The first omega alephs: from simplices to trees of trees to higher walks

The point of departure for the present work is Barry Mitchell's 1972 theorem that the cohomological dimension of $\aleph_n$ is $n+1$. We record a new proof and mild strengthening of this theorem; our more fundamental aim, though, is some clarification of the higher-dimensional infinitary combinatorics lying at its core. In the course of this work, we describe simplicial characterizations of the ordinals $\omega_n$, higher-dimensional generalizations of coherent Aronszajn trees, bases for critical inverse systems over large index sets, nontrivial $n$-coherent families of functions, and higher-dimensional generalizations of portions of Todorcevic's walks technique. These constructions and arguments are undertaken entirely within a $\mathsf{ZFC}$ framework; at their heart is a simple, finitely iterable technique of compounding $C$-sequences.


Introduction
In 1972, building on the work of Barbara Osofsky and others, Barry Mitchell showed that for each nonnegative integer n the cohomological dimension of the cardinal ℵ n is n + 1 [Mit72]. This is data that the set-theoretic community -i.e., the mathematical community most foundationally interested in the transfinite -has yet to fully exploit or absorb. This fact, in turn, is all the more striking when we consider that Mitchell's results, properly understood, fall squarely within the long-active area of set-theoretic research into incompactness phenomena, that is, into behaviors of size-ℵ α structures which sharply differ from those of their smaller substructures. Mitchell's results attest the existence of a family of (n + 1)-dimensional incompactness principles, each of which first holds at the cardinal ℵ n .
There have been several reasons for this neglect. First among these is the abstract and nonconstructive nature of Mitchell's original proof: in essence, Mitchell adapted to the setting of functor categories an inductive sequence of arguments by contradiction which had been pioneered in ring-theoretic contexts, where the homological significance of ℵ n had first been perceived (see [Oso74]). Close reading of this proof, in other words, has probably tended to reinforce a perception of Mitchell's result as fundamentally algebraic in import, difficult and possibly pointless to disentangle from its original framework. (For the reader's convenience we sketch both Mitchell's original argument and its broader historical contexts in an appendix.) Secondly, it would be another decade before the critical template for what Mitchell's result could in fact be about would emerge within the field of set theory. This would be Todorcevic's method of minimal walks [Tod87]. As we will argue below, these comprise the essential content of the n = 1 case of Mitchell's theorem. This recognition, in turn, situates the now-classical walks apparatus as only the first in a family of higher-dimensional analogues. Extracting these higher-dimensional analogues from Mitchell's theorem is one main object of our work below, and we conclude by describing their basic form. These higher walks are almost certain to be of independent interest, and indeed, their fuller analysis, quite apart from this their original setting, is a focus of ongoing work.
Our argument in fact amounts to a new proof and slight strengthening of Mitchell's theorem: it upgrades the characterization of the key inverse system in its original proof from projective to free, and derives from each (suitably conditioned) C-sequence on ω n a canonical witness to the n th instance of the theorem. As hinted above, when n = 1 this witness is little other than the associated system of walks on countable ordinals, algebraically recast; more generally, all these witnesses are, in a suitable sense, recursive on the input of a C-sequence, in strong contrast to the nonconstructive arguments alluded to above. The core of our strengthened version of the theorem also admits the following pithy reformulation: Theorem. Let n be a positive integer. Then ω n is the least ordinal supporting no n-dimensional tail-acyclic simplicial complex.
A simplicial complex B whose vertices are the elements of an ordinal γ is tail-acyclic if its restrictions to tails [β, γ) of γ are each acyclic, i.e., if the reduced simplicial homology groupsH ∆ n of those restrictions all vanish, as we will explain in greater detail below. So concise a characterization of the ordinals ω n in terms of dimensional constraints on objects of a geometric flavor is suggestive in its own right; indeed, below, this theorem will initiate a series of related recognitions in which dimension, in cohomological and even strikingly spatial senses, emerges as a fundamental motif and structuring principle of the ZFC combinatorics of ω n (n ∈ ω). The spatial flavor of these recognitions is reflected in the number of accompanying figures, later sections even assuming something of the character of a "picture book." These pictures are meant to bolster intuitions in the face of the accruing coordinates and abstractions that higher-dimensional combinatorics tend to entail; they foreground frameworks within which not only walks but also related classical objects like coherent Aronszajn trees appear as only the first, initiating instance in a family of multidimensional generalizations.
We now describe this paper's organization. As the above account might suggest, set theorists do form this work's primary intended audience, but only by a gentle margin. We naturally hope for a more diverse readership and accordingly adopt fairly minimal assumptions about our readers' backgrounds in either set theory or homological algebra; this is feasible because our arguments are so generally and deliberately elementary in nature. Hence in Section 2 we review what basics we will need about walks and C-sequences, simplicial complexes, and free and projective inverse systems, relating the latter with Aronszajn trees; we also introduce a notion of internal walks and conclude the section with a statement of Mitchell's theorem. In Section 3 we define tailacyclic simplicial complexes and state our strengthened variant of Mitchell's theorem, as well as its translation to a statement about the existence of bases of inverse systems of abelian groups. Section 4 introduces the idea at the heart of all our constructions, namely a simple, finitely iterable technique of compounding C-sequences, and applies this technique to describe bases of inverse systems with n-dimensional generating sets. Section 5 derives a family of n-coherent functions f n : [ω n ] n+2 → Z from these bases and reduces the proof of Mitchell's theorem (and its strengthening) to showing these functions to be nontrivial; this nontriviality is then the argument of Section 6. We endeavor in the remaining sections to reconnect these functions with more familiar mathematical objects. We begin Section 7, for example, by deducing from the functions f n that ω n is the least ordinal with a nonzero constant-sheaf n thČ ech cohomology group. We describe as well how these functions may be viewed as n-dimensional generalizations of coherent Aronszajn trees, the so-called "trees of trees" of the present work's title. We turn in Section 8 to the n-dimensional walks lying at the heart of the functions f n , treating the representative n = 1 and n = 2 cases in some detail. In Section 9, we conclude with several open questions alongside a brief survey of other n-dimensional phenomena arising among the cardinals ℵ n . In part for the further light it sheds on these phenomena, a sketch of Mitchell's original argument of his theorem is included in an appendix, along with some discussion of its antecedents. The proof of one theorem from the main text, being laborious, is deferred to the appendix as well, along with supplementary details of the proof of another.
One final word before beginning: Sections 6 and 7 (and to a lesser degree Sections 5 and 4) demand markedly more of the reader than the others; there simply is no ready language for the structures f n lying at their core. First-time readers, accordingly, should feel free to skim these sections, reading them even mainly through their figures and captions, noting results like Theorem 7.6 and Corollary 7.11 along the way, and more conventionally re-engaging the text in Section 8. Most essential more generally for any number of non-linear approaches to the text will be Definition 4.1 and the notational conventions listed at the end of Section 2.3.

Background material and conventions
2.1. Walks and C-sequences. Among the most consequential developments in the study of infinitary combinatorics over the past forty years have been the arrival and elaboration of Todorcevic's method of minimal walks ([Tod87; Tod07; Tod10]). As indicated above, this method -or, more precisely, the question of its ultimate scope and meaning -forms the shaping inspiration of the present work. In this section we describe its basic features.
We first establish some conventions. Below, the early Greek letters will always denote ordinals, with only ε in a few clearly marked exceptions denoting a more general partial order. In these contexts, topological references are always to the order topology on a given α, i.e., to that generated by the initial and terminal segments of α.
Definition 2.1. A subset Q of a partial order P is cofinal in P if for all p ∈ P there exists a q ∈ Q with p ≤ q. The cofinality cf(P ) of a partial order P is the least cardinality of a cofinal Q ⊆ P . It will occasionally be convenient to adopt the convention that the cofinality of a partial order possessing a maximum element is ℵ −1 .
We write S n k for the set {α < ω n | cf(α) = ℵ k }. We write Lim and Succ for the classes of limit and successor ordinals, respectively.
We write [γ] n for the collection of size-n subsets of an ordinal γ, frequently identifying this collection with that of the strictly increasing n-tuples of elements of γ without further comment.
A C-sequence on γ is a family C = {C β | β < γ} in which each C β is a closed cofinal subset of β. For concision, we will often term closed cofinal subsets of ordinals clubs.
We now describe the fundamentals of minimal walks; unless otherwise indicated, this material is standard and drawn from the references cited just above. With respect to some fixed C-sequence C = {C β | β ∈ γ}, for any α ≤ β < γ the upper trace of the walk from β down to α is recursively defined as follows: Tr(α, β) = {β} ∪ Tr(α, min(C β \α)), with the boundary condition that Tr(α, α) = {α} for all α < γ. The walk from β to α is loosely identified with its upper trace, or with the collection of steps between successive elements thereof, which is typically pictured as a series of arcs cascading in a downwards left direction, as on the left-hand side of Figure 1 below. The number of steps function ρ 2 sends any α and β as above to |Tr(α, β)| − 1. When γ = ω 1 and each C β in C is of minimal possible order-type, the ρ 2 fiber maps ϕ β ( · ) := ρ 2 ( · , β) : β → Z form a nontrivial coherent family of functions. More precisely, under these assumptions ϕ β α − ϕ α = 0 modulo locally constant functions (1) for all α ≤ β < γ, but there exists no ϕ : γ → Z such that ϕ α − ϕ α = 0 modulo locally constant functions (2) for all α < γ (see [BL19b,Corollary 2.7]). Coherence broadly refers to relations like the first above; nontriviality refers to relations like the second. The ρ 2 fiber maps, for example, also exhibit the following nontrivial coherence relations, closely related but not identical to (1) and (2): for all α ≤ β < γ, but there exists no ϕ : γ → Z such that ϕ α − ϕ α = 0 modulo bounded functions (4) for all α < γ.
Just as for ρ 2 , the fiber maps ϕ β ( · ) := ρ 1 ( · , β) form a family of functions which is nontrivially coherent -i.e., which follows the pattern of relations in (1) and (2) above -but in this case it is with respect to the modulus of finitely supported functions.
As the above might suggest, walks and nontrivial coherence relations exhibit a particular affinity for the ordinal ω 1 , one, indeed, which it would be difficult to overstate. 1 Observe most immediately, for example, that by virtue of their nontrivial coherence the families of fiber maps associated to either ρ 1 or ρ 2 above determine that most characteristic of combinatorial structures on ω 1 : in either case, ({ϕ β α | α < β < ω 1 }, ⊆) forms an ω 1 -Aronszajn tree.
Definition 2.2. A tree T = (T, ⊳) consists of a set T of nodes and a partial order ⊳ thereon with the property that ⊳ well-orders the set {s ∈ T | s ⊳ t} for each t ∈ T . The height ht(t) of a node t is the order-type of {s ∈ T | s ⊳ t}; the height ht(T ) of T is min{α | {t ∈ T | ht(t) = α} = ∅}. A branch is a maximal ⊳-linearly ordered subset of T ; a branch b is cofinal if min{α | {t ∈ b | ht(t) = α} = ∅} = ht(T ). A κ-tree T is a tree of height κ satisfying |{t ∈ T | ht(t) = α}| < κ for all α < κ. A κ-tree is Aronszajn if it possesses no cofinal branch, and coherent if it admits representation as the set of initial segments of a coherent family (modulo finite differences, typically) of functions, ordered by inclusion.
We describe higher-order variants of these structures in Section 7.2 below. We should perhaps emphasize, though, that we do not pursue higher-dimensional combinatorics quite for their own sake. The heart of the matter, rather, is this: walks techniques' extreme successes in capturing or consolidating the ZFC combinatorics of ω 1 are unmatched at any higher ω n , and it is natural to wonder why. Broadly speaking, there have evolved two main strategies for extending these techniques' reach to higher cardinals κ. In the first, assumptions along the lines of the combinatorial principle (κ) on the underlying C-sequence do translate much of what's so productive in the ω 1 setting to higher κ (see [Jen72b;Tod87;Tod07]); this has the simple disadvantage of involving assumptions supplementary to the ZFC axioms. The second, on the other hand, involves relaxations of requirements, on the modulus of coherence, for example, from mod finite to mod countable or mod κ. Although initially avoiding assumptions beyond the ZFC axioms, this approach tends subsequently to need them, in the form of cardinal arithmetic conditions, for results of real force.
Against this background, Mitchell's theorem is provocative, particularly once one recognizes its n = 1 case as fundamentally that of minimal walks on the ordinals of ω 1 (see Sections 6.1 and especially 8.1 below). While remaining well within the framework of the ZFC axioms, it records substantial generalizations of that case to each of the higher ordinals ω n (n ∈ ω). It thereby 1 "Despite its simplicity, [the method of minimal walks] can be used to derive virtually all known other structures that have been defined so far on ω 1 " [Tod07,p. 19]; see also the remarks at page 7 therein.
suggests a third way of extending walks techniques, one which bypasses expansions of assumptions via expansions of dimension.
2.2. Internal walks. As the above account may have suggested, walks tend to engender multiple forms of nontrivial coherence. This observation formalizes as follows: (1) Nontrivial coherent families with respect to various moduli witness the nonvanishing of the cohomology groupsȞ 1 (γ; P) with respect to various corresponding presheaves P. 2 (2) For any limit ordinal γ and n ≥ 1, the groupsȞ n (γ; P) with respect to the presheaves P corresponding to the most combinatorially prominent moduli -namely, those of the finitely supported functions and the locally constant functions to an abelian group A, as above -are isomorphic. In other words, nontrivial coherent families of the first sort exist on γ if and only if nontrivial coherent families of the second sort exist on γ as well. Both of the aforementioned moduli will feature in more general settings below. For example, by way of its fiber maps the function ρ 2 above may be viewed as representing a nontrivial element ofȞ 1 (ω 1 ; Z), where Z denotes the sheaf of locally constant functions to the integers (see [BL19b]; see Section 7.1 below for further discussion of higher cohomology groups); by way of the above isomorphism, we may identify ρ 1 with such an element as well.
We now describe a simple mechanism for extending the observation thatȞ 1 (ω 1 ; Z) = 0 to the result thatȞ 1 (δ; Z) = 0 all ordinals δ of cofinality ℵ 1 . This mechanism is both a critical component of higher walks and a useful heuristic for the relativizations appearing more generally below.
Here and throughout, terms like C δ will always denote closed cofinal subsets of δ of minimal possible order-type. We will refer to their elements via increasing enumerations η δ i | i ∈ cf(δ) (or η i | i ∈ cf(δ) when δ is clear) and require moreover that cf(η δ i ) = cf(i) for any limit ordinal δ and i ∈ cf(δ). We will generally assume some fixed C-sequence to be defined over whatever ordinals we are working with.
To ground the recursion, let Tr δ (α, min(C δ \α)) = {min(C δ \α)} for all α < δ. (Similarly for β ∈ C δ , though the notation grows cluttered. 3 ) Rather than mapping a walk on the countable ordinals to 2 Readers unfamiliar with these frameworks are both referred to [BL19b] and reassured that they will play no essential role in our main argument; they function mainly as a convenient shorthand below. 3 As the referee has noted, under present definitions, Tr δ (β, β) = {β, min C δ \β} when β ∈ C δ , with the consequence that ρ 2 [δ](β, β) = 1, in contrast with the classical fact that ρ 2 (β, β) is always zero. Superficial modifications to these definitions would eliminate this effect without affecting this section's broader argument, and indeed, the term internal walk may be taken to refer to any of a family of minor variations on the single basic idea we are describing. The Figure 1. The order-isomorphism π : ω 1 → C δ : k → η k translating a C-sequence on ω 1 , and thereby a walk, to one on C δ . On the left is the standard picture of a walk determined by a C-sequence (drawn in gray; notches depict representative elements of the associated C γ ) on ω 1 ; we term the walk on the right-hand side C δ -internal. For initial inputs β / ∈ C δ such walks require a first step "up into" C δ ; this we have depicted as well.
one on those of C δ , this second framing maps the underlying C-sequence on the countable ordinals to the ordinals of C δ , and walks thereon. See Figure 1.
In one view, these internal walks are the material of walks of the next higher order; these we describe in Sections 8.2 and 8.3 below. The basic idea of these translations, though, suffuses this work very generally.
2.3. Simplicial complexes and the systems P n (ε). By simplicial complex B on β we mean a simplicial complex whose vertices are the elements of β. We may more generally identify any n-dimensional face of B with the size-(n + 1) set of its vertices. Best suited for our purposes, in other words, are abstract simplicial complexes on β: ⊆-downward-closed collections of finite subsets of β. Writing B n for the set of n-dimensional faces of B, we then have variant we have recorded here has the virtue of corresponding to the values arising at the second or inner coordinate along the internal branches of the three-coordinate higher walks; see Section 8.2.
The dimension of a simplicial complex B is sup{n | B n = ∅}. For any such B let and for all α ∈ B n write α for the associated generator of C n (B). Writing α i for the (n − 1)-face of B obtained by omitting the i th element of α, the maps α → i≤n (−1) i α i then induce boundary homomorphisms ∂ n : C n (B) → C n−1 (B) and simplicial homology groups for each n ≥ 0. Here C −1 (B), and hence ∂ 0 , equals zero. The reduced simplicial homology groups H ∆ n (B) are similarly defined, but with C −1 (B) = Z and ∂ 0 : α → 1 for all α ∈ β. Observe that the complex B (or, more precisely, its geometric realization) is connected if and only if H ∆ 0 (B) = Z, or equivalently if and only ifH ∆ 0 (B) = 0. When β is of cofinality ℵ k , its order-structure manifests as a k-dimensional combinatorialtopological condition on the family of simplicial complexes B on β. This is the content of Theorem 3.5 below. The mechanism of this surprising rapport is a grading of simplicial complexes, for which inverse systems are a convenient framework.
Definition 2.3. Let C be a category. An inverse system in C over I consists of a partially ordered index-set I, terms X i (i ∈ I), and bonding maps x ij : X j → X i satisfying x ik = x ij x jk for all i ≤ j ≤ k in I, where the terms and bonding maps are objects and morphisms in the category C, respectively. We will typically represent such systems as triples (X i , x ij , I) and more abstractly denote inverse systems by boldfaced variables like X; we will also take C to be the category Ab of abelian groups except where otherwise indicated below. A morphism between two inverse systems The terms of our central examples are of the following forms: for n ≥ 0 and A a collection of ordinals, let P n (A) = In the framework of (5) above, P n (A) is C n (B), where B is the complete n-dimensional simplicial complex on A. For both P n (A) and R n (A), again write α for the generator associated to α ∈ [A] n+1 . Again boundary maps on these α determine maps for n ≥ 1. For any ordinal ε and n ≥ 0 define then the inverse system P n (ε) = (P n ([α, ε)), p αβ , ε) with p αβ : P n ([β, ε)) → P n ([α, ε)) the natural inclusion map, for α ≤ β < ε, and define R n analogously. (Here and below we denote intervals of ordinals just as we would intervals of reals; [α, ε) = {ξ ∈ Ord | α ≤ ξ < ε}, for example.) Observe that p αβ and d n commute; hence the maps d n determine in turn a mapping of inverse systems This map may be regarded as a natural transformation between contravariant functors (α → P n ([α, ε)) and α → P n−1 ([α, ε)), respectively) from the partial order ε, viewed as a category, to the category of abelian groups. We write Ab ε op for the category with such functors as objects and natural transformations as morphisms. Observe that Ab ε op is an abelian category; in particular, sums and kernels and quotients and, hence, exact sequences exist therein, and are evaluated pointwise (e.g., the terms of a quotient are the quotients of the corresponding terms). We write ∆ ε ( · ) for the diagonal functor A → (A, id, ε) embedding Ab into Ab ε op ; in particular, ∆ ε (Z) is the inverse system (Z, id, ε). The aforementioned objects then assemble in the following exact sequence: We term this sequence the standard projective resolution of ∆ ε (Z). Simple as it might appear, the sequence P(ε) will be a main object of study below. A main part of our argument, in other words, will frequently be the manipulation of algebraic relations between n-tuples of ordinals. For this work, a clear but flexible notation is critical; we therefore pause to collect and augment its more scattered description above: (1) For A a collection of ordinals, we write β ∈ [A] n to mean that β is an increasing n-tuple (β 0 , . . . , β n−1 ) of ordinals in A. We will typically write a 1-tuple (β) as β. For β ∈ [A] n and 0 ≤ i < n, we write β i for β with the i th coordinate removed. If β is a 1-tuple, then β 0 = ∅. As we did when defining simplicial complexes above, we will sometimes simply view β as an n-element subset of A; we write α < β to mean that every element of α is less than every element of β. We apply the restriction-notation B X both to functions and to simplicial complexes; in the latter case, it denotes the simplicial complex comprised of those x ∈ B satisfying x ⊆ X.
(2) As for d n above, we will define maps among inverse systems largely by way of their actions on generators α ; at times, we will conflate maps between terms (like d n ) and maps between inverse systems (like d n ) as well. Relatedly, we will tend not to formally distinguish between a generator γ ∈ P n ([β, ε)) and its images p αβ ( γ ). When we do, it will be to regard γ as an element of the "highest possible" term of P n (ε) -namely, P n ([γ 0 , ε)).
(3) We write α, β for α ⌢ β ; we will also at times write sums of generators inside the angled brackets, preferring expressions like d k α, β to k i=0 (−1) i α i , β . As they do here, commas can render such expressions more readable. In subscripts, however, such commas typically have more of an effect of clutter. In these cases we omit them, denoting concatenations of coordinates, as in (β 0 , . . . , β m−1 , γ 0 , . . . , γ n−1 ), as concatenations of tuples, as in β γ. Putting all this together: the tuple (β 0 , β 2 , δ) would typically appear in a subscript as C β 1 δ , for example; it would appear in a generator probably as β 1 , δ . Lastly, an expression like d k B means {d k α | α ∈ B}.

Free and projective inverse systems.
Definition 2.4. For any object P in Ab ε op , let id denote the identity morphism. P is projective if for any epimorphism e : R → P there exists a morphism s : P → R such that e s = id. We will sometimes term such a right-inverse to an epimorphism a section. Dually, we will sometimes term a left-inverse r to a monomorphism m a retract.
An object X in Ab ε op is free if there exists some B ⊆ α<ε X α such that any x in any X α has a unique B-decomposition Example 2.5. The system ∆ ε (Z) is free if and only if ε is a successor, i.e., if cf(ε) = 1. The system P n (ε) is free, on the other hand, for any ordinal ε and n ∈ ω. By an argument exactly as in more standard settings, it follows that every P n (ε) is projective as well. 4 The reverse question of whether a projective system is free (or, conversely, of whether a nonfree system is nonprojective) is in general much subtler. Even the simplest instance is less than obvious: Let ε be a limit ordinal. Is ∆ ε (Z) projective?
The question involves a different order of quantification from that of freeness: it quantifies over the collection of morphisms in Ab ε op . Arguably the obscurity -or, in another view, the power -of the notion of projective consists, simply, in this quantification. To see that a question like (6) is as much about the ambient category as it is about the object itself, consider the following: Definition 2.6. For any infinite cardinal κ let κ-Ab denote the category of abelian groups of size less than κ, and let κ.g.-Ab denote the category of abelian groups with generating sets of size less than κ. When κ = ω, of course, these are the categories of finite abelian groups and of finitely generated abelian groups, respectively; note also that ω is the only infinite cardinal κ for which the categories κ-Ab and κ.g.-Ab are distinct.
It is straightforward to verify that each of the categories defined above is abelian. Recall that a cardinal κ has the tree property if there exist no κ-Aronsjazn trees, i.e., if every tree of height κ and level-widths all less than κ possesses a cofinal branch. By Theorem 2.7 and König's Infinity Lemma, ∆ ω (Z) is projective in the category of height-ω inverse systems of finite abelian groups. ∆ ω (Z) is not projective, however, in the wider category of height-ω inverse systems of abelian groups. We will argue this latter fact as the base case in the inductive proof of Proposition A.2; it will follow as well from our results in Section 6. A subtler point complicating the statement of Theorem 2.7 is the fact that ∆ ω (Z) is not projective in the category of height-ω inverse systems of finitely generated abelian groups. This subtlety is sufficiently removed from our main concerns, though, that we record just the contours of a counterexample in a footnote. 5 4 The interested reader is encouraged to verify these assertions directly; for the first and last of them, though, see also [Mar00,Example 11.17] and Lemma 2.8 below, respectively. 5 Perhaps the simplest counterexample is the inverse sequence of finitely generated abelian groups Q = (Q i , q ij , ω) in which each Q i = Z ⊕ Z and each q i,i+1 : Q i+1 → Q i is given by the maps (1, 0) → (2, 0) and (0, 1) → (1, 1). Define an epimorphism e : Q → ∆ω(Z) by letting each e i be the map determined by (1, 0) → 0 and (0, 1) → 1. If s were right-inverse to e then s 0 (1) would need to fall in {(1 + 2 n , 1) + 2 n+1 Z ⊕ {0} | n ∈ ω}, as the reader may verify. But this intersection is empty.
The above remarks and theorem were something of a digression, meant to help frame the recognition below that a number of projective inverse systems are free. 6 To apply this recognition, we will want the following standard lemma: Lemma 2.8. An inverse system X is projective if and only if X is a direct summand of a free inverse system Y.
Proof. For the only if direction, fix an epimorphism e from a free system Y to X. As X is projective, e admits a right-inverse s, so that Y ∼ = s(X) ⊕ ker(e) ∼ = X ⊕ ker(e). For the if direction, observe that if Y = X ⊕ Z then any epimorphism e : R → X naturally extends to an epimorphism e ′ : R ⊕ Z → X ⊕ Z. As Y is free, e ′ has a right-inverse s ′ , which restricts to an s : X → R right-inverse to e.
As the lemma suggests, it is not in general true that subsystems of free inverse systems of abelian groups are free, or even projective. A central concern below, in fact, is the question of whether the subsystem d n P n (ε) of the free system P n−1 (ε) is projective. This question, as we will see, is fundamentally a question about the cofinality of ε. Observe in this connection that we may truncate the exact sequence P(ε) at any d n P n (ε) to form a shorter exact sequence as follows: The systems we consider are indeed "big," so we are recording a fact somewhat described by Hyman Bass's 1963 title Big projective modules are free [Bas63]. Note that by Theorem 2.7, though, that title is far from describing the situation for inverse systems in any unqualified generality: assume the tree property of some infinite cardinal κ (readers wary of large cardinals may take κ to be ω). Then ∆κ(Z) is a projective system in (κ-Ab) κ op which, by Claim A.3 below, is not projective in Ab κ op , and therefore cannot be free. Mitchell cites the question of when projective objects in categories Ab C are free as motivating [EM65] (see [Mit72,p. 5]). Note also that the proof of Theorem 2.7 shows that lim Q = 0 for the system Q associated therein to an Aronszajn tree T . The question of the values of higher limits of Q-like constructions is a subtler matter and could conceivably shape a productive approach to the study of various set-theoretic trees.
If d n P n (ε) is projective then (7) shares with P(ε) the property that all terms except possibly the "target" ∆ ε (Z) are projective.
Definition 2.9. A projective resolution of an inverse system X is an exact sequence ending in X as in P(ε) or (7), above, in which all nonzero terms except possibly the rightmost are projective. Such resolutions are sometimes written P → X → 0. The length of P is the supremum of the indices of its nonzero terms -where P's terms are indexed, as above, from right to left, beginning with zero. Possibly all of P's terms are nonzero; its length in this case is ∞. The projective dimension of X, written pd(X), is the minimal length of a projective resolution of X. An equivalent definition is the following: given any projective resolution P = P n , d n | n ∈ ω of X, the projective dimension of X is the least n such that d n P n is projective.
Example 2.10. X is projective if and only if · · · → 0 → X id −→ X → 0 is a projective resolution, if and only if pd(X) = 0. More generally, pd(X) may be read as quantifying how "far" a system X is from being projective.
We conclude this section with several summary remarks and with a statement of Mitchell's theorem. Our interest is in projective resolutions of ∆ ε (Z), for two related reasons: (i) They translate order-theoretic information into algebraic information.
(ii) They are of computational value.
(See [Mar00, Section 12.2]). Here X = (X α , x αβ , ε) is any system in Ab ε op and P is any projective resolution of ∆ ε (Z), such as P(ε) or (7) above. Ext n and lim n are the higher derived functors of the functors Hom and lim, respectively (see just below or [Wei94] or [Mar00] or [Jen72a] for further discussion). Via equation 8, the standard projective resolution P(ε) of ∆ ε (Z) uniformizes the computation of higher derived limits, providing, in particular, explicit formulae for lim n ( · ), as we will now describe.
Above, Hom(P, X) denotes the cochain complex with n th term Hom(P n , X), where P n is the n th term of P; the associated coboundary maps are those induced by the boundary maps of P. If P = P(ε) then since P n (ε) is free, elements of Hom(P n (ε), X) amount simply to a choice of map for each basis-element α of P n (ε). It follows that lim n X may be computed as the n th cohomology group of the cochain complex we denote K(X), with cochain groups and coboundary maps d j : We will apply this description in Section 5 below.
Again note, on the other hand, that other resolutions P of ∆ ε (Z) might be taken in place of P(ε) in equation 8. In particular, the eventual zeros of a resolution like (7) will translate in equation 8 to vanishing cohomology groups, and hence to vanishing higher derived limits for any inverse system indexed by ε, for all n above some finite m. More precisely: Lemma 2.11. The projective dimension of ∆ ε (Z) is n ∈ N if and only if n is the largest integer for which lim n ( · ) is nonvanishing, i.e., for which there exists some X in Ab ε op with lim n (X) = 0. 7 Sensitivities of pd(∆ ε (Z)) to the cofinality of ε transmit in this manner to functors of broad application and importance, namely, the higher derived limits lim n . These are of sufficient significance, for example, to warrant the following definition: Definition 2.12 ( [LM74]). The cohomological dimension cd(ε) of a partial order ε (or more generally of any small category ε) is the supremum of {n | lim n : Ab ε op → Ab does not equal 0}. The supremum of N is denoted by ∞. 8 Before continuing, we pause to recall the most essential feature of higher derived limits: their interrelations in long exact sequences deriving from short exact sequences in Ab ε op . Higher limits are an artifact of the lim functor's "failure to be exact"; for example, the lim-image of the short exact sequence may itself be only half or left exact, meaning that while lim i will inherit the injectivity of i, lim q may fail to be surjective. On the other hand, a long exact sequence extending the lim-image of (10) and comprised of higher limits will conserve the exactness of (10); its form is the following: (We forego discussion of the connecting morphisms, but these also, like each lim n , are in the proper sense functorial.) The basic heuristic for this phenomenon is that higher limits array, in group form, the information of inverse systems, in a graded and coordinated fashion (with the caveat that some such information may, in the process, be lost).
We turn now, at long last, to Mitchell's theorem.
Theorem 2.13 ( [Mit72]). Let ε be a linear order of cofinality ℵ ξ . If ξ is finite then the cohomological dimension of ε is ξ + 1. If ξ is infinite then the cohomological dimension of ε is ∞.
The theorem holds even with the convention that the cofinality of a partial order which has a maximum element is ℵ −1 . Mitchell extended his theorem in the following year to the generality of directed partial orders ε [Mit73]; not unrelatedly, the functors lim n extend to the category pro-Ab, as described in [Mar00,Section 15]. This family of results organizes, in other words, into a core -the combinatorics of the cardinals ℵ ξ , particularly when ξ is finite -and various techniques of extension. Our interest is in that core; as will grow clearer, we regard it as at heart expressing the cofinality interrelations among the ordinals, interrelations which C-sequences instantiate. For this reason we will focus on the case of Theorem 2.13 when ε is an ordinal; its extension to linear orders will require little more than a comment in Section 5.
To sum up: the main content of Mitchell's theorem is that objects like ∆ ε (Z) and P(ε) exhibit significant sensitivities to order-theoretic considerations. Our aim is to better understand in what these sensitivities consist. Our guiding interest in all that follows, in other words, is point (i) above, in question form: What is it in the ordinals -the ordinals ω n , in particular -that these algebraic structures are capturing?

Tail-acyclic simplicial complexes
It is natural to consider, for a given γ ∈ ω 1 , the family of all walks Tr(α, γ) from γ down to some α < γ. Such a family is most concisely conceived, perhaps, as the graph α<γ Tr 2 (α, γ) (12) on γ + 1, where Tr 2 (α, γ) records the steps of Tr(α, γ) as edges {{γ i+1 , γ i } | i < ρ 2 (α, γ) − 1}. It is an effect of the fact that Tr(β, γ) is an initial segment of any Tr(α, γ) passing through β, together with the fact that walks' steps are always "from above", that any such graph is well-behaved or good in the following sense: Example 3.2. Consider the following graphs on the ordinals 4 and 3, respectively: G 0 is a good graph. On the other hand, G 1 [1,3) is disconnected, so G 1 is not good.
In fact, G 1 is the forbidden configuration: a connected graph is good if and only if it contains no copy of G 1 (i.e., contains no {{α, β}, {α, γ}}, for α < β < γ). A consequence is the following theorem, one measure of the difficulty of extending the technique of minimal walks beyond the countable ordinals. Proof. Suppose for contradiction that ω 1 admitted a good graph G. As G is connected, there exists for each γ ∈ Lim ∩ ω 1 some least γ 1 ≥ γ such that {ξ, γ 1 } ∈ G for some ξ < γ. Let γ 0 denote the least such ξ. The function γ → γ 0 is then a regressive function and hence, by the Pressing Down Lemma, constantly α on some stationary S ⊆ Lim ∩ ω 1 . For any β and γ in S above α with β 1 = γ 1 , then, {{α, β 1 }, {α, γ 1 }} is a copy of G 1 in G -a contradiction.
On the other hand, (12) defines a good graph on any countable successor ordinal, and a natural variant of its definition handles the countable limit case as well. In fact, the more elementary {{α, min(C γ \(α + 1))} | α < γ} defines a good graph on any γ of countable cofinality (with G 0 , above, as a simple instance).
These phenomena generalize.
Definition 3.4. An n-dimensional simplicial complex G on an ordinal γ is tail-acyclic if G n−1 = [γ] n and for all α < γ and Tail-acyclic n-dimensional G on γ, in other words, have a complete (n − 1)-skeleton and are connected and acyclic on any tail of γ; good graphs are simply the n = 1 case of this definition. 9 Theorem 3.5. Let n be a positive integer. Then ω n is the least ordinal supporting no n-dimensional tail-acyclic simplicial complex.
In particular, there is some least number of dimensions -namely, n + 1 -in which ω n can support a tail-acyclic simplicial complex.
We will argue Theorem 3.5 by way of an algebraic translation, which we motivate as follows: In other words, d 1 I is a basis for d 1 P 1 (ω). Pictorially, edges {i, i + 1} connect the points j and k as below: These edges evidently define a unique path between any two points in ω. Put differently, d 1 I defines a good, or tail-acyclic, graph G I on ω. More precisely, the spanning and linear independence properties of d 1 I manifest as the connectedness of, and lack of cycles in, G I , respectively. "Goodness" captures the fact that these properties persist on any restriction of d 1 I and d 1 P 1 (ω) to a tail [n, ω) of ω -the fact, in other words, that d 1 I defines a basis for the inverse system d 1 P 1 (ω). These seemingly rudimentary considerations are surprisingly sensitive to the cofinality of the index-set of d n P n (γ). For example: by Theorems 3.3 and 3.7 below, the least ordinal γ for which d 1 P 1 (γ) is not free is ω 1 ; in fact, d 1 P 1 (ω 1 ) is not even projective.
Theorem 3.7. For n ≥ 1, the system d n P n (γ) is free if and only if γ admits a tail-acyclic ndimensional simplicial complex.
Proof. For the forward direction of the proof, suppose that d n P n (γ) is free. We will want the following fact: Fact 3.8. If d n P n (γ) is projective and cf(γ) = ℵ ξ , then ξ < n. 9 The requirement of a complete (n − 1)-skeleton in Definition 3.1 streamlines the argument of Theorem 3.7; if every n-dimensional G satisfyingH ∆ k G [α,γ) = 0 for all α < γ and k ≥ 0 extends to a tail-acyclic G ′ ⊇ G (i.e., extends to one with a complete (n − 1)-skeleton) then this requirement is unnecessary. This may in fact be the case, as it clearly is when n = 2, but for more general n it is at the very least quite tedious to rigorously argue. Hence for now, for simplicity's sake, we adopt this requirement and record the extension problem as one of our concluding questions.
This fact is immediate from Theorem 6.1 together with Lemma 2.11, or Proposition A.2 below. In the following section, we construct for any d n P n (γ) as in Fact 3.8 a basis of the form d n B = {d n α | α ∈ B}, with B ⊆ [γ] n+1 . Let d n P n (γ) be free and fix such a basis, and write B for the ⊆-downward closure of B. In other words, B is the natural interpretation of B as a simplicial complex. We show that B is tail-acyclic. As d n B is linearly independent, Proof. Towards contradiction, suppose instead that β i ∈ [γ] n \B n−1 for some β ∈ [γ] n+1 . Then no linear combination of elements of d n B can supply the summand β i of d n β , hence d n B does not span d n P n (γ).
for 0 < k < n is nothing other than the homology of the chain complex As noted in Section 2.3, this sequence is exact, soH ∆ k (B) = 0. By definition these arguments hold on any tail of γ; in consequence, B is tail-acyclic. For the reverse direction of the proof, simply observe that the above argument is reversible. In other words, any tail-acyclic n-dimensional simplicial complex is determined by its collection, B, of n-faces, which in turn define a basis d n B = {d n α | α ∈ B} for d n P n (γ), just as above.
To see that d n B spans d n P n (γ), observe that any may be identified with an element e of C n−1 (B). Since ∂ n−1 e = 0 andH ∆ n−1 (B) = 0, we must have ∂ n f = e for some f ∈ C n (B); this ∂ n f naturally corresponds to a linear combination of elements of d n B, showing that the latter indeed does span the group d n P n ([0, γ)). Since B is tail-acyclic, this argument applies to the restriction of B to any tail of γ, hence d n B spans the inverse system d n P n (γ), as desired.
To see that d n B is linearly independent, observe that via an identification just as above; in consequence, if the α i are all distinct, then the coefficients a i must all equal zero, sinceH ∆ n (B) = ker(∂ n ) = 0. Theorem 3.5 then takes the following form: Theorem 3.10. For n ≥ 1, ω n is the least ordinal ε such that d n P n (ε) is not free.
Theorem 3.10 is in fact true even for n = 0. This theorem, like 3.3 and 3.5 above, conjoins both positive and negative statements, namely that (1) d n P n (ε) does admit a basis, for ε < ω n , while (2) d n P n (ω n ) does not.
We argue (1) and (2), respectively, in Sections 4 then 5 and 6 below. To show (1), we define an explicit basis for each eligible d n P n (ε); this we do by an expanded, or compound, use of Csequences. We then describe "pullbacks" f n of these n-dimensional basis-systems which exhibit nontrivial n-dimensional coherence relations, and thereby witness the (hitherto abstract) fact that pd(∆ ωn (Z)) > n; from this point (2) follows. In the process, we will have shown the core of Mitchell's theorem and a bit more: for each ε < ω n , that theorem only implies that d n P n (ε) is projective, whereas just below, we show in a very concrete fashion that it is free.

Bases from C-sequences
In this section, we will define for each ε of cofinality ℵ k and positive n > k a B ⊆ P n (ε) such that d n B is a basis for d n P n (ε). Clubs C β on β ∈ ε ∩ {γ | cf(γ) < cf(ε)} will structure the construction of B. Hence, as above, we begin by fixing for each relevant β a closed cofinal C β ⊆ β such that otp(C β ) = cf(β), and such that for any limit β and i < cf(β) the cofinality of the i th element of C β is cf(i). In particular, C 0 = ∅; we will also assume that C κ = κ for any infinite regular cardinal κ.
One may continue in this fashion, defining C γ by induction on the length of γ much as above: Suppose C γ is defined and β ∈ C γ . Let α = otp(C γ ∩ β) and let π : Observe that if β = ω k then its role in points (1)-(3) is superficial; C αi+1...αnβ = C αi+1...αn , for example. We will sometimes in this case omit mention of C β , terming an α as above internal, simply.
Observation 4.2. The following observations are straightforward: • If ε is a successor ordinal, then C αε = ∅ in the one case in which it is defined, namely when ε = α + 1. • If β 0 is a limit ordinal above α, then C β (α) is defined and is a successor ordinal.
• More generally, if β is a tail of γ then if C γ is defined then C β is as well and C γ ⊆ C β .
• If ε, the largest cardinal involved, is of cofinality ℵ k , then the recursions of Definition 4.1 are only meaningful for k + 2 many steps. β is internal to C ε , in particular, implies that The next definition is crucial.
Definition 4.3. For ε of cofinality ℵ k and positive n > k, let B n (ε) denote the collection of α, β satisfying the following: Note as in Observation 4.2 that our C-sequence conventions entail that any such β 0 is a successor ordinal. Where we wish to emphasize the choice of the parameter C ε in the above definition, we Example 4.4. For any n > 0 and ε a successor ordinal, e.g., ε = δ + 1, Here there is only one possibility for C ε , and the only β which is internal to C ε is the 1-tuple δ.
Lemma 4.5. If ε is a successor ordinal and n > 0 then d n B n (ε) is a basis for d n P n (ε).
Proof. Here and below it will suffice to check that d n B n (ε) is linearly independent and decomposes each generator d n β of d n P n (ε). Again let ε = δ + 1. Linear independence follows from the fact that for any collection of pairwise distinct α j , Hence the only d n B n (ε) decomposition of 0 is the trivial one. Therefore, since d n d n+1 β, δ = 0 for any β ∈ [δ] n+1 , As any other β ∈ [ε] n+1 contains δ and consequently falls in B n (ε), this completes the argument.
Theorem 4.6. Fix a positive integer n. Then for every ordinal ε for which cf(ε) < ℵ n , the collection d n (B n (ε)[C ε ]) is a basis for d n P n (ε). In particular, every such d n P n (ε) is free.
Proof. The proof is by induction on ε. Denote the following inductive hypothesis IH(ε): If δ < ε and cf(δ) = ℵ k and n > max (0, k), then for any club C δ on δ, Notice that if ε is a limit ordinal and IH(ξ) holds for all ξ < ε, then IH(ε) holds. Hence we only need to show that IH(ε) implies IH(ε + 1). If ε is a successor ordinal, then IH(ε + 1) follows from IH(ε) by the preceding example and lemma. If ε is a limit ordinal of cofinality greater than ℵ ω , then IH(ε + 1) and IH(ε) are equivalent assertions. This leaves just one case of interest: limit ε of cofinality less than ℵ ω . First, a lemma: We return to the proof of the theorem. Assume IH(ε), with ε a limit ordinal of cofinality less than ℵ ω . Enumerate the elements of C ε as {η ε i | i ∈ cf(ε)}. Claim 1. d n B n (ε) is linearly independent.
Proof of Claim 1. Towards contradiction, suppose instead that for nonzero coefficients z j (j < ℓ) and { α j , β j | j < ℓ} ⊂ B n (ε). Let δ = max{β j | j < ℓ}, and let J = {j | β j = δ}. Note that all β j are elements of C ε . In particular, δ is an element of C ε . Note also that (without loss of generality) the α j indexed by J are all distinct. By (14), Case 1: δ is a limit ordinal. Then together with the induction hypothesis, J z j d n−1 α j , δ = 0, which follows from (15), contradicts Lemma 4.7.
Case 2: δ is a successor ordinal: and we may conclude as in equation (13) that j∈J z j α n−1 j = 0, a contradiction.
Proof of Claim 2. We argue by induction on δ ∈ C ε . Let We show that if d n B n (ε) γ+1 generates d n P n (γ + 1) for all γ ∈ δ ∩ C ε , then d n B n (ε) δ+1 generates d n P n (δ + 1). The base case, δ = η ε 0 , is exactly as in Example 4.4. Case 1: δ is a limit ordinal. Consider then d n α, δ ∈ d n P n (δ + 1). Let By Lemma 4.7, the left-hand summands of the last line are all from d n B n (ε) δ+1 , while the rightmost sum is in d n P n (η + 1) for some η ∈ δ ∩ C ε . By our induction hypothesis, this concludes Case 1.
Again the rightmost summand decomposes by hypothesis, while each summand of We will be done if we show that, for any α ∈ [δ] n+1 with α n > η ε i , d n α has a B n (ε) δ+1 decomposition. Again, though, since d n d n+1 α, δ = 0, and all summands on the right are as discussed above: either of type (17), or from B n (ε) δ+1 directly. This concludes the proof of Claim 2.
Lemma 4.7 in the above argument bears comparison with square principles (see again [Jen72b; Tod87; CFM01]): structuring both is a certain uniformity at the limit points In our basis construction, it comes at the cost of an additional coordinate: Hence these additional coordinates accrue. In other words, more room is needed to carry out the construction on higher cofinality ε; this is one heuristic for the associated rise in cohomological dimension. Internal tails record these accruing coordinates and are the key to our further constructions. More particularly, C ε -internal tails organize the d n B n (ε)-decomposition of d n P n (ε) to such a degree that the associated map d n P n (ε) → P n (ε) (see equation 18 below) extends to all of P n−1 (ε), as we describe in the following section. The basic principle is the following: a fact used in the proof of Lemma 4.7 is that any γ has a maximal proper internal tail β. Hence γ = α, β for some i and α ∈ [ε] i+1 , and γ then has some "nearest" basis element b( γ), if C βε (α i ) is defined: Definition 4.8. Given an ε and C ε as in the above construction, the maximal proper internal tail of γ ∈ P n (ε) is the longest tail β of γ which is internal to C ε and not all of γ. In this case, if The function b will feature centrally in the following sections. Note that every element of B n (ε) is of the form b( γ) for some γ ∈ [ε] n . Observe finally that Theorem 4.6 implies the "upper bound half" of Mitchell's theorem (due, in fact, originally to Goblot [Gob70]): Corollary 4.9. If the cofinality of an ordinal ε is ℵ k then the cohomological dimension of ε is at most k + 1.

Translation to mod finite settings
In this section we describe how the basis constructions of Section 4 induce functions whose nontriviality implies the "lower bound half," and hence the entirety, of Theorems 2.13 (Mitchell's), 3.5, and 3.10. We then define these functions explicitly; we argue their nontriviality in Section 6 below.
5.1. The argument of the remainder of Mitchell's theorem. We begin by returning our attention to the projective resolution We "telescope" the interval P n+1 (ω n ) → P n (ω n ) to record the existence of a right-inverse, or section, s : d n+1 P n+1 (ω n ) → P n+1 (ω n ) of the map d n+1 : Here i is the inclusion map. The existence of a section s as above follows from the fact that d n+1 P n+1 (ω n ) is projective, which follows in turn from Lemma 2.8 and Theorem 4.6. In fact that theorem affords us more; namely, it affords us an explicit description of such an s: simply let One of our chief interests below will be functions f out of P n (ω n ) which extend s, i.e., which satisfy f i[dn+1Pn+1(ωn)] = s. It will follow from our analysis in Section 6 that no such f can map into P n+1 (ω n ): if f did, then d n+1 f would define a retract r of the map i, but if such an r exists then P n (ω n ) ∼ = d n+1 P n+1 (ω n ) d n P n (ω n ). As P n (ω n ) is free, this would imply that d n P n (ω n ) is projective (by Lemma 2.8), which results below will contradict. We therefore have the following diagram.
The issue, as we will see momentarily, is not that there exist no natural extensions f of s : d n+1 P n+1 (ω n ) → P n+1 (ω n ), but that these extensions all output values taking infinite support. In other words, these extensions require a concomitant expansion of the target system from P n+1 (ω n ) to R n+1 (ω n ) (here and below, see again Sections 2.3 and 2.4 for definitions).
Here j is the inclusion map; below we will define functions f n ∈ Hom(P n (ω n ), R n+1 (ω n )) making the above quadrilateral commute, i.e., satisfying j s = f n i. As noted in Section 2.4, these functions may be identified with elements f n of the cochain group K n (R n+1 (ω n )) via the simple equation f n ( α) = f n ( α ). These in turn determine elements [f n ] of the cochain group K n (R n+1 (ω n )/P n+1 (ω n )) (this quotient should be read as encoding mod finite relations, thereby reconnecting with the material of Section 2.1, as we will see). Moreover, since for all α ∈ [ω n ] n+2 , then [f n ] will represent a nonzero element of Showing (20) is our main object in Section 6. Therein, in analogy with Section 2.1, we will often term the phenomena of equations (20) and (19) nontriviality and coherence relations, respectively.
We describe now how showing lim n (R n+1 (ω n )/P n+1 (ω n )) = 0 for all n in fact implies the remainder of Theorems 2.13, 3.5, and 3.10. We focus first on the cases of ε = ω n . As Theorem 4.6 implies that cd(ω n ) ≤ n + 1 for all n ∈ ω, we need only to show that cd(ω n ) ≥ n + 1 for all n ∈ ω; for this it will suffice to show for each such n that lim n+1 X = 0 for some X indexed by ω n , as argued in Section 2.4. For the n = 0 case, simply observe, for example, that together with the existence of the short exact sequence For, letting k = n + 1 and ε = ω n in the long exact sequence (11), we see that is exact. It is not difficult to see that lim j R n+1 (ω n ) = 0 for all j > 0 (direct computational verification via K(R n+1 (ω n )) is straightforward). Exactness then entails that lim n R n+1 (ω n )/P n+1 (ω n ) ∼ = lim n+1 P n+1 (ω n ), hence P n+1 (ω n ) is just such an X as we had desired. In fact, the nontrivial functions f n we define below will correspond under this isomorphism to s d n+1 , underscoring the canonical relationship between these two functions (and their canonical relationship, in turn, with the underlying choice of C-sequence) and certifying the latter as very concrete witnesses to cd(ω n ) ≥ n + 1. Observe also that by way of Definitions 2.9 and 2.12 and Lemma 2.11, the fact that cd(ω n ) ≥ n + 1 for all n ∈ ω together with Theorem 4.6 immediately implies Theorem 3.10 and, hence, Theorem 3.5 as well.
Extending our results to the full statement of Theorem 2.13 is then straightforward; there are two broad cases to check: (1) Linear orders ε of cofinality ℵ n for some finite n. As the reader may verify, the argument of Theorem 4.6 is easily adapted to apply to linear orders ε, implying that cd(ε) ≤ n + 1. Witnesses to lim n+1 P n+1 (ω n ) = 0 readily relativize to any ordertype-ω n subset δ of ε; if δ is cofinal in ε then they extend in turn to systems indexed by ε, implying that cd(ε) ≥ n + 1.
Alternately, having shown that cd(ω n ) = n + 1 for all n ∈ ω, appeal to a theorem like [Mar00, Theorem 15.5] will immediately extend the result to linear orders ε of cofinality ℵ n .
(2) Linear orders ε of cofinality κ ≥ ℵ ω . Suppose that cd(ε) = n < ∞ and hence that d n P n (ε) is projective. We may then construct a direct summand d n P n (X) of d n P n (ε) with cf(X) = ℵ n ; by Lemma 2.8, d n P n (X) is then projective, i.e., cd(X) ≤ n, contradicting point (1) above. (This is a mild abuse: under our conventions, d n P n (X) isn't an object of Ab ε op , but it naturally identifies with one.) It suffices in fact to take any X ⊆ ε of cofinality ℵ n such that P n (X) is closed with respect to the function s d n , where s is the section d n P n (ε) → P n (ε) furnished by our assumption. For then, letting p : P n (ε) → P n (X) denote the natural projection, d n p s defines a retract of the natural inclusion j : d n P n (X) → d n P n (ε), implying that d n P n (X) is a direct summand of d n P n (ε), as desired. This argument resembles nearly enough the first part of the induction step of Proposition A.2 below that the reader is referred there for further details and a diagram.
5.2. The functions f n . We now describe the functions f n that will form the focus of the remainder of the paper, beginning with the case of n = 0.
Example 5.1. The case of ω : for any j < ω. This, though, amounts to a definition: the implicit formula Hence the formula entirely determines the "0-column" of f ( 0 ). Similarly, the formula entirely determines the "1-column" of f ( 1 ) and hence, by (21), that of f ( 0 ) as well -and so on. This defines f on { j | j ∈ ω} and therefore on all of P 0 (ω); as described in the previous subsection, it determines a function f 0 ∈ K 0 (R 1 (ω)) as well. It is not difficult to see, in fact, that f 0 represents a nonzero element of lim(R 1 (ω)/P 1 (ω)). 10 This technique very generally applies. Recall from Definition 4.8 the function b, which via the addition of a single coordinate into α, if possible, converts α to some "nearest" b( α) ∈ B. 11 Recall also that s was defined so that s d n+1 Recall that d n+1 B n+1 defines a basis for d n+1 P n+1 (ω n ), and that every element of B n+1 is of the form b( α) for some α. It follows that equation 22 is both a sufficient and a necessary condition for f n to extend s, and may even, as in Example 5.1, be read as defining such an f n : let α = ( β, γ) ∈ [ω n ] n+1 , with | β| = j + 1 and γ the maximal proper internal tail of α, so that b( α) is either 0 or β, C γ (β j ), γ . In the former case, let Unlike in Example 5.1, equations 23 and 24 alone do not fully determine f n . However, these equations do share with that of Example 5.1 a canonical solution, namely the function associating to α just those generators b( · ) ∈ B n+1 (ω n ) appearing in the full formal expansion of equation 24. More precisely, we identify the function f n ( α) with the pointwise limit of the generator-sums appearing in the possibly infinitely many steps of the recursive expansion of equation 24. Below, we duly argue that this operation is meaningful, but first-time readers might proceed directly to Section 6; therein, its nature should rapidly grow intuitively clear.
Technically speaking, f n is an , and we will tend to do so below. Observe that statements like "f n ( α) ∈ R n+1 ([α 0 , ω n ))" convert under this convention to statements about the support of f n ( α). The legitimacy of the previous paragraph's definition of f n derives from the following lemmas, which collect some useful general 10 It is gratuitous but tempting and possibly clarifying to write the relationship of these functions as follows: is a generator, not an n-tuple, the notation b( α) i hereabouts is a minor abuse; still, its meaning should be clear.
information about these functions along the way. By a branch of the formal expansion of equation 24 we mean a sequence of the form where each k i ≤ n + 1 is other than the index of the coordinate added by the i th application of b.
Lemma 5.2. Any generator b( α) ∈ [ωn] n+2 Z appears at most once in any branch of the (possibly infinite) formal expansion of equation 24.
Proof. We will show slightly more, namely that (25) is nonrepeating. Begin much as before, by letting α = ( β, γ) with β and γ of lengths ℓ + 1 and m + 1 respectively and γ the maximum proper internal tail of α, Observe first that if j = m then the coordinate γ m will never reappear in the sequence (25); in consequence, neither will α nor b( α). Now suppose j < m and let γ −1 = C γ (β ℓ ) and observe that as (γ j−1 , γ j+1 , . . . , γ m ) is internal, so long as this tuple remains a tail of the entries in the sequence (25), no application of b can recover the coordinate γ j . Observe also that if only γ j−1 is ever removed from this tuple then subsequent applications of b will only ever introduce coordinates ξ ≤ γ j−1 < γ j to the sequence (25). Hence only if some k i removes a coordinate γ j ′ > γ j may the coordinate γ j , and hence α or b( α), possibly reappear in the sequence (25) -but this argument then applies to γ j ′ , and so on, and can only end with the coordinate γ m . As we noted at the outset of this case, though, γ m , once lost, is irrecoverable.
This concludes the proof.
The above reasoning carries yet stronger implications, namely:

Lemma 5.3. A branch in the formal expansion of equation 24 either
(1) terminates, in the sense that b( β) = 0 for some β in its sequence, or (2) stabilizes, in the sense that there exists an m ∈ ω such that k j = 0 for all j ≥ m.
To see this, observe that if item (1) of the lemma fails, then there must exist an m(n + 1) such that j > m(n + 1) implies k j < n + 1: if there were not, then the last coordinates of the terms of (25) would contain an infinite descending sequence of ordinals. One may then similarly deduce that k j < n for all j above some m(n) ≥ m(n + 1), and so on, down to n = 1; let m = m(1) (we are light on the details here because they so resemble those of the arguments of Lemmas 5.2 and 5.4; see particularly the close of the proof of Lemma 5.4). Note that as the argument of m descends or "moves to the left," so too does a stable internal tail within the elements of (25), so that we can be quite concrete about the eventual form of Lemma 5.3's case (2); ultimately, it takes the shape of wherein β is the maximal internal tail of each (η j , β). Since cf(β n−1 ) < ℵ n , the cofinality of β 0 is ℵ 0 (if it were less, (26) would terminate), and it is the supremum of the sequence (η j ) j∈ω ⊆ C β .
We will apply the following lemma to conclude that the function f n is well-defined.
We apply the lemma as follows: suppose for contradiction that the formal expansion of some f n ( α) outputs the generator γ infinitely often. Then this expansion must contain an infinite branch of type (5.2) whose elements all count γ among their descendents. By Lemma 5.3, this branch eventually assumes the form (26); by Lemma 5.4 then, γ 0 < β 0 . This, though, is a contradiction, since once η j > γ 0 there is no longer any path of type (25) from (η j , β) to γ .
Proof of Lemma 5.4. Begin with α = ( υ, ε), where ε is the maximal internal tail of α and the length of υ is ℓ+1, so that b( α), if nonzero, is υ, C ε (υ ℓ ), ε . A key point is that by the definition of C ε (υ ℓ ), the branch (25) can continue -meaning b(b( α) k ) = 0 -only if k removes either an element of ε, or υ ℓ . In the former case, the initial segment υ of α is unaffected; in particular, all elements of the sequence α → b( α) → b( α) k satisfy the conclusion of the lemma. Hence it's only via steps of the latter sort that that conclusion may conceivably fail; suppose therefore that k removes υ ℓ . There are then two possibilities, depending on the length of υ. If ℓ > 0 then, letting η = (C ε (υ ℓ ), ε), satisfy the conclusion of the lemma; note furthermore that each coordinate of b(b( α) k ) k ′ is less than or equal to the corresponding coordinate of α (with at least one coordinate strictly less). This leaves only the case of ℓ = 0. Here again though, clearly, the conclusion of the lemma holds throughout the sequence α → b( α) → b( α) k . As the analysis we've just described will reapply to the last term of each of the sequences we've considered, this concludes the argument.
In its course, we showed that in the "ℓ > 0 case" corresponding coordinates descend, some strictly, in the passage from α to b(b( α) k ) k ′ . This also clearly holds of the passage from α to b( α) k in what was termed "the former case" above, and these two recognitions together suffice for the argument of Lemma 5.3.
Note in conclusion that we might read Lemma 5.3 as pointing to something "essentially finitary" about the function f( α), in the sense that its expansion has no truly interesting infinite branches, and that we might in turn read the most fundamental implication of the above arguments -namely, that the coefficient of any generator γ in f n ( α) may be computed in finitely many steps -as among its effects. This is a perspective that the higher walks of this paper's Section 8 may be regarded as formalizing.
To recapitulate: the primary task of the following section is to show that the functions f n are nontrivial in the sense of formula 20 above. This fact together with Theorem 4.6 will then immediately imply Theorems 2.13, 3.5, and 3.10, in just the fashion described in Subsection 5.1. In the process, the walks material of Sections 2.1 and 2.2 will begin to reappear, along with its higher-order analogues.

The coherence and nontriviality of the functions f n
In what follows, the letters x, y, and z will correspond to the first, second, and third coordinateplaces in ordered triples; more generally, z will denote the last coordinate-position in any ordered n-tuple below, with prior coordinate-positions then labeled in descending alphabetic order. We write = * to denote equality modulo finite differences. As discussed above, although f n is an element of K n (R n+1 (ω n )), we nevertheless regard each f n ( α) as a function [ω n ] n+2 → Z; similarly for each e n−1 ( α). This approach entails minor abuses, but appears to be the simplest. If any function in the equations below is restricted, then the comparison = * should be read as taking place on that restriction, but in this section it will be equally valid, and sometimes more telling, to read an expression like f A as the function [ω n ] n+2 → Z coinciding with f on A and outputting zero elsewhere, and to read = * , in conjunction, as applying over all of [ω n ] n+2 .
6.1. The case of n = 1. Here the function f n of the previous section specializes to a function f 1 ∈ K 1 (R 2 (ω 1 )) with the property that In fact this difference from zero is precisely s d 2 ( α, β, γ ). The coherence, in other words, of the system f 1 amounts simply to the fact that s d 2 -images have finite supports. What remains to be shown is its nontriviality, namely, the fact that no e 0 ∈ K 0 (R 2 (ω 1 )) satisfies the following property: As noted, this will establish the case n = 1 of Mitchell's theorem, and its argument will furnish the template for the cases of higher n. As noted as well, statements like "f 1 ∈ K 1 (R 2 (ω 1 ))" describe the supports of f 1 (i.e., f 1 (α, β) may be identified with an element of R 2 ([α, ω 1 )) for all α < β < ω 1 ), but we can be much more precise: when n = 1, the definition of f n via equations 23 and 24 assumes a particularly straightforward form: It follows immediately that This facilitates sufficiently "spatial" readings that we introduce the following notation: for A, B ⊆ [ξ] <ω , let A ⊗ B denote the collection of tuples ( α, β) ∈ A × B for which α < β. Extensions of this notation should be self-explanatory. For example, it follows from equations 27 and 30 that and, hence, that It follows also from equation 30 that for any "trivializing" e 0 as in (28), hence the data of such an e 0 is entirely present (mod finite) in e 0 (0). In other words, there exists an e 0 as in (28) if and only if for some e 0 (0) (For the "if" direction, let e 0 (β) = e 0 (0) [β,ω1)⊗[ω1] 2 .) We will derive a contradiction from the existence of such an e 0 (0); this will conclude the n = 1 step of our proof of Theorems 2.13, 3.5, and 3.10.  Figure 2); by equation 29, these have the following general forms: Line 35, restricted to either the 2 nd or 3 rd coordinate, bears copies (minus the first or last element, respectively) of the walk from β down to α + 1. Line 36, similarly, is an image of the club C β above α (here our convention that otp(C β ) ≤ ω for all countable β is essential). For limit β, of course, these clubs C β are infinite; by equation 31, f 1 (α, γ) must contain all but finitely much of each of these C β -images, where β ranges through (α, γ) ∩ Lim. This is a requirement in some tension with equation 32, a tension manifesting as the nontriviality of the system f 1 .
This nontriviality may be seen in either of (at least) two ways; both generalize to higher dimensions. For the first, observe that, by equations 31 and 36, we may define the following function for any β < γ < ω 1 in which β is a limit ordinal: Figure 3. Schematic profile views of f 1 (0, γ) and a candidate trivialization e 0 (0). In each, the lighter shaded box denotes the restricted domain α ⊗ ω 1 ⊗ (α, ω 1 ). By equation 32, the support therein of f 1 (0, γ) (depicted as black dots) is finite for any α < γ, but the support therein of any trivializing e 0 (0) must be uncountable for some α < ω 1 , entailing contradiction.
These two arguments are, arguably, in principle the same. Each merely emphasizes a different bar to triviality; in the latter case, lower-order nontriviality in z = γ hyperplanes (γ ∈ S 2 1 ) enforces disagreement thereon between some lower-index e 1 (0, · ) terms, which the Pressing Down Lemma then concentrates on a pair of terms for a contradiction. In the other argument, the eventual finitude of the disagreements between pairs of e 1 (0, · ) terms allows for closing-off arguments, entailing contradiction when they intersect with S 2 1 . In each case, the nontriviality principle is one first appearing at ω 2 ; as the reader may verify, all the "initial" families {f 2 (0, β) | β ∈ [δ\{0}] 2 } are trivial in the sense of (45). 12 In the interests of clarity, we conclude this section by simply outlining the argument of the nontriviality of the functions f n (n > 2); at least on a first reading, the greater detail in which it is recorded in Appendix B may serve only to obscure the key points. The argument in each case begins with a coherence equation (as in (27), (39)) and the nontriviality relation (as in (28), (40)) which we aim to prove; each derives in a straightforward way from the argument and equations 19 and 20 of Section 5.1. Similarly, explicit recursive definitions of the functions f n (as in (29), (42)) derive from equation 24; support considerations then afford us "degree reductions" of the coherence and nontriviality relations at hand (as in (31), (34), (44), (45)). More importantly, as a comparison of the explicit f n and f n−1 definitions makes plain, for any γ ∈ S n n−1 the family {f n ( β, γ) | β ∈ [γ] n } is in essence a relativization to C γ of the family {f n ( β) | β ∈ [ω n−1 ] n }. 13 It will be our inductive assumption that the latter family, and hence the former, is nontrivial. The second argument for the nontriviality of f 2 is then the simplest to generalize: for each γ ∈ S n n−1 the nontriviality of {f n ( β, γ) | β ∈ [γ] n } implies that there exists some α γ ∈ [C γ ] n such that, just as in equation 47, (Here α γ (0) denotes the minimum element of α γ .) Hence, just as before, by the Pressing Down Lemma there exists a stationary S ⊆ S n n−1 and α ∈ [ω n ] n such that α γ = α for all γ ∈ S. However, also just as before, our triviality equations will imply that err( α) := γ > α | This contradiction shows that no e n−1 can trivialize f n . As argued in Section 5.1, we have shown the following: Theorem 6.1. For all k ∈ ω the cohomological dimension of ω k is greater than or equal to k + 1.
As described, this theorem together with Corollary 4.9 then concludes the proof of Theorems 2.13, 3.5, and 3.10.
In the remaining sections we foreground some of the more intriguing combinatorial phenomena manifesting in or by way of the functions f n and the higher-order variants of familiar objects which they articulate.

Trees of trees, cohomology, and higher coherence in various guises
7.1. The cohomology of the ordinals. It will be useful henceforth to adopt a more systematic usage of the terms coherence and triviality, and of their order-n instances. The following definition generalizes the mod finite nontrivial coherence relations of Section 2.1; here and in all subsequent definitions, A denotes an arbitrary abelian group.
or, in other words, if ϕ β is finitely supported, for every β < ε. If ϕ itself is finitely supported then it is 0-trivial. Observe that this definition of 0-trivial readily applies to functions of more general domain as well.
(For readability, here and below we've suppressed restriction-notations, understanding equations to hold on the intersection of their constituent functions' domains). The family Φ n is n-trivial (or simply trivial, when the n is clear) if there exists a Ψ n−1 = {ψ α : We call any of the aforementioned functions or families of functions A-valued, and of height ε.
Observe that the operation of pointwise addition determines group structures on the both the set coh(n, A, ε) of n-coherent A-valued height-ε families of functions and the set triv(n, A, ε) of n-trivial A-valued height-ε families of functions; observe moreover that the latter then forms a subgroup of the former. The following is shown in [BL19b, Theorem 2.30]: Theorem 7.2. LetȞ n (ε; A) denote the n thČ ech cohomology group of the ordinal ε (endowed with its usual order-topology) with respect to the sheaf A of locally constant functions to A. Then for all ordinals ε and positive integers n and abelian groups A, H n (ε; A) ∼ = coh(n, A, ε) triv(n, A, ε) .
In particular,Ȟ n (ε; A) = 0 if and only if there exists an A-valued nontrivial n-coherent family of functions of height ε. In [BL19b] the following is shown as well: 14 Theorem 7.3. For any abelian group A and positive integer n and ordinal ε of cofinality less than ℵ n , we haveȞ n (ε; A) = 0.
In [BL19b] it is also asserted that there exist groups A for whichȞ n (ω n ; A) = 0, and hence that ω n is the least ordinal with nonvanishing constant-sheafȞ n , but no proof is given. We now show that from the work of the previous section, this deduction is easy.
Recall first that a first step in the analysis of the functions f n above was a reduction in "coherence degree"; f 1 , for example, satisfies relations (27) closest in form to the 2 -coherence of Definition 7.1 above, but is best regarded as a collection of 1-coherent families of functions, as in equation 31. This nontrivial 1-coherence permeates f 1 "in every direction"; for example, let for α < β < ω 1 . By equation 31 and the non-existence of an e 0 as in (34), Lemma 7.4. {ϕ x β | β ∈ ω 1 } is a nontrivial coherent family of functions. The reason for this is that any ϕ trivializing this family naturally identifies with anê 0 (0) ∈ [ω1] 3 Z such that for all limit β < ω 1 and all but a finite set a(β) of α < β, By the Pressing Down Lemma, there then exists a stationary S ⊆ ω 1 and finite set a such that a(β) = a for all β ∈ S. Let γ = max a (do nothing if a = ∅) and modifyê 0 (0) on {{α} ⊗ [ω 1 ] 2 | α ∈ a} to agree with f 1 (0, γ + 1); this defines an e 0 (0) as in (34), a contradiction.
In particular,Ȟ 1 (ω 1 ; A) = 0 for A = Z/2Z. For higher n, conversions patterned on (49) deriving from bijections θ : ω n → [ω n ] n+2 omit more and more of the data of f n ; consequently, though such 15 The point is that although ϕ θ β records only a strictly "initial tetrahedron" of the output of f 1 (0, β) for any β ∈ E θ , coherence ensures that any trivialization of Φ indeed translates, via θ, to a trivialization of f 1 . Note that ϕ θ β does fully record that tetrahedron since the range of any f 1 (0, β), viewed as a function [β + 1] 3 → Z, is {−1, 0}, as is immediate from equations (29) and (30); in other words, there is no information loss in the passage from Z to Z/2Z per se.
functions will be n-coherent, they no longer so clearly inherit the nontriviality of f n . Just as in Lemma 7.4, however, families of functions recording first-coordinate slices of f n , Theorem 7.6. {ϕ ⋆ β | β ∈ [ω n ] n } is a nontrivial n-coherent family of functions. In particular, H n (ω n ; A) = 0 for A = ωn Z. Hence ω n is the least ordinal with a nontrivialČech groupȞ n with respect to any constant sheaf A.
We return to the question of the integral cohomology groups of the ordinals ω n in our conclusion below. Here we remark simply that it is consistent with the ZFC axioms thatȞ n (ω n ; A) = 0 for A = Z and all n ∈ ω (this follows from ZFC + V = L, for example; see [BL19b]); the question of whether this is a ZFC theorem seems a very good test of our understanding of the higherdimensional combinatorics that form the present work's theme. Unsurprisingly, this question may also be phrased in terms of higher derived limits; it is in fact equivalent to the following: Question 7.7. Let Q(ε) be the inverse system (Q α , q αβ , ε) with Q α = α Z and q αβ : Q β → Q α the natural projection. Is it a ZFC theorem that lim n Q(ω n ) = 0 for each n ∈ ω?
This question in turn appears closely related to the sensitivity of the vanishing of lim n A to the dominating number d = cf( ω ω, ≤) and its relation to the cardinal ℵ n , where A is the inverse system featuring centrally in [MP88; DSV89; Tod98; Ber17; BL19a]. Indeed, this sensitivity was a main initial motivation for the present line of investigation.
7.2. Trees of trees. In this section, a mild modification of Definition 7.1 will facilitate description of n-dimensional generalizations of coherent Aronszajn trees, each of which makes its first ZFC appearance at ω n ; classical coherent Aronszajn trees themselves comprise the n = 1 case. The modification is simply to allow more general domains (still depending on γ 0 ) for the functions ϕ γ ; as before, the comparison of such functions will always take place on the intersection of their domains.
For motivation, observe that the family itself defines a coherent ω 1 -Aronszajn tree: simply view its elements, i.e. the nodes of any level α of the tree, as functions [α] 3 → Z, and order these nodes by inclusion. As a cofinal branch in this tree would render f 1 trivial, this tree is Aronszajn.
On the left-hand side of Figure 5 below is the more or less standard visualization of such a tree. On the right is a complementary visualization, one organized to foreground the essential mechanics of the nontrivial coherence of the system f 1 (0, · ). As the analysis of Section 6.1 made clear, those mechanics concentrate on the planes {z = β | β ∈ Lim ∩ ω 1 }; more precisely, they concentrate on distinguished copies of C β therein. Schematically, then, we might view {f 1 (0, γ) z=β | β ≤ γ} as a family On the left is the standard view: T is defined by its branches s 1 γ (γ < ω 1 ), i.e., by height-γ functions which encode the data of f 1 (0, γ). On the right is an alternative visualization: here s 1 γ is pictured as the wedge to the left of the vertical line at γ. At each β ≤ γ, the function s 1 γ outputs a 0-coherent function, namely the characteristic function of a tail of C β . Moreover, any s 1 γ and s 1 δ disagree on only finitely many columns in their common domain (and disagree only finitely often thereupon as well), a relationship to the functions s 1 γ which no length-ω 1 t 1 can globally replicate. ordinals β ≤ γ. Any of several approaches might effect this sort of identification; in perhaps the simplest, 1 if β is a limit ordinal and α ∈ C β \m(β, γ) 1 if β is a successor ordinal and β = α + 1 0 otherwise for each α < β; here m(β, γ) is the function defined in (37) above. This abstraction of f 1 (0, γ), of course, is essentially that of Figure 3 reflected through the graph of z = x. The point of all this is simply the following: {s 1 γ | γ < ω 1 } is a natural recasting of the family {f 1 (0, γ) | γ < ω 1 } in which: (1) Each {s 1 γ ( · , β) | β ≤ γ} is a family of 0-coherent functions, each of which is non-0-trivial when β is a limit.
(2') For any γ ≤ δ < ω n the functions s n δ (γ+1)×[γ] n−1 and s n γ differ by an (n−1)-trivial function. More precisely, the families (3') There exists no t n such that for all γ < ω n the functions t n (γ+1)×[γ] n−1 and s 1 γ differ by an (n − 1)-trivial function. Put differently, there exists no t n = s n ωn such that for all γ < ω n the family ωn induces the function ϕ ωn α in the manner described in the paragraph preceding (1'). Cumbersome as the notations do grow, the idea of these families is in fact very simple, as Figure 6 is meant to convey. There and below, we focus on the case of n = 2.
As mentioned, the argument of the higher-n cases involves no conceptual novelty, but does entail rising notational costs (again functions e n−1 trivializing f n are defined from the trivializations t n , but in more "pieces" together involving a longer series of derived trivializations u). In its place we describe a cleaner mild variation of the "trees of trees" structures above and show that nontrivial instances of this variation coincide with instances of the higher-order nontrivial coherence of Definition 7.1. Our argument of this fact is close enough in spirit to the one we omit that little content is ultimately lost.
We turn now to the definition of this variation. Though our terminology will overlap with that of Definition 7.1 above, context will generally indicate which of the two notions of n-coherence we have in mind; in cases of potential ambiguity, we denote the notions in Definitions 7.1 and 7.8 as ncoherence I and n-coherence II , respectively (this overlap is also somewhat justified by Theorem 7.10). It will streamline discussion, also, to tacitly identify families of functions {s n γ : [γ] n → A | γ < κ} with single functions s n : [κ] n+1 → A via the equation s n ( α, γ) = s n γ ( α). Such an identification is operative at each step of the following inductive definition: For any n > 0 a family of functions {s n γ : Such a family is n-trivial if there exists a t n : [ε] n → A such that t n [γ] n − s n γ is (n − 1)-trivial for all γ < ε.
Observe that any n-coherent family S n = {s n γ | γ < ε} induces a tree Observe also that if S n is non-n-trivial then T (S n ) has no cofinal branch. In fact, S n is non-n-trivial if and only if the uniform closure T * (S n ) of T (S n ) has no cofinal branch; this notion generalizes that of [Tod07, Definition 4.1.2]: Definition 7.9. An n-coherent tree T (S n ) is one of the form (53) for some n-coherent family of functions S n = {s n γ : [γ] n → A | γ < ε}. The uniform closure of T (S n ) is the tree An n-coherent tree T (S n ) is nontrivial if its uniform closure T * (S n ) has no cofinal branch.
The ways or degrees to which n-coherence materializes at the level of the tree-structures of the trees T (S n ) (as it very consequentially does in the case of n = 1; see [Tod07,Chapter 4]) remains an interesting question.
As hinted above, the terminological overlap of Definitions 7.8 and 7.1 reflects a degree of equivalence between the notions they describe. This is a relationship made precise by Theorem 7.10. Although at first glance, particularly for low n, this relationship may appear to consist in little more than notational rearrangements (as is indeed the case in one direction; see Lemma C.1), the theorem in its full generality is quite subtle, and its proof is sufficiently tedious that we defer it to our appendix (Section C). In Appendix C, we show something slightly stronger than this, namely that for every n > 0 and ordinal ε and abelian group A, These quotients are the obvious modifications of that of Theorem 7.2 in light of Definition 7.8. From Theorems 7.10 and 7.6 the following is immediate: Corollary 7.11. For all n > 0 the least ordinal ε for which there exists a nontrivial n-coherent tree of height ε is ω n .
8. The trace functions at the heart of the functions f n 8.1. The case of n = 1. An organizing question in Sections 6.1 and 7.2 was that of the behavior of the function f 1 on the planes z = β for limit ordinals β. For any such β and α < β < γ this question amounts essentially to that of f 1 (α, γ)'s "point of arrival" to z = β: by equation 29, once β appears as second coordinate in an output − ξ, β, δ in the iterative expansion of f 1 (α, γ), some tail of the sequence (36) will follow by way of the term f 1 (ξ, β). As β is a limit, under our standing C-sequence assumptions (see Section 4), the aforementioned δ can only have been β + 1; if γ > β + 1 then this output must have been preceded in the f 1 (α, γ)-expansion by some − ξ ′ , β + 1, δ ′ , with ξ ′ ≤ ξ. Reasoning along these lines focuses our attention on the region f 1 (α, γ) [α,β)⊗[(β,γ]] 2 , the prism (minus the y = β plane) depicted in the α = 0 case of Figure 7 below. Note that falls in this region only if ξ L i < β < ξ T i = C δi (ξ L i ) < δ i and that in this case the indexing makes sense; by this we mean that there is a next value in As above, these considerations have maximum scope when α = 0; therefore let it. Beginning with f 1 (α, γ), then, the right, left, and middle coordinates of the above-described sequence (ending when the middle coordinate is β) are more than a little familiar:  (56) below. More precisely, this phase begins at the point (max(C γ ∩ β), min(C γ \β), γ), moving down and forward until β appears as y-coordinate, in the starred node (max(L(β, γ)), β, β + 1) (for concreteness, here we take both β and γ to be limit ordinals). It then outputs an image of the tail of the club C β (as in (36)) above max(L(β, γ)).
region, we have δ 0 = γ and ξ T 0 = min C γ \β and ξ L 0 = max C γ ∩ β (equaling 0 if this intersection is empty) and, more generally, where In other words, the restrictions of f 0 (α, γ) to the regions β ⊗ [[β, ω 1 )] 2 isolate sequences (54) which coordinatewise are precisely the upper and lower traces of the walk from γ down to β (see again Section 2.1 for definitions). As 0<β<ω1 β ⊗ [[β, ω 1 )] 2 = [ω 1 ] 3 , each element of the support of any f 1 (α, γ) will fall in some such prism and, hence, in some sequence of the form (54); in consequence, f 1 may reasonably be regarded as little other than a knitting together, in a strikingly comprehensive fashion, of the upper and lower traces of pairs of countable ordinals. See Figure 7.
These recognitions can illuminate the classical: by (57), for example, for all α < β < γ < ω 1 . In other words, for any such α, β, γ the difference between ρ 2 (α, γ) and ρ 2 (α, β) manifests as difference between f 1 (0, γ)↾ β⊗[ω1] 2 and f 1 (0, β), which is finitely supported by equation 31. This imposes a uniform bound, immediately implying the mod bounded coherence relations of ρ 2 recorded in equation 3. Observe in contrast, however, that the mod bounded nontriviality relations of ρ 2 are considerably less spatially obvious within the f 1 system. These latter relations appear only really to be accessible via a deeper Ramsey-theoretic analysis of the functions ρ 2 , a point of more general significance below.
The materialization of so much of the classical walks apparatus -ρ 0 , ρ 1 , ρ 2 , Tr, L, for example, even the clubs C β themselves -as elementary spatial features of the f 1 system may not itself be altogether surprising, in light of equation 29. The interesting point is that f 1 is only the first in an infinite series of nontrivial n-coherent systems f n of broadly similar spatial organization.
8.2. The basic form tr n . As the reader may recall from Section 6.1, classical walks also appear in the x = β planes of the functions f 1 (β, γ) (see the planes x = 0 and x = α of Figure 2). These in fact are the projections ] 2 or, more precisely, of all but the last element of the associated sequence (54).
From the perspective of the f 1 system, then, the classical upper trace Tr(β, γ) manifests at once in or as ( (3) the key finitary constituents (together with L(β, γ)) of the nontrivial coherent family of countably supported functions f 1 (0, γ) (γ < ω 1 ). A related view is of the functions Tr and L as distilling away the redundancies of the f 1 system, as in lines 55 and 56 above. These interrelated conditions generalize: in this and the following section we describe functions Tr n , each recursively defined on the pattern of Tr 1 := Tr, satisfying the higher-n analogues of items (1) through (3) above. Conditions (1) and/or (2) tell us computationally what these higher-order traces should be. Several minor choices arise as to how precisely to render this data, a point we return to below. For concreteness, though, we first record what we propose as the basic form tr 2 of the degree-2 upper trace, one which is broadly "read off" from either f 2 (α, β, γ) w=α or f 2 (0, β, γ) α⊗[[α,ω2)] 3 in the manner of (1) or (2) above, as the reader may verify. 16 The form is 16 Readers are encouraged to compare this definition with equation 42, from which it is derived. Observe that either of the determining restrictions just cited will entail that the f 2 (C βγ (α), β, γ) (if β ∈ Cγ ) and f 2 (β, C γ (β), γ) (if β ∈ Cγ ) terms disappear in the translation to the tr 2 form. This is because each of those terms exits the aforementioned restrictions (in either approach), and does not return to them in any subsequent step of its expansion. This in itself does not represent data loss, however, as the supports of those terms are, in aggregate, recovered by evaluations at other values of tr 2 . This phenomenon is visible already in the relations between the "2-branching" expansion-trees of f 1 and the "1-branching" form of the classical Tr, which readers also may find it valuable to compare.
(α, β, γ) Figure 8. First steps of tr 2 (α, β, γ). Associated to any tr 2 -input such as (α, β, γ) are an ordinal output, like β ∅ , and two further tr 2 -inputs, like (α, β ∅ , γ) and (α, β, β ∅ ). Shaping the above diagram are the assumptions (made somewhat at random, simply for concreteness) that β < γ and the outputs β ∅ , β 0 , and β 1 defined as above are each meaningful and not equal to β. Lower outputs β σ may well correspond to undefined expressions, in which case they should be regarded as the empty set, marking the end of a branch. recursively defined as follows: for all α ≤ β ≤ γ < ω 2 , Grounding this recursion are the following boundary conditions: • if β ∈ C γ and C βγ \α = ∅ then tr 2 (α, β, γ) = ∅; • if β = γ then tr 2 (α, β, γ) = ∅. Several remarks are immediately in order: (i) Any tr 2 (α, β, γ) is naturally viewed as a binary tree, one in fact generalizing the 1-branching tree (i.e., the walk) associated to Tr(α, β). Depicted in Figure 8 are the first two levels of the tree associated to tr 2 (α, β, γ) under generic assumptions on α, β, and γ. The nodes of this tree are labeled with two sorts of data: as with the classical Tr, the recursively defined function tr 2 records an ordinal (appearing in the lower half of a node) then proceeds to new inputs (appearing in the top halves of successor nodes); each of these displayed in Figure 8 above. This is because unlike in the classical case, the collection of ordinals output and the collection of tuples input in the course of a higher-dimensional walk are no longer informationally equivalent; while the former may be better suited to combinatorial applications, deductions per se will often require the fuller data of the latter. This also is the reason for the disjoint unions appearing in equation 59: a novelty of the higher-dimensional traces is the possibility of output-ordinals arising therein repeatedly.
(ii) As mentioned, the form of tr 2 derives from that of f 2 (0, β, γ) α⊗[[α,ω2)] 3 together with multiple presentational choices. 17 One example of such a choice is the following: coordinates both of the form min(C βγ \α) and min(C γ \β + 1) may arise among the supports of f 2 (0, β, γ) α⊗[[α,ω2)] 3 . Outside the contexts of R n (ε) or P n (ε), however, steps to min(C γ \β) are more natural than steps to min(C γ \β + 1), and the combinatorial effects of uniformly adjusting definitions in this direction are superficial. Similar comments apply to the expansion of domain to inputs α ≤ β ≤ γ < ω 2 , to the choice of boundary conditions, and to the choice of which ordinals to record as outputs: the fact is, a number of variations on the basic idea hold fairly equal title to the name of "higher walk"; what counts in any of them is that they record the data of the systems f n with some of the concision (i.e., finitude) and versatility of classical walks.
We now describe the fundamental features of the function tr 2 , framing our discussion in terms of Figure 8. Observe that in the passage from any node to one directly below, exactly one element of the associated coordinate-triple is replaced. Observe also that this element is never the least one, α. Descending along the leftmost branch, for example, it is the second coordinate that is always changing; observe that its pattern β, β ∅ , β 0 , . . . is that of the C γ -internal walk from β down to α (see Section 2.2). Along the rightmost branch, on the other hand, it is the third coordinate that is in motion; its pattern is visibly that of the classical walk from γ down to β. Here it is natural to term the walks associated to rightwards paths through the tree external. Any tr 2 (α, β, γ) may then be regarded as a structured family either of internal walks or of external walks, insofar as any binary tree is, as a set, simply the union either of its leftwards or rightwards branches.
Here a word of clarification is in order. While leftwards paths through tr 2 (α, β, γ) correspond precisely to internal walks, rightwards paths may properly contain classical walks in the following way: let β ′ = min(Tr(β, γ)\{β}). If α ′ = min(C ββ ′ \α) is defined then the node below and to the right of (α, β, β ′ ) is (α, α ′ , β). The rightwards path out of this node will then describe a classical walk from β down to α ′ , possibly again initiating a further walk out of α ′ upon arrival, and so on. External walks correspond in this way to iterated descending chains of classical walks ("walks of walks").
The derivation, via the generalized principles of (1) and (2) above, of a tr 2 as in (59) readily extends to define functions tr n sending (n + 1)-tuples of ordinals to n-branching trees; details are left to the reader. By essentially the same argument as that for Lemma 8.1, these trees are all finite. Also as above, "hyperplanes" through these trees (i.e., subtrees of smaller branching number determined by some fixed rule of descent) correspond to lower-order walks-structures and their relativizations. To give something of the flavor of these generalizations, the first step of such a tr 3 (α, β, γ, δ) would depend on whether γ ∈ C δ and, if so, on whether β ∈ C γδ and, if so, on the value of min(C βγδ \α).
8.3. The functions Tr n and ρ n 2 . Mirroring the n-branching descending trees tr n from below are higher-order lower trace functions L n tracking the movement of the first coordinate of the associated outputs of f n (0, · ) α⊗[[α,ωn)] n+1 ; these n-branching trees increase, like L 1 , along each of their branches. Together, the functions tr n and L n essentially encompass the structured coordinate data of the systems f n , just as in the case of n = 1. For higher n, however, these f n systems carry the additional data of sign; the full-fledged upper trace functions Tr n consist simply in the addition of this data to the functions tr n . To record this data, these functions take as inputs signed n-tuples of ordinals, and output signed ordinals, as in the example of Tr 2 : The boundary conditions are just as before. Observe that, its outputs' signs being unvarying (and its outputs' ordinals all distinct), is informationally equivalent to the classical Tr, and naturally identifies therewith (strict identification would require revision of the ordinal outputs to begin with β and revision of the boundary values Tr 1 (±, α, α) to {±α}, but these are minor points). Just as subdivision of n-simplices is a reasonable heuristic for the iterative processes of tr n (i.e., new n-tuples are formed from a new ordinal conjoined with elements of the boundary of the old one, viewed as a simplex), geometric notions of orientation form a natural heuristic for the signs arising in Tr n ; as in multivariable calculus or geometry, for example, these signs or orientations only assume their proper significance in settings of more than two coordinates. These signs exhibit useful and interesting organizing effects within Tr n . For the duration of the down-and-rightwards movement in tr 2 that we identified above with a classical walk, for example, inputs' and outputs' signs are both constant, until the last step. If here a further walk is initiated, signs flip, remaining constant for its duration, and so on. Similarly, inputs' signs along any leftwards, internal walk of Tr 2 are constant on the branch's full length; outputs' signs are constant after a possible first step "up into" the internal walk (see again Figure 1, caption). Hence one may speak not only of the signs of nodes, but of the signs of eventually-rightwards branches as well.
These observations should begin to suggest the number of interesting characteristics, "statistics," or ρ-functions which higher walks admit. Clearly, their fuller analysis falls beyond the scope of the present paper. We close, therefore, with a description of one of the simplest of these, a generalization of the function ρ 2 , to better indicate where we believe the main next work to lie. For increasing (n + 1)-tuples of ordinals α, define the function ρ n 2 ( α) as the "negative charge" of Tr n ( α), i.e., as the "minuses minus the pluses": In particular, under the boundary condition parenthesized above, ρ 1 2 (α, β) = |Tr 1 (+, α, β)| = |Tr(α, β)| − 1 = ρ 2 (α, β) for all α ≤ β. The reasoning at (58) also readily generalizes; in other words, the following may be deduced either from the coherence of f n or by arguments generalizing the classical: Proposition 8.2. For each n > 0 let Φ n = {ϕ β : β 0 → Z | β ∈ [ω n ] n } denote the family of fiber maps ϕ β (α) = ρ n 2 (α, β). Then Φ n is an n-coherent family of functions, with respect to either of the moduli of bounded functions or of locally constant functions.
Here n-coherence refers to the relations described in Definition 7.1, with the modulus adjusted accordingly. To see the proposition's conclusion with respect to bounded functions, for example, observe that for all β ∈ [ω n ] n+1 and α < β 0 , by the same reasoning as at (58). Observe then that the right-hand side is finite by equation 65 of Appendix B.
As in the case of n = 1, establishing non-n-trivial n-coherence is a taller order, typically calling on Ramsey-theoretic facts like the following: for every i < j in ω 1 . Then for any ℓ ∈ N there exists a cofinal Γ ⊆ ω 1 such that ρ 2 (β i , γ j ) > ℓ for any i < j in Γ.
We term this property ρ 2 -unboundedness below; see [Tod07,Lemma 2.4.3] for a stronger version of this theorem. This property is not far in spirit from one of the first and most celebrated applications of walks, namely the construction in [Tod87] of witnesses to the negative partition relation Here the subscript could even be taken to be ℵ 1 ; our choice is mainly for parallelism with the following ZFC result from [Tod94], ℵ0 , together with this relation's higher-order variants ℵ n → [ℵ 1 ] n+1 ℵ0 , implicit therein. This in turn suggests that one guiding question for future work should be whether functions deriving from higher walks witness higher-dimensional versions of the ρ 2 -unboundedness of Theorem 8.3. This question does not appear altogether easy. Simply to motivate this work and define its central objects has been among this paper's main aims.

Conclusion
In the above, we described for each n > 0 a number of interrelated n-dimensional (or (n + 1)dimensional) combinatorial phenomena correlating closely with the ordinal ω n . Much of the interest of these principles lies in their being ZFC phenomena, and indeed, our account has very deliberately avoided any appeal to additional set-theoretic assumptions. Having named these phenomena, however, the effects of such assumptions form some of the most immediate next questions. Answers to these questions are well-understood in the n = 1 case: (κ), for example, ensures the existence of nontrivial coherent families of functions (and hence of nontrivial 1-coherent trees) of height κ, while the P-Ideal Dichotomy implies that there are no such families on any ordinal of cofinality other than ℵ 1 . One framing of the second part of Question 9.1 is the question of whether it is consistent thatȞ 2 (ω 3 ; A) = 0 for all abelian groups A; any positive result seems certain to require large cardinal assumptions (see [BL19b]).
Question 9.1 in part addresses the conspicuous question of "beyond ℵ ω ", but there are other senses in which we might: To what degree may we regard the techniques of this paper as more general steppingup principles, translating combinatorial phenomena on any ℵ α to higher-dimensional phenomena on ℵ α+k ?
The question of whether families of combinatorial phenomena indexed by the natural numbers might extend into the transfinite is too nebulous to record formally, but cannot be altogether dismissed; Hjorth in [Hjo02] has described characterizations of all the alephs of countable subscript, for example.
In broadest senses, the present work pursues a study of set-theoretic incompactness principles of higher dimension; it thereby raises questions of how this shaping notion of dimension might in these contexts be made precise. The cohomological dimension of Mitchell's Theorem is one way it might be, but a hallmark of classical dimension theory is the provable equivalence in "nice" settings of several otherwise distinct notions (see, e.g., [HW41,Introduction]). The structures we have highlighted are far from isolated; n-dimensional combinatorial phenomena on ω n manifest with increasing frequency in a variety of mathematical fields, often to considerable effect. The following is a somewhat haphazard survey: • Kuratowski's Free Set Theorem ([Sie51; Kur51; Sik51]; see [Erd+84]) may be the bestknown of results relating the cardinals ℵ n to their subscripts. The theorem in recent decades has found application to longstanding problems in both model theory [MS18] (see also [BKL17;LS93]) and lattice theory [Weh98; Weh07; Ruz08]. • Results in which the combinatorics of ℵ n make an appearance via assumptions like 2 ℵ0 = ℵ n or even n = ℵ n are too numerous to even begin to list here. We do note, though, that the decisive precedent [Oso68] for Mitchell's Theorem was of this form (see the discussion in the appendix below); we note as well the importance of such assumptions in infinite combinatorics of an additive and/or spatial character [Kom94; Sch00; Fox07], much as the present work's are. • In an application to Banach space theory, Lopez-Abad and Todorcevic construct length-ω n normalized weakly-null sequences without unconditional subsequences via compoundings (of ρ functions) not far in spirit from those pervading our work above [LAT13]. • In the note [Lar17], Larson records combinatorial features of finite subsets of the ordinals strongly evocative of Lebesgue covering dimension: associated to each ω n are collections whose subsets "reduce" to sets of size n + 1, but not in general to sets which are smaller. Finer invariants than cohomological dimension are particularČech cohomology computations, as we have described in Section 7.1 above. The outstanding question in this area is the following: Question 9.3. Is it a ZFC theorem thatȞ n (ω n ; Z) = 0 for all n ≥ 0?
The question is in some sense perverse: throughout our account above, combinatorial structures on ω n (particularly for n > 1) have appeared "most at home" in the wide berths of settings like [ω n ] n+2 or ωn Z. Hence one of the most natural and persuasive ways of certifying the ndimensionality of ω n , namely, a ZFC argument thatȞ n (ω n ; Z) = 0, would entail overcoming the very affinity through which it was first perceived, i.e., it would entail realizing these "wide-angle" combinatorics on the much more restricted setting of Z. It would be equally interesting if this is not possible in the ZFC framework, but it seems likelier that this task is simply a test of the depth of our understanding of the higher-dimensional combinatorics of the ordinals ω n . Higher walks seem to us both a potential resource and motivation for developing just this sort of understanding.
Question 9.4. Do functions deriving from higher walks exhibit higher-dimensional negative partition properties? In particular, do any exhibit higher-dimensional versions of ρ 2 -unboundedness?
As indicated, the main result of [Tod94] does suggest the existence of underexplored combinatorics in this direction.
We close with a simplified version of a question lingering from Section 3 (see footnote 9): Question 9.5. Call a simplicial complex X acyclic ifH ∆ k (X) = 0 for all k. Is it the case that, for any n > 0, any n-dimensional acyclic simplicial complex X on a set S may be extended to an n-dimensional acyclic simplicial complex Y on S with a complete (n − 1)-skeleton, i.e., satisfying X n−1 = [S] n ?
We do not expect this question to involve any deep set-theoretic considerations; its answer does, however, a bit surprisingly, appear to be unknown.
The question also cues the following reflection: we labored above to access some of the concrete content of Mitchell's abstract category-theoretic result; its nature in turn suggests that abstract simplicial or homotopical techniques could conceivably play a role in its further development. Particularly attractive would be a homotopical framework in which cardinal succession figures as suspension, helping to account, for example, for the chart concluding [BL19b,p. 44]. Suggestive in this direction is how the "space between" one cardinal and the next accommodates a cone construction at the heart of Mitchell's original argument; this is one further reason we record that argument in our appendix, just below.

Appendix A. Mitchell's original argument and its background
For simplicity, we restate Mitchell's theorem just in terms of ordinals: Mit72]). If ε > 0 is an ordinal of cofinality ℵ ξ and ξ is finite, then the projective dimension of ∆ ε (Z) is ξ + 1. If ξ is infinite, then the projective dimension of ∆ ε (Z) is ∞.
That pd(∆ ε (Z)) ≤ ξ + 1 for such an ε was known at the time, due to Goblot [Gob70]. Hence the novelty of Mitchell's result was its computation of lower bounds for pd(∆ ε (Z)); this part of the theorem may be rephrased as follows.
Proposition A.2. Let ε be of cofinality ℵ ξ . Then d n P n (ε) is not projective for any finite ordinal n ≤ ξ.
Below we sketch the original argument of Proposition A.2, referring the reader to [Mit72] or [Mar00] for details. Some words of context, though, seem to be in order before beginning.
Fundamental to all our arguments and constructions above were functor or presheaf categories Ab ε op ; the name "presheaf" derives from the centrality of Ab τ (X) op toČech or sheaf cohomology computations, relevant in our context as well (here τ (X) op denotes the collection of open subsets of a topological space X, reverse-ordered by inclusion). More generally, let P denote any partial order; Mitchell's point of departure was the resemblance of Ab P to R-module categories R Mod ∼ = Ab R , where R is a ring, construed on the right as a one-object additive category. Under this view, just as module theory is the representation theory of rings R, the study of the category Ab P might be thought of as "the representation theory of orders," and all of the foregoing may be viewed as a study of several of the most fundamental objects of these categories, ∆ ε (Z) and P n (ε) (n ∈ ω).
Put differently, the theorem that has formed our focus first emerged within a larger project of translating "noncommutative homological ring theory [...] to (pre)additive category theory" and as such incorporates multiple prior recognitions of the homological significance of the cardinals ℵ n [Mit72, p. 2]. We would heartily recommended Osofsky's 1974 survey The subscript of ℵ n , projective dimension, and the vanishing of lim n to any reader interested in that background, as well as Husainov's wider-ranging 2002 survey of Mitchell's work and its wake [Oso74;Hus02]. We record here just a few of the more noteworthy points: (1) "The first irrefutable indications that cardinality was intimately tied up with projective dimension came in 1967 in two separate papers where lower bounds as well as upper bounds on dimensions were calculated in terms of subscripts of cardinalities" [Oso74, p. 14]. These were [Pie67] and [Oso67]; rough outlines of the argument we will sketch below are legible in each.
(2) Very shortly thereafter, Barbara Osofsky published [Oso68]; this is the acknowledged template for Mitchell's theorem [Mit72,p. 6]. It seems telling that these first "indications" all emerged in work on rings which articulate orders: Boolean rings, valuation rings, and directed rings, respectively. In [Oso68], cardinal arithmetic assumptions transfer features of the cardinals ℵ n to rings of size continuum. Perhaps the best-known result in this line (cited in [Wei94,p. 98], for instance), for example, is that the global dimension of ω C is k + 1 if and only if 2 ℵ0 = ℵ k . Most striking from our perspective, though, is the appendix of [Oso68]: therein, following the lead of [Bas63], Osofsky constructs bases for projective modules d 1 P 1 and d 2 P 2 quite close in spirit to the n = 1 and n = 2 cases of our more general constructions above. Osofsky summarizes her survey as follows: "What began as a study of dimension via derived functors branched off into a study of dimension via cardinality and came back to a study of derived functors via cardinality" [Oso74, p. 8]. As should be clear, the present work pursues a fourth combination in this sequence: the study of cardinality via derived functors and dimension.
Sketch of proof of Proposition A.2. The argument is by induction on ξ. The base case ξ = 0 consists in verifying that d 0 P 0 (ε) ∼ = ∆ ε (Z) is not projective if ε is a limit ordinal. The mechanism of the induction is an argument that if d n−1 P n−1 (δ) is not projective for any δ < ε with cf(δ) < cf(ε) then d n P n (ε) is not projective either.
The base case: By the following claim, if ε is a limit ordinal then the epimorphism d 0 : P 0 (ε) → ∆ ε (Z) has no right-inverse. Hence ∆ ε (Z) is not projective. Claim A.3. Let ε be a limit ordinal. Then the only morphism f : ∆ ε (Z) → P 0 (ε) is the zero morphism.
The induction step: This consists in showing that if d n P n (ε) is projective and n > 0 then for any regular κ < cf(ε) there exists a δ ∈ Cof(κ) ∩ ε with d n−1 P n−1 (δ) projective as well. This is argued via the following diagram: Rows are telescopings of the projective resolutions of ∆ ε (Z) and ∆ δ (Z), respectively; they are, in other words, the natural decompositions of the differentials d n : P n → P n−1 into P n ։ d n P n followed by d n P n ֒→ P n−1 . Natural projections p : P n (ε) → P n (δ) and inclusions i : P n (δ) → P n (ε) connect pairs P n (ε) and P n (δ). Similarly, d n P n (δ) naturally includes into d n P n (ε); what is perhaps surprising is that this inclusion j may have no left-inverse. 18 This is the first key observation in the induction: if d n P n (ε) is projective and, hence, admits some section s of the map d n , then at closure points δ of s d n , a left-inverse to j does exist -namely, q = d n p s.
The second key observation is that q, together with the space in ε above δ, may be used to define a retract r of the inclusion d n P n (δ) ֒→ P n−1 (δ). The existence of such an r will imply that P n−1 (δ) ∼ = d n P n (δ) ⊕ d n−1 P n−1 (δ), by the exactness of the sequence Hence d n−1 P n−1 (δ) is a summand of the free system P n−1 (δ). By Lemma 2.8, d n−1 P n−1 (δ) is therefore projective. If we have shown that pd(∆ δ (Z)) ≥ n, then this is a contradiction, hence our assumption that d n P n (ε) is projective was false. In consequence, pd(∆ ε (Z)) ≥ n + 1.
In the preceding paragraph, we referenced "the space in ε above δ": fix ξ ∈ ε\δ. The key device in this second part of the argument -i.e., in the derivation of a retract r from q -is the formation of a cone over P n−1 (δ) in P n (ε). By this we mean the following: let Q n (δ, ξ) be the subsystem of P n (ε) generated by { α, ξ | α ∈ [δ] n }. As the reader may verify, there are natural inclusion-relations between d n Q n (δ, ξ) and several of the main terms in the diagram 60. These we denote t, u, and v in the diagram below; to see that t, for example, is meaningful, observe that 18 j : d 1 P 1 (5) → d 1 P 1 (ω), for example, does have a left-inverse, while j : d 1 P 1 (ω) → d 1 P 1 (ω 1 ) does not, as the reader is encouraged to verify.
What the cone construction critically affords us is a retract, w, of v. This is defined as follows: for β ∈ [ε] n , let For a generator d n α, ξ of d n Q n (δ, ξ), Hence w is a retract of v, as desired. The point is the following: Write a for d n P n (δ) ֒→ P n−1 (δ), as in diagram 61 above. Then given a q left-inverse to j, the map r = quwi is left-inverse to a: The equation records a diagram-chase on (61) above, together with the fact that wv = id. It shows that r is indeed a retract of a, and thereby concludes the induction step.
Appendix B. Details of the proof that f n is nontrivial Here we record in greater notational detail the argument outlined at the conclusion of Section 6.2 that the functions f n are nontrivial. We follow that outline closely. In particular, our argument is by induction on n; therefore assume the claim true for all f m with m < n, recalling that we handled the base cases of n = 1 and n = 2 in Sections 6.1 and Sections 6.2, respectively.
Consider now the function f n ∈ K n (R n+1 (ω n )) derived, in the manner described in Section 5.2, from equation 24. Equation 19 takes the form of the following coherence condition: Our aim is to show that there exists no e n−1 ∈ K n−1 (R n+1 (ω n )) trivializing f n in the manner of equation 20, i.e., satisfying To that end, we begin by deducing from equation 24 that and hence, by (62), that This is the "degree reduction" in coherence relations alluded to in the argument's outline in Section 5.2. Similarly, by definition, supp(e n−1 ( β)) ⊆ [ [β 0 , ω n ] ] n+2 for all β ∈ [ω n ] n for any e n−1 ∈ K n−1 (R n+1 (ω n )); this implies the relation for any e n−1 as in equation 63. Hence to show that f n is nontrivial in the sense of equation 63, it will suffice to show that there exists no e n−1 as in equation 66.
For this purpose the key point is the existence of nontrivial (n − 1)-coherent families of functions within the restriction, for each γ ∈ S n n−1 , of f n ( · , γ) to the hyperplane z = γ; it is here that we make use of our inductive hypothesis. Before proceeding, though, we will require some additional notation.
Fix γ ∈ S n n−1 and let π denote the order-isomorphism ω n−1 → C γ . It is then immediate from definitions that for any β ∈ [C γ ] n , the maximal proper internal tail of ( β, γ) is ( α, γ), where ξ is the maximal proper initial tail of π −1 [ β] and α = π[ ξ]. It follows that b( β, γ) = π[b(π −1 [ β])], γ for all such β. Here we've committed the mild abuse of applying π to a group element, but our meaning should be clear; more generally, for any a ∈ [ωn−1] n+1 Z, let π[a] denote the element of [γ] n+1 Z induced by the order-isomorphism π together with the inclusion of C γ into γ. Also, for any element a of [γ] n+1 Z, define the element a * γ of [ωn] n+1 ⊗{γ} Z by a * γ ( α, γ) = a( α). In particular, π[0] = 0 and 0 * γ = 0. Our central assertion may now be stated as follows: Put differently, the order-isomorphism π translates the nontrivial coherence of f n−1 to the restriction of the f n ( · , γ)-images of [C γ ] n to the hyperplane z = γ. To see (67), observe first that f n ( β, γ) = 0 if and only if f n−1 (π −1 ( β)) = 0, for any β ∈ [C γ ] n . If for some such β this is not the case then for some j ≤ n − 2, by equation 24. Note that the term f n (b( β, γ) n+1 ) of the latter has disappeared from the above sum, on the grounds that it contributes nothing to the restricted range [ω n ] n+1 ⊗{γ}. Observe that the branches of the expansion (in the sense of Lemma 5.2) of (68) are of the form where each k i ≤ n is other than the index of the coordinate added by the i th application of b. It is then easy to see that these branches are precisely those which may be written as π[B] * γ, where B is a branch of the expansion of f n−1 (π −1 [ β]). More precisely, the π and * γ operations induce a correspondence between the expansion-trees, and thus between the expansions, and, hence, between the values of f n−1 (π −1 [ β]) and f n ( β, γ) [ωn] n+1 ⊗{γ} , in exactly the sense recorded in equation 67.
This correspondence underlies the following lemma, from which the remainder of step n of our inductive argument rapidly follows.
Since we have shown that (67) holds at all γ ∈ S n n−1 , Lemma 69 implies that there exists a stationary S ⊆ S n n−1 and α ∈ [ω n ] n such that α γ = α for all γ ∈ S. However, (64) and (66) together imply that This is the desired contradiction, showing that no e n−1 ∈ K n−1 (R n+1 (ω n )) can trivialize f n in the sense of equation 66, and thereby concluding our argument.
Proof of Lemma B.1. It is our inductive assumption that there exists no e n−2 ∈ K n−2 (R n (ω n−1 )) trivializing f n−1 in the (lower degree) sense of (66). We will show that if the conclusion of Lemma B.1 failed, then the e n−1 under discussion would induce an e n−2 which contradicts this assumption. Our argument consists in two claims. It is convenient at the outset to assume that 0 ∈ C γ ; there is no loss of generality in doing so, as in any case f n (0, β, γ) will "expand into" f n [Cγ ] n ⊗{γ} , within which setting our arguments would, without this assumption, apply with only cosmetic changes. Proof. One approach is to observe that functions e n−1 satisfying the negation of (69) for all α γ ∈ [C γ \{0}] n define cocycles in cohomology groups corresponding to lim n−2 of a flasque inverse system, in the sense of Jensen's [Jen72a, page 5]; Theorem 1.8 therein then converts in our context to the existence of a trivializing g in precisely the sense of (71). A more computational argument would consist in verifying that letting for all α ∈ [C γ \{0}] n−2 with α 0 ∈ Lim and ξ < α 0 partially defines a function g which then canonically extends to satisfy the conclusion of the claim on the domain described. Proof. The e n−2 in question is the π −1 -translation of the function e n−1 (0, · , γ) + (−1) n−1 g(0, · ). This works as asserted by equation 67, together with the fact that for all α ∈ [C γ \{0}] n−1 . The first and second equalities follow from our claim's first and second premises, respectively, and the equality of the first and third lines should be read as a relativization to the z = γ hyperplane of the e n−2 variant of equation 66.
Appendix C. A proof of Theorem 7.10 We first recall the statement of the theorem: Theorem. There exists a height-ε A-valued nontrivial n-coherent I family of functions if and only if there exists a height-ε A-valued nontrivial n-coherent II family of functions.
As remarked above, we will in fact show that for every n > 0 and ordinal ε and abelian group A, coh I (n, A, ε) triv I (n, A, ε) ∼ = coh II (n, A, ε) triv II (n, A, ε) .
In this context we write [Φ n ] I and [S n ] II for the cosets associated to n-coherent I families Φ n and n-coherent II families S n , respectively, omitting the subscripts when they are clear from context.
Proof. We will define functions a n and b n from the left side to the right and from the right side to the left, respectively, of equation 72 such that a n b n and b n a n are each the identity map. A few preliminary observations will be useful: • For n > 0, any n-coherent I family of successor height ε is n-trivial I ; similarly for any ncoherent II family of successor height ε. Hence we may, and will, restrict our attention below to limit ordinals ε. • If X is a cofinal subset of ε then the class [Φ n ] I of an n-coherent I family Φ n is determined by Φ n X := {ϕ β | β ∈ [X] n }. Put differently, if Φ n X is n-trivial I , then so too is Φ n . Similarly for n-coherence II . Hence we may, and will, define b n -images as classes [Ψ n X ] I , where X ⊆ ε is the cofinal set ε ∩ Succ. • A family's initial segments are all n-trivial I or n-trivial II if and only if that family is ncoherent I or n-coherent II , respectively. We now define the maps a n and b n . It is both convenient and illustrative to first handle the cases of n = 1 and n = 2. Throughout this discussion, assume the abelian group A in question to be fixed.
It is tempting now to define the map b 2 as that induced by the reverse ofã 2 , but this would be misguided: above n = 1, this operation can fail to send 2-coherent II families to 2coherent I ones. The correct approach is rather to let b 2 ([S 2 ]) = [Ψ 2 = {ψ β | β ∈ [ε∩Succ] 2 }], where each ψ γδ is a trivializing function t 1 γδ as in (73) above. Cancellations of the s-terms imply that (t 1 δη − t 1 γη + t 1 γδ ) β = * 0 for all β < γ with γ < δ < η in ε ∩ Succ, implying in turn that Ψ 2 is 2-coherent I , as desired (note the importance of the fact that cf(γ) = ℵ 0 to this reasoning). Below, in the paragraph beginning with (⋆(δ)), we argue more generally that this map does not depend on the choices of t 1 γδ ; since in the case of S 2 =ã 2 (Φ 2 ) each t 1 γδ may be taken to be ϕ γδ , the composition b 2 a 2 is therefore indeed the identity, as claimed.
The simplest steps of the argument concern the functions a n . As above, a n is defined to be the function induced by a more concreteã n amounting to a rearrangement operation on the arguments and indices of any family of functions Φ n . By the following lemma, this operation converts ncoherence I and n-triviality I to n-coherence II and n-triviality II , respectively, ensuring, in particular, that a n is well-defined.
Cascading families of trivializing functions like the t k β appearing above form a main motif in analyses of n-coherence II and n-triviality II . They arise, for example, within a slightly different schema in the definition of b n appearing below. In light of such variety, we adopt a somewhat open definition: Definition C.2. A cascading family of trivializations is a collection of the form {t k γ : [γ 0 ] k → A | γ ∈ [X] n−k+1 and 1 ≤ k < n} for some set of ordinals X and n > 2, in which, for each k > 1, each t k−1 γ trivializes II an expression containing t k β for some β ⊆ γ. As above, such collections are often defined in the presence of a distinguished family of functions Φ n ; in such cases, definitions of t k γ via t k γ (α, β) = ϕ β γ (α) for all (α, β) ∈ [γ 0 ] k and γ ∈ [ε] n−k+1 will be called the natural assignments with respect to Φ n .
We now define the maps b n . As in the case of n = 2, the b n -image of any [S n ] will be a [Ψ n = {ψ β | β ∈ [ε ∩ Succ] n }] given by ψ β (α) = t 1 β (α), but for higher n the derivation of t 1 β is more elaborate. First observe that as S n is n-coherent II , for each (γ, δ) ∈ [ε ∩ Succ] 2 there exist t n−1 γδ such that equation 75 holds with γ = β n−1 and δ = β n . It follows that (t n−1 δη − t n−1 γη + t n−1 γδ ) [β] n−1 is (n − 2)-trivial II for all γ < δ < η in ε ∩ Succ and β < γ; as γ is a successor, t n−1 δη − t n−1 γη + t n−1 γδ itself admits an (n − 2)-trivialization II t n−2 γδη . Reasoning just as above, families t n−2 δηξ − t n−2 γηξ + t n−2 γδξ − t n−2 γδη will then admit (n − 3)-trivializations II t n−3 γδηξ , and so on: this sequence of t n−i ends with the t 1 β ( β ∈ [ε ∩ Succ] n ) we desire. It is easy to see that Ψ n so defined is n-coherent I . This follows from the fact that each for all α < β 0 ; as β 0 is a successor, the parameter α may be dropped. We then have n j=0 (−1) j t 1 β j = * n j=0 (−1) j n−1 i=0 (−1) i t 2 ( β j ) i = * 0 for all β ∈ [ε ∩ Succ] n+1 , as desired. Suppose now that S n is theã n -image of some n-coherent I family of functions Φ n . Then as the reader may verify, natural assignments for the cascading family of trivializations t k β defining b n would have been valid, ending with t 1 β = ϕ β . In consequence, once we have shown that the above procedure indeed determines a well-defined map b n : coh II (n, A, ε) triv II (n, A, ε) → coh I (n, A, ε) triv I (n, A, ε) it will follow almost immediately that b n a n is indeed the identity, as claimed. We therefore focus on showing the former.