Tame parametrised chain complexes

Persistent homology has proven to be a useful tool to extract information from data sets. Its method can be summarised by a standard workflow: start with data, build the chain complex of a simplicial complex modelling the data, apply homology obtaining the so-called persistent module, and retrieve topological information using invariants. Complete, and thus most discriminative, invariants are given by the indecomposables of the persistent modules. However, such invariants can be retrieved only for the objects of finite representation type whose decomposition is efficiently computed. In addition, homology might be an overkill, and some information may be lost while applying it to the chain complexes. The starting point of our investigation is the idea that a direct study of the chain complex can address these issues. Therefore, we investigate the category of tame parametrised chain complexes, which are chain complexes evolving according to one real parameter. Such a category is quite rich and includes many interesting types of objects, such as parametrised vector spaces, commutative ladders and zigzag modules. We define a model category structure on the category of tame parametrised chain complexes. This setting is quite natural since chain complexes admit a model category structure themselves. Moreover, we can exploit the rich theory of model category to extract invariants. In general, in a model category, there are special objects called cofibrant objects, that can be used to study any other object in the category by approximating it through them. After identifying the cofibrant objects in the category of tame parametrised chain complexes, we study their indecomposables. We find that, despite in general tame parametrised chain complexes are of wild representation type, the indecomposables of cofibrant objects can be fully described. We then approximate every tame parametrised chain complex using two cofibrant objects, called the minimal cover and the minimal representative. Such objects are crucial since they are invariants. In particular, the minimal cover is a homological invariant, and the minimal representative is a homotopical invariant. Thus, these two objects are retrieving all the topological information of the objects they are approximating. In conclusion, we prove that it is possible to analyse data using a new workflow: start with data, build the chain complex of a simplicial complex modelling the data, associate to it either a minimal cover or a minimal representative, and decompose the chosen one to retrieve a summary of the information in the data.