Tame functors from posets to chain complexes
speaker: Claudia Landi (Università di Modena e Reggio Emilia)abstract: Tame functors indexed by posets with values in the category of non-negative chain complexes are a case of generalized persistence modules convenient for encoding homotopical and homological properties of objects in topological data analysis. We will start discussing some circumstances in which the category of such functors admits both an Abelian and a model structure, highlighting the case of posets of dimension 1. Next, we will consider structure theorems in this category. We will see that the structure theorem for filtered (i.e. with internal maps that are monomorphisms) tame functors indexed by non-negative reals gives insights into the usual persistent homology barcoding algorithm. In the case of factored (i.e. with internal maps that are epimorphisms) tame functors, again indexed by non-negative reals, the structure theorem yields the construction of barcodes for Morse-Smale vector fields. Generally, when functors need not be filtered nor factored, the family of indecomposables is wild. However, if the indexing poset is of dimension 1, any functor admits a cofibrant replacement for which a structure theorem will be presented.
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