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This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors ˇH𝑛 on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor ˇH𝑛 def taking values in the category GPC of groups with a Polish cover (a category first introduced in this work's predecessor), followed by (ii) a forgetful functor from GPC to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes of d-spheres or d-tori for any 𝑑 ≥ 1, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functors ˇH𝑛 def to show that a seminal problem in the development of algebraic topology - namely, Borsuk and Eilenberg's 1936 problem of classifying, up to homotopy, the maps from a solenoid complement 𝑆3\Σ to the 2-sphere - is essentially hyperfinite but not smooth. Fundamental to our analysis is the fact that the Čech cohomology functors 𝑋 ↦→ ˇH𝑛(𝑋;𝐺) admit two main formulations: a more combinatorial one and a more homotopical formulation as the group [𝑋, 𝑃] of homotopy classes of maps from X to a polyhedral 𝐾(𝐺, 𝑛) space P. We describe the Borel-definable content of each of these formulations and prove a definable version of Huber's theorem reconciling the two. In the course of this work, we record definable versions of Urysohn's Lemma and the simplicial approximation and homotopy extension theorems, along with a definable Milnor-type short exact sequence decomposition of the space Map(𝑋, 𝑃) in terms of its subset of phantom maps; relatedly, we provide a topological characterization of this set for any locally compact Polish space X and polyhedron P. In aggregate, this work may be more broadly construed as laying foundations for the descriptive set theoretic study of the homotopy relation on such spaces Map(𝑋, 𝑃), a relation which, together with the more combinatorial incarnation of ˇH𝑛, embodies a substantial variety of classification problems arising throughout mathematics. We show, in particular, that if P is a polyhedral H-group, then this relation is both Borel and idealistic. In consequence, [𝑋, 𝑃] falls in the category of definable groups, an extension of the category GPC introduced herein for its regularity properties, which facilitate several of the aforementioned computations. [ABSTRACT FROM AUTHOR]

A question dating to Mardešić and Prasolov's 1988 work [S. Mardešić and A. V. Prasolov, Strong homology is not additive, Trans. Amer. Math. Soc. 307(2) (1988) 725–744], and motivating a considerable amount of set theoretic work in the years since, is that of whether it is consistent with the ZFC axioms for the higher derived limits lim n   (n > 0) of a certain inverse system A indexed by ω ω to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all n -coherent families of functions indexed by ω ω to be trivial. In this paper, we prove that, in any forcing extension given by adjoining ℶ ω -many Cohen reals, lim n A vanishes for all n > 0. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher-dimensional Δ -system lemmas. This work removes all large cardinal hypotheses from the main result of [J. Bergfalk and C. Lambie-Hanson, Simultaneously vanishing higher derived limits, Forum Math. Pi 9 (2021) e4] and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of lim n A for all n > 0. [ABSTRACT FROM AUTHOR]

In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system A with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that lim=A (the nth derived limit of A) vanishes for every = > 0. Since that time, the question of whether it is consistent with the ZFC axioms that lim=A = 0 for every = > 0 has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the ZFC axioms that lim=A = 0 for all = > 0. We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to lim=A = 0 will hold for each = > 0. This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions N2 ↦ Z which are indexed in turn by n-tuples of functions 5 : N ↦ N. The triviality and coherence in question here generalise the classical and well-studied case of = = 1. [ABSTRACT FROM AUTHOR]

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable P -points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which a = u = i = ℵ 1 < 2 ℵ 0 = ℵ 2 , each of a , u , i has a Π 1 1 witness and there is a Δ 3 1 well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a Δ 3 1 well-order of the reals is consistent with c = ℵ 2 and each of the following: a = u < i , a = i < u , a < u = i , where the smaller cardinal characteristics have co-analytic witnesses. Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type. [ABSTRACT FROM AUTHOR]

The point of departure for the present work is Barry Mitchell's 1972 theorem that the cohomological dimension of ℵ 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 ω 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 ZFC framework; at their heart is a simple, finitely iterable technique of compounding C -sequences. [ABSTRACT FROM AUTHOR]