Homological Algebra for Persistence Modules

Abstract

We useČech closure spaces, also known as pretopological spaces, to develop a uniform framework that encompasses the discrete homology of metric spaces, the singular homology of topological spaces, and the homology of (directed) clique complexes, along with their respective homotopy theories. We obtain six homology and homotopy theories of closure spaces. 1 Closure spacesIn this section we provide background on EduardČech's closure spaces [9,25].2.1. Elementary definitions. We start with the elementary definitions we need to work with closure spaces. Definition 2.1. Let X be a set and let P(X) denote the collection of subsets of X. A function c : P(X) → P(X) is called a closure operation (or just closure) for X if the following axioms are satisfied:for all A, B ⊆ X. An ordered pair (X, c) where X is a set and c a closure for X is called a (Čech) closure space. Elements of X are called points.Lemma 2.2. Let (X, c) be a closure space. If A ⊆ B are subsets of X then c(A) ⊆ c(B). Proof. Note thatExample 2.3. Let X be a set. Then the identity map 1 P(X) : P(X) → P(X) is a closure operation for X. It is called the discrete closure for X. The closure operation defined by the map A → X for A = ∅ and ∅ → ∅ is called the indiscrete closure for X.Definition 2.5. Given a set X and a closure c for X we can associate to it an interior operation for X, int c : P(X) → P(X) given by int c (A) := X − c(X − A).The interior operation satisfies the following:1) int c (X) = X, 2) For all A ⊆ X, int c (A) ⊆ A, 3) For all A, B ⊆ X, int c (A ∩ B) = int c (A) ∩ int c (B). If int : P(X) → P(X) is a function satisfying the 3 conditions in Definition 2.5 and one defines c int : P(X) → P(X) by c int (A) := X − int(X − A), then c int is a closure operation for X.Proposition 2.6. [9, 14.A.12] A subset A of a closure space (X, c) is open if and only if int c (A) = A.Definition 2.7. [9, Definition 14.B.1] Let (X, c) be a closure space. Let A ⊆ X. We say B ⊆ X is a neighborhood of A if A ⊆ X − c(X − B) = int c (B). In the case that A = {x}, we say B is a neighborhood of x. The neighborhood system of A is the collection of all neighborhoods of A. Definition 2.8. [9, Definition 16.A.1] Let (X, c X ) and (Y, c Y ) be closure spaces. A map). If f is continuous at every point of X we say f is continuous. Equivalently, f is continuous if for every A ⊆ X, f (c X (A)) ⊆ c Y (f (A)). A continuous map f is called a homeomorphism if f is a bijection with a continuous inverse.

Zig-Zag Modules: Cosheaves and K-Theory

“…It would be interesting to apply Quillen's Q-construction to these categories of persistence modules and compare the resulting K-theories to our computations below. (We note that [7] contains much more than we just outlined, e.g., the authors prove an embedding theorem in the vain of the Gabriel-Popescu Theorem. )…”

“…Through the work of Bubenik and collaborators, K-theoretic considerations have started to appear in the TDA literature. In [7], Bubenick and Milićević show that the category of persistence modules over any preorder is Abelian. The key ideawhich we use below as well-is that functor categories inherit many of the properties of the target category, so if the target is Abelian or Grothendieck, i.e., AB5 with a generator, then the functor category with domain a preorder (or any small category) is Abelian or Grothendieck.…”

Multiparameter persistent homology via generalized Morse theory

“…Thus we have an inclusion of categories ∆ 2 × R → R 3 where R 3 denotes the category corresponding to the poset (R 3 , ≤). For a poset P and p ∈ P , let U p = {q ∈ P | p ≤ q}, called the up-set of p. Then our persistence modules may also be considered to be R 3 -graded modules over the monoid ring K[U 0 ], where U 0 is the up-set of 0 ∈ R 3 (see [3,23]).…”

“…However, this module does not decompose into one-dimensional summands, and there is no complete invariant analogous to the persistence diagram [5]. Much of the recent work on multiparameter persistent homology focuses on either its algebraic structure, for example [12,3,23], or its computational challenges, for example [21,7,19,26]. For papers related to this one that take a more geometric approach, see [22,10].…”

Persistent sheaf Laplacians

“…In the last decade, R. Ghrist and coworkers and many other researchers have revived and expanded the cellular sheaf theory with applications in science and engineering [8,12,16,29]. Tools from sheaf theory have also been developed to study persistence modules [1,2,8,15]. Roughly speaking, a cellular sheaf consists of a simplicial complex, an assignment of vector spaces for each cell, and an assignment of linear maps for each face relation, satisfying certain rules so that it gives rise to a sheaf cochain complex.…”