Related papers: Strong Homology, Derived Limits, and Set Theory
The derived functors $\lim^n$ of the inverse limit are widely studied for their topological applications, among which are some repercussions on the additivity of strong homology. Set theory has proven useful in dealing with these functors,…
In 1988, Sibe Marde\v{s}i\'{c} and Andrei Prasolov isolated an inverse system $\mathbf{A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail…
We show that the fixed point subnet of a strongly additive conformal net under the action of a compact group is strongly additive. Using the idea of the proof we define the notion of strong additivity for a pair of conformal nets and we…
Recent advances in our understanding of higher derived limits carry multiple implications in the fields of condensed and pyknotic mathematics, as well as for the study of strong homology. These implications are thematically diverse,…
We show that it is consistent relative to a weakly compact cardinal that strong homology is additive and compactly supported within the class of locally compact separable metric spaces. This complements work of Marde\v{s}i\'{c} and Prasolov…
In the study of strong homology Marde\v{s}i\'c and Prasolov isolated a certain inverse system of abelian groups $\mathbf A$ indexed by elements of $\omega^\omega$. They showed that if strong homology is additive on a class of spaces…
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…
We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We…
Let $f(n)$ be a strongly additive complex valued arithmetic function. Under mild conditions on $f$, we prove the following weighted strong law of large numbers: if $ X,X_1,X_2,... $ is any sequence of integrable i.i.d. random variables,…
We show that the vanishing of higher derived limits of the system $\mathbf{A}_\kappa$ implies the additivity of strong homology on the class of locally compact metric spaces of weight at most $\kappa$, thereby establishing a converse to a…
In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a…
We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo…
We study the relation between additivity and deduction theorems in the algebraic semantics of congruential modal logic. Additivity of the modal operator is well-known to imply the local deduction-detachment theorem. Our main theme is that…
We prove a general version of the homological perturbation lemma which works in the presence of curvature, and without the restriction to strong deformation retracts, building on work of Markl. A key observation is that the notion of strong…
We endow the set of persistence diagrams with the strong topology (the topology of countable direct limit of increasing sequence of bounded subsets considered in the bottleneck distance). The topology of the obtained space is described.…
We give sufficient conditions which ensure that a functor of finite length from an additive category to finite-dimensional vector spaces has a projective resolution whose terms are finitely generated. For polynomial functors, we study also…
The current work is motivated by the papers $[B_3]$, $[B_6]$, $[Be]$, $[Be-Tu]$. In particular, using Theorem 3.7 of $[B_3]$ and methods developed in this paper, the spectral and strong homology groups of continuous maps were defined and…
We present a new aspect of the study of higher derived limits. More precisely, we introduce a complexity measure for the elements of higher derived limits over the directed set $\Omega$ of functions from $\mathbb{N}$ to $\mathbb{N}$ and…
A folk theorem says higher order arithmetic has the proof theoretic strength of set theory with limited power set. This paper makes the theorem precise in terms of several axiom system based on ZF.