≤ 3) along Ψ ′ to obtain a "potential mapping diagram" as in (3.15): Note that as in §3.16, we may choose F 1 to be a product of free path spaces, so we can think of ϕ as a choice of homotopies between the various decompositions in Y 2 of maps 3 → 0 in ∆. As the right vertical rectangular pullback has horizontal fibrations, we can apply Lemma A.5 and the fact that the original outermost diagram commutes up to homotopy (because Y 3 2 is homotopy commutative) to deduce that there is a map ϕ in the correct homotopy class, yielding κ as indicated.
The question is whether µκ ∼ γη 1 . By Corollary A.10, our secondary operation consists precisely of those [θ] satisfying θ ∼ µ • κ. Thus, the question is answered in the affirmative precisely when our secondary operation Y 3 2 vanishes. In that case, by Lemma A.5 applied to the upper left square, with µ a fibration, we can find κ ′ ∼ κ satisfying µ • κ ′ = γ • η 1 , so inducing the dotted α 2 by the pullback property. We then alter the map labeled (d 0 , d 1 , d 2 , d 3 ) within its homotopy class by instead using r ′ 2 • κ ′ , which will make the entire diagram now commute, since <2 . Thus, we obtain a full 3-truncated ∆-simplicial object Y 3 (if we wish to proceed further, we take a Reedy fibrant replacement).
If Y 3 2 does not vanish, then there is no way to extend this Y 2 to a full 3-truncated object. 5.6. Remark. As with any obstruction theory, when Y 3 2 does not vanish, we need to backtrack, and see if we can get our obstruction to vanish by modifying previous choices. We observe that in special cases, given a truncated ∆-simplicial object, there is a formal procedure for adding degeneracies to obtain a full (similarly truncated) simplicial object (see, e.g., [B1, §6]).

Pointed higher operations
Most familiar examples of higher homotopy operations are pointed, so we now describe the modifications needed in our general setup when the indexing category J , as well as the model category E, are pointed (see §2.B). This will also cover "hybrid" cases, where certain composites in the diagram are required to be zero in E, rather than just null homotopic.
For (a), note that if g ∈ J , Y(g) must be the zero map, but otherwise any choice of lifting will do, since J x 0 has no non-trivial compositions. For (b), follow the proof of Lemma 3.3 with J replacing J , using reduced matching spaces and Definition 2.21 for the fibrancy.
We also have the following version of Lemma 3.9: 6.2. Lemma. Assuming 2 ≤ k ≤ n < |x|, any pointed functor Y : J n → E * with a pointed extension to J x k−1 induces a pullback grid with natural dashed maps : We then deduce the following analogue of Proposition 3.11 (with a similar proof): 6.4. Proposition. Assuming 2 ≤ k ≤ n < |x|, any pointed functor Y k : ∂J x k → E as in §3.1 induces maps into a pullback grid: Again, the dashed map only makes the outermost diagram commute up to homotopy. Furthermore, the dotted map m x k exists (after altering σ x =k ( Y x k ) within its homotopy class) if and only if there is a dotted map α k such that p k−1 α k = η k−1 and r k α k ≃ σ x =k ( Y x k ).
With this at hand, we may modify Definition 3.16 as follows to obtain a sequence of obstructions to extending pointed diagrams: 6.6. Total Pointed Higher Homotopy Operations.
Assume given pointed and pull back the right column of (6.5) along Ψ ′ to obtain the following pullback grid: As in §3.16, Lemma A.5 allows us to modify ϕ so as to obtain a map κ : Y(x) → F 3 into the pullback.
6.8. Definition. We define the total pointed higher homotopy operation for x to be the set Y x k−1 of homotopy classes of maps θ : vanishes if it vanishes at some value θ.
6.9. Remark. In many cases of interest we will have Q x k−1 ≃ * , in which case the pointed operation Y x k−1 vanishes at θ precisely when θ ∼ * , as one might expect, so the subset vanishes precisely when it contains the zero class.
We have chosen our definitions so as to have the following analogue of Proposition 3.18: does not vanish, then there is no choice of ϕ for which such a lift exists, and so there is no pointed extension compatible with the given choices. 6.11. Remark. Given pointed functors Y x k : J x k → ho(E * ), Y x k−1 : J x k−1 → E * and a pointed Reedy fibrant Y k : ∂J x k → E * as in §3.1(II) for |x| > n ≥ k ≥ 2, we may define separated pointed higher homotopy operations Y x k−1 j+1 for x as in Definition 4.8, using a refinement of (6.7) constructed mutatis mutandis with products over J (x, s) replaced everywhere by products over J(x, s).
Separation Lemma 4.2 is stated in sufficient generality to apply here, too, with Remark 4.5 modified accordingly, yielding the following variant of Theorem 4.24: Then the total pointed higher homotopy operation separates into a sequence of k − 1 pointed operations, and the following are equivalent: ( (1 ≤ j < k) of separated pointed higher homotopy operations of §4.8 vanish (so in particular each in turn is defined).

Long Toda Brackets and Massey Products
We are finally in a position to apply our general theory to the two most familiar examples of higher order operations: (long) Toda brackets and (higher) Massey products. Since both are cases of the (pointed) higher operations fully described in Sections 3-4 and 6, we thought it would be easier for the reader to consider two specific examples in detail, briefly indicating what needs to be done for the higher version.

7.A. Right justified Toda brackets
Since the ordinary Toda bracket (of length 3) was treated in Section 1, we start with the next case, the Toda bracket of length 4 (the first example of a long Toda bracket in the sense of [Wa]).
Thus, if E * is a pointed model category, assume given a diagram Y : J → ho E * of the form with each adjacent composite null-homotopic: that is, a chain complex of length 4 in ho E * , as in Example 2.13 (compare (1.2)). Without loss of generality, we can assume all objects involved are both cofibrant and fibrant. Applying the double induction procedure of §3.1, we see that we must deal with chain complexes of length n ≤ 4, as follows: (a) When n = 0, we have no inner induction, and making the result Reedy fibrant consists of factoring the representative to produce a fibration f : Y(1) → → Y(0) in the specified class [f ].
(b) When n = 1, note that J(x, t) is empty if |x| − |t| > 1, for this pointed indexing category, so as a consequence M In this case, the indexing set for products in the right column of (6.7) is the singleton J(3, 2), while the forgetful map in the bottom row of (6.5) is the identity of the zero object, with Ψ the zero map.
Factoring Ψ as a trivial cofibration ι followed by a fibration Ψ ′ , as in the bottom row of (6.7), and pulling back the right column yields the diagram: Thus F 1 is a model for the reduced path space on Y(1), with Ψ ′ the path fibration. However, since f was chosen above to be a fibration, the composite F 1 → Y(0) is a fibration, too, with F 1 contractible, so we see that F 2 , being the pullback of the dotted rectangle, is a model for the loop space ΩY(0), which we denote by Ω ′ Y(0). Similarly, M 2 1 is a model for Fib(f ).
Our total secondary pointed homotopy operation Y 3 1 (cf. §6.8) is thus a set of maps θ : Y(3) → Ω ′ Y(0), and it vanishes when this set contains the zero map (cf. Remark 6.9). This is our usual Toda bracket f, g, h , described in the language of Section 6. (d) In order for our four-fold Toda bracket f, g, h, k (denoted by Y 4 2 above) to be defined, Y 3 1 must vanish. This allows us to choose a pointed extension Y 3 : J 3 → E * of Y 2 which realizes Y | J 3 . The fact that the diagram Y 3 has realized Y through filtration degree 3 means that each of the maps g and h factors through the fiber of the previous one, as in the following solid commutative diagram: Making Y 3 pointed Reedy fibrant ( §2.21) just means ensuring that the maps h 1 and g 1 are fibrations. The separation grid of Lemma 4.2 then takes the form: where we have extended the pullback grid downwards, and to the right, to show how it was constructed from the previous case (diagram (7.2)) using Lemma 4.17. We have also indicated how (representatives of) the maps of (7.3) fit in.
As in Step (c) above, we can identify (using horizontal fibrations in the larger square beneath it), and F 2,3 4,3 is a model for Ω 2 Y(0) (now using the rectangle with diagonal corners F 2,3 4,3 and F 1,2 3,2 , along with the previous identification of the latter). Similarly, M 3 2 is a model for Fib(g 1 ) of (7.3), while P 4 3 is Fib(h 1 ) (which is also the homotopy fiber). See (7.11) below for the full identification.
Therefore, the final obstruction to having a dotted lift k 1 in (7.3) (or (7.4)) is the composite k • h 1 .
Note that there are no factors of type G k,ℓ i,j as in (4.18) here, since we can always choose the zero map as our factorization of the zero map between zero objects. 7.5. Remark. Our total pointed tertiary homotopy operation Y 4 2 is a set of homotopy classes θ : Y(4) → ΩM 2 1 . However, using Lemma 4.2, we can replace it by two separated higher homotopy operations for 4, in the sense of §4.8: vanishes, the third order operation Y 4 2 3 ⊆ [Y(4), Ω 2 Y(0)] is defined, and serves as the final obstruction to lifting Y . By definition, this is our four-fold Toda bracket f, g, h, k . 7.6. Lemma. Given a pointed Reedy fibrant diagram Y 3 realizing (7.1) through filtration 3, the associated second order separated higher homotopy operation Y 4 2 2 is our usual Toda bracket g, h, k .
Proof. Note that F 1,3 3,2 is a model for the homotopy fiber of g : Y(2) → Y(1) (which is not-itself a fibration). Thus, the rectangle with corners F 1,3 3,2 and Y(1) in (7.4) is a homotopy invariant version of the rectangle with corners F 2 and Y(0) in (7.2), used to define our Toda bracket in Step (c) above -this time, applied to the left 3 maps in (7.1). The map corresponding to θ in (7.2) -the value of the Toda bracket -is the map Y(4) → F 1,2 4,3 obtained by composing κ with the vertical maps F 1,4 4,3 → F 1,2 4,3 , which is indeed the definition of the value of Y 4 2 2 associated to our choices (see Definition 4.8). 7.7. Aside. Note that if the dotted forgetful map M 2 1 → Y(1) in (7.4) were a fibration, the horizontal dotted map above it would be a fibration, too, so right properness would imply that the vertical map M 3 2 → F 1,3 3,2 would be a weak equivalence. 7.8. Length n Toda brackets.
The general procedure described in Section 6 tells us what needs to be done for Toda diagrams (chain complexes Y in ho E * ): of arbitrary length n. We sketch the main features of the general construction, already discernible in the case n = 4 described above: In the double induction of §3.1, we can concentrate on the last stage -assuming the vanishing of shorter brackets on the right, which guarantees the existence of a solid diagram (7.10) analogous to (7.3); our length n Toda bracket, f 1 , f 2 , . . . f n−1 , f n , will be the final obstruction to finding the dotted map g n in (7.10), perhaps after altering f n within its homotopy class. The existence of the fibrations g k for 2 ≤ k < n, and the fact that f 1 is a fibration, mean that we have a lifting Y n−1 : J n−1 → E * of Y | J n−1 , which we have made pointed Reedy fibrant. The underlining in the notation represents our intention to leave that portion fixed.
The construction of the separation grid for Y n−1 ( §4.23) greatly simplifies, in this case, as we see in comparing (7.2) to (7.4): at each step, one writes the previous separation grid vertically (instead of horizontally) on the right (after changing the previously chosen g n−1 into a fibration, thus altering Y(n − 1) up to homotopy). We then factor the zero map Ψ and pull back the leftmost existing column to form a new column to its left. Factoring the zero map from Q x k−1 to the second place from the bottom in this new column and again pulling back, we note that the intermediate object produced by this factorization is a reduced path object, so by induction the entry immediately above it is a loop object (being the pullback over a fibration with upper right and lower left corners contractible -one because it is the reduced path object, and the other by induction). Moreover, the number of loops increases as we move up and to the left (see Lemma 8.3).
Repeat this step until the new column involves just two maps (so the second object from the bottom is at the same height as the product of the objects M s k−1 on the right). The pullback in the upper left corner is now the actual fiber of g n−1 . To illustrate, we reproduce diagram (7.4) with the pieces identified up to homotopy: Note that while not all the pullbacks in the grid can be easily identified, the targets of the separated operations (boxed) are iterated loop spaces on the original objects of (7.9), as one would expect for long Toda brackets. This last obstruction, consisting of a subset of the homotopy classes of maps into the top left iterated loop space, then represents our length n Toda bracket, f 1 , f 2 , . . . f n−1 , f n , with the lower separated higher homotopy operations corresponding to the vanishing of the lower obstructions necessary in order to define it (together with those already assumed to vanish in order to build the current commuting diagram).
The corresponding pointed diagram Y : J → ho T * has products of Eilenberg-Mac Lane spaces K i := K(R, i) in all but the top slot: x x r r r r r r r r r r r r r r r x r r r r r r r r r r r r r r where the central diamond represents associativity of the cup product maps µ; π 1 and π 2 are the two projections; and we have omitted the zero map from top to bottom that appears in (7.12) in the interest of clarity.
Choose a strictly associative model of the Eilenberg-Mac Lane Ω-spectrum in question (cf. [Ro]), with strictly pointed multiplication, so in particular at each level K r is a simplicial (or topological) abelian group. We can then make all of (7.13) below Y(g) (involving only the cup product maps) strictly commutative. Our Massey product will be the total pointed higher homotopy operation Y g 1 (for n = k = 2). From §2.16 we see that if we let K : The pullback grid of (6.7) then takes the form: (7.14) Thus a point in F 2 is given by (U, V, x, u, v, z, W ) ∈ P K r+s ×P K s+t ×M f 1 ×P K r+s+t with U : u ∼ * , V : v ∼ * , and W : xu = vz ∼ * . We thus have a natural map λ : F 2 → ΩK r+s+t × ΩK r+s+t sending (U, V, x, u, v, z, W ) to (xU − W, V z − W ). Postcomposition with the difference map d : ΩK r+s+t × ΩK r+s+t → ΩK r+s+t yields (xU − V z). Now Y(g) maps into the top right corner of (7.14) by (a lift of) (α, β, γ), and thereby on to M f 1 , and into the bottom middle term by ϕ := F, G, α, µ(β, γ), µ(α, β), γ, L , with L some nullhomotopy of µ(α, β, γ). Together these two maps induce the map θ : Y(g) → F 2 of §6.8.
The two factors of λ•θ merely give the usual indeterminacy for the Massey product, as we can see by choosing L := µ(F, γ) or L := µ(α, G).

7.15.
Remark. An alternative definition of the usual (higher) Massey products, more in line with that given for the Toda bracket, appears in [BBG,§4.1].

Fully reduced diagrams
Ultimately, we would like to develop an "algebra of higher order operations," along the lines of Toda's original juggling lemmas (see [T2, §1]). As a first step in this direction, we consider a special type of pointed diagram, which most closely resembles the long Toda diagram of (7.9).
The most useful property of the separated higher operations associated to Toda diagrams is that we can often identify their targets F j,j+1 x,k as loop spaces (as we saw in (7.11)).
It turns out the property of the pointed indexing category J needed for this to happen is the following: 8.1. Definition. A pointed indexing category J as in §2.12 is called fully reduced if any morphism decreasing degree by at least 2 lies in J .
Furthermore, the map forget of §2.16 factors through J(s,t),|t|=|s|−1 Y(t), so no factors of type G k+1,j x,k (cf. (4.20)) are needed when constructing the separation grid (4.18). This also implies that F j,j x,k is contractible for j < k, which is the key ingredient for identifying the targets of the separated operations as loop spaces.
Our key decomposition result is the following. 8.3. Lemma. If J is a fully reduced pointed indexing category and n ≥ k ≥ j ≥ 2, we have: in (4.18), where each f i is a non-identity map in J, with target of degree i.
Proof. We prove this by induction on k (for fixed n and x), as in Lemma 4.17. In each case, we combine two pullbacks over fibrations, one of which has fiber identified at an earlier stage, with two corners contractible; the upper left corner (source) is then homotopy equivalent to the loop space on the lower right corner, (see Step (e) of §7.A).
For 2 = j < k, we use the basic pullback rectangle to construct the pullback rectangle (8.5) where the vertical maps are fibrations, and both J(x,s) |s|=3 F 1,1 s,2 and F 1,1 x,3 contractible, as in Remark 8.2.
For 2 < j < k, we similarly use the pullback rectangle (8.6) in which the vertical maps are fibrations, together with the fact that F j−1,j−1 x,k and each F j−1,j−1 s,k−1 are contractible, to prove the claim by induction on j (since loops commute with products).
For 2 ≤ j = k, recall that when |s| = 2 the first non-trivial case (with k −1 = 1) involves the first pullback diagram For 2 < j = k we have the second pullback diagram and combining (products of) either type into (8.9) yields a pullback with horizontal fibrations and with F k−1,k−1 x,k (and of course * ) contractible, so the result (with 2 ≤ j = k) also follows by induction.
With these conventions, each factor in the product Y(x) → Ω j−1 Y(v) is a j-ary Toda bracket by construction, and vanishing of the product is equivalent to vanishing of each factor. 8.10. Theorem. In the fully reduced case, all higher operations decompose into a sequence of Toda brackets of order no greater than the degree of the first target object in the string.

Appendix A. Background Material
We collect here a number of basic facts about model categories needed in this paper and one non-standard lemma included for ease of reference elsewhere. We refer the reader to [Hir, for the basics on model categories and homotopy assumed for this appendix.
A.1. Notation. Given two maps f, g : X → Y , we write f ∼ r g if the maps are right homotopic, and f ∼ l g if the maps are left homotopic.
A.2. Lemma (Homotopy Lifting Property). Suppose we have the solid diagram with q a fibration and T cofibrant: Dually, if Z is fibrant and f is a cofibration then there is a homotopy ψ ∼ r q • f precisely when there is a map q ′ : Y → Z with a homotopy q ′ ∼ r q such that be a factorization of the fold map T ∐ T 1 T ∐1 T −→ T such that i 1 ⊔ i 2 is a cofibration and p is a weak equivalence. Cofibrancy of T implies i 1 : T → Cyl(T ) is an acyclic cofibration by [Hir,7.3.7]. Given a homotopy H : Cyl(T ) → Z with H • i 1 = q • f and H • i 2 = ψ, we may use the left lifting property in to factor H as q •Ĥ, and set f ′ :=Ĥ • i 2 . If f is instead a cofibration, use the dual argument.
A.5. Lemma (Homotopy Pullback Property). Suppose we have the following solid diagram where the square is a pullback, T is cofibrant, and the two vertical maps are fibrations. (A.6) Then there is a dotted map f : T → Y with a homotopy q • f ∼ l ip precisely when there is a dotted map g : T → W with a homotopy j • g ∼ l f and r • g = p.
Proof. Suppose there is a homotopy q • f ∼ l i • p. Since T is cofibrant and q is a fibration, the Homotopy Lifting Property (with Since the square is a pullback, there is a map g : T → W such that j • g = f ′ and r • g = p.
Since f ∼ l f ′ , we conclude that f ∼ l j • g.

Consider the following three statements:
(1) There is a map κ : T → U such that σ = r • κ, and there are (left) applying the Homotopy Lifting Property (with q = p and f = θ), to ψ = t • σ there exists θ ′ ∼ l θ with p • θ ′ = t • σ.
A.10. Corollary. In (A.9) assume again that the squares are pullbacks, T is cofibrant, and the horizontal maps are fibrations. Assume further that u • ϕ ∼ l s • t • σ. Then we have the following: (1) There exists a map κ : T → U such that σ = r • κ and ϕ ∼ l q • Φ • κ.
(2) There exists a map θ : T → W such that ϕ ∼ l q • θ and p • θ = t • σ. Moreover, if s is additionally a monomorphism then there is a homotopy θ ∼ l Φ • κ.
For (2), we have u • ϕ ∼ l s • t • σ. Again, by the Homotopy Pullback Property, there is a map ϕ ′ ∼ l ϕ such that u • ϕ ′ = s • t • σ, so since the bottom square is a pullback, there is a map θ : T → W with t • σ = p • θ and ϕ ′ = q • θ, and so ϕ ∼ l q • θ. Finally We have the duals of Lemma A.5, Corollary A.7, Lemma A.8 and Corollary A.10: A.11. Lemma. Suppose the following square is a pushout, V is fibrant, and the two horizontal maps are cofibrations: Then there is a dotted map f : Y → V with a homotopy p • α ∼ r f • i precisely when there is a dotted map g : Z → V with a homotopy g • β ∼ r f and g • j = p.
A.13. Corollary. If k : X → Y is a pointed cofibration and h : Y → Z is any pointed map with Z fibrant, the composite h • k : X → Z is null-homotopic if and only if there exists a map h ′ : Y → Z, right homotopic to h, which factors through cof(k).
A.14. Lemma. Suppose we are given the following diagram in which both squares are (strict) pushouts, T is fibrant, the indicated horizontal maps are cofibrations, and the outer diagram commutes up to homotopy: Consider the following three statements: (1) There exists a map κ : Z → T such that σ = κ • u and there are (right) homotopies θ ∼ r κ • s and ϕ ∼ r κ • s • t.
(2) ϕ ∼ r θ • t and there is a map θ ′ : X → T , homotopic to θ, such that There is a map θ ′ : X → T homotopic to θ such that ϕ is homotopic to (1) There exists a map κ : Z → T such that σ = κ • u and ϕ ∼ r q • κ • s • t.
Moreover, if Φ is additionally an epimorphism then there is a homotopy θ ∼ r κ • s.
We define the reduced path object P W associated to a pointed object W by the pullback pr 2 •(p 1 ×p 2 ) * G G W Since W is fibrant, the right hand vertical map is a trivial fibration, by [Hir,7.3.7]. Hence the left hand vertical map is a trivial fibration, by [Hir,7.2.12]. Thus P W is weakly contractible.
If f : X → W is null-homotopic, there is a map H : X → Path(W ) with p 1 • H = f and p 2 • H = 0. From the first factorization, and the pullback property of (A.17), there is a map φ : X → P W such that f = p W • φ.
We similarly define the reduced cone CX on a pointed object X by the pushout (A.20) extends to the full diagram above, with ψ a null homotopy for h • k and F g the pullback of g along p W .
Proof. Suppose the composite g • h • k is null-homotopic. Then Lemma A.18 gives a factorization g • h • k = p W • φ in (A.23). Since p W is a fibration, so is i.
If h • k is also null-homotopic then this composite factors as h • k = p Z • κ, for some κ : X → P Z. Now factor κ as X κ ′ → V q → P Z, with κ ′ a cofibration and q a trivial fibration. Since X is cofibrant and P Z is weakly contractible by Lemma A.18, * → V is a trivial cofibration. Therefore, p Z • q lifts to a map η : V → P F g with i• p Fg • η = p Z • q. Setting ψ := η • κ ′ makes the whole diagram commute.
X k G G Y extends to the full diagram above, with ψ giving a null homotopy for k • h and M g the pushout of g along i W .

Appendix B. Indeterminacy
For most higher homotopy operations, one cannot expect a closed formula for the indeterminacy of operations of the type provided by [T2, Lemma 1.1] for the classical (secondary) Toda bracket. This is because tertiary and higher operations depend on choices made for the vanishing of the lower order operations, and the amount of choice remaining might vary for different sets of earlier choices.
However, if we take these earlier choices as given, within the inductive framework described here the only remaining source of indeterminacy is in the choice of the specific map ϕ ′ which makes the outer diagram in (A.9) commute on the nose, and how that choice affects the resulting lift θ ′ . Note that the homotopy class [ϕ ′ ] = [ϕ] is then fixed, as is the actual map u • ϕ ′ = s • t • σ : T → Z. To help keep track of all this, in this appendix ϕ will denote our initial choice of the map with the induced lift θ, while ϕ ′ will denote some other choice, with induced lift θ ′ . We now investigate how changing ϕ to ϕ ′ changes θ to θ ′ , as maps T → W : Given ϕ, a choice of ϕ ′ such that u • ϕ = u • ϕ ′ corresponds uniquely to a map into the pullback (B.1) while a choice of such a map ϕ ′ equipped with a (right) homotopy H : ϕ ∼ r ϕ ′ corresponds to a map into the pullback we find that the image of the left vertical map p ′ is essentially the indeterminacy (see Corollary B.10 below).
Note that there is a canonical choice of induced map ψ : T → Y u in (B.1), corresponding to ϕ ′ = ϕ, and a similar canonical choice of induced map ψ : T → Y u in (B.2), corresponding to the canonical self-homotopy H ϕ of ϕ (namely, the composite T ϕ − → Y iy − → Path(Y )), which will be used below. Given a map u : Y → Z, consider the following pullback grid: (a) The pointed set {ϕ ′ : T → Y | u • ϕ ′ = u • ϕ}, based at ϕ itself, will be denoted by Var u (ϕ).
(b) The pointed set {H : T → Path(Y ) | H : ϕ ∼ r ϕ ′ , u • ϕ ′ = u • ϕ} of (right) homotopies, based at H ϕ , will be denoted by Var u (ϕ). (c) The set {σ : A → B | v • σ = ρ} of lifts of ρ with respect to v will be denoted by Lift v (ρ). In accordance with Remark 3.2, we can disregard the distinction between the left homotopies appearing in the first half of Appendix A and the right homotopies we have here. B.6. Remark. From the pullback properties of the constructions above we see that there are natural bijections of pointed sets Var u (ϕ) ∼ = Lift u ′ (ϕ) and Var u (ϕ) ∼ = Lift u ′ (ϕ), where Lift u ′ (ϕ) is based at ψ and Lift u ′ (ϕ) is based at ψ.
Proof. We may expand (B.4) into: y y r r r r r r r r r r r 1×p pr 2 x x q q q q q q q q q q Since the rightmost face is a pullback (by assumption), as are both the front and left long rectangular vertical faces (by construction), the lower leftmost face, and hence the upper leftmost face, are pullbacks, too. We define P rel by making the upper rightmost face a pullback, so that the back upper vertical face is, too. We think of ϕ : T → Y as mapping to the front lower left Y , and θ : T → W to the back lower left W , with ϕ ′ : T → Y mapping to the front right Y , and θ ′ : T → W to the back right W . Since u • ϕ ′ = u • ϕ, the lower pullback rectangle in (B.4) implies that (ϕ, ϕ ′ ) induce a map F : T → Y u and thus F : T → W p . Since also u • ϕ = s • p • θ = s • t • σ and a right homotopy H : ϕ ∼ r ϕ ′ is a map H : T → Path(Y ) which, together with ϕ⊤θ ′ : T → Y × W , induces H : T → P rel , together with F , these induce a lift of θ along p ′ . Conversely, any lift θ : T → W p, u of θ along p ′ yields H, and thus H, by projecting along the structure maps of the top pullback square.
B.9. Remark. When Y ∼ * , we have Path(Y ) ∼ → Y × Y , so Y u ∼ → Y u ≃ ΩZ and W p, u ≃ W × ΩZ. In this case a map T → W p, u thus corresponds up to homotopy, to a choice of map θ, together with a homotopy class in [T, ΩZ] (adjoint to the indeterminacy construction of [Sp1,§1]). Note that each of the vertical faces in (B.8) is a pullback over a fibration, so they are homotopy-meaningful.
The indeterminacy of our operations is then described by the following. B.10. Corollary. Given ϕ = q • θ : T → Y (also satisfying p • θ = t • σ) in (A.9), the indeterminacy in our operation produced by varying ϕ lies in the image of p ′′ # : [T, W p, u ] → [T, W ], where p ′′ = p ′′ • p.
In fact, we can restrict to the fiber of p ′ # over [θ] (the subset consisting of those homotopy classes containing an element of Lift p ′ (θ)).