Invariants for tame parametrised chain complexes (Q2240611)

This paper proposes a novel and promising approach to topological data analysis, and more specifically persistence modules arising from a single parameter dataset, which takes greater advantage of homotopy theoretic techniques and concepts than existing theory. In doing so, the proposed approach combines under a unified framework types of persistence modules that have so far been handled separately -- specifically, commutative ladders and zigzag modules. The basic idea is to consider as a basic object the chain complex rather than its homology. Chain complexes arising from simplicial complexes and other types of topological constructions are much richer objects than their homology, which is the invariant typically used in topological data analysis. On the other hand, to compute persistent homology one must first extract a chain complex from the data. Thus, it is natural to ask whether it makes more sense to work with the complexes rather than their homology. Persistence module theory typically considers functors from some poset (or more generally a small category) into the category of vector spaces over a field. The general setup however allows the target category to be more general. In this paper the poset is the nonnegative real line and the target category is the category of chain complexes over a field. The paper concentrates on the subcategory of tame \([0,\infty)\) parametrised chain complexes. This category is shown to admit a model structure in which the cofibrant indecomposable objects are shown to satisfy a simple decomposition theorem. This is quite surprising in view of the fact that the entire category is of wild representation type. In practical terms, the paper thus proposes that following turning data into a parametrised chain complex, the next step is extraction of a minimal cofibrant approximation of the parametrised chain complex, and finally representing the minimal cofibrant approximation by what the paper calls a Betti diagram. The paper provides the theoretical background that would be required, in principle, to make this new approach computable and hence advantageous to potential users.