Decomposing multipersistence modules using functor calculus

Abstract: Multiparameter persistent homology has attracted growing interest in the topological data analysis community, in part due to its ability to handle noisy data. However, unlike the single-parameter case, multipersistence modules do not generally admit an interval decomposition, which makes the multiparameter setting considerably more complicated. Nevertheless, there exist certain sufficient conditions that guarantee interval decomposability, such as a locally defined condition called middle exactness. In this talk, I introduce poset cocalculus, which is a variant of functor (co)calculus that is defined for functors from a poset to a model category. The motivation for this framework lies in the relevance of functors from posets to the model category of chain complexes over a field, as any multipersistence module is the homology of such a functor. Poset cocalculus provides tools for relating local conditions on these functors to their global structure. I apply this to give a novel, more synthetic proof of the fact that middle exactness implies interval decomposability.

algebraic topologycategory theory

Audience: researchers in the topic

Bilkent Topology Seminar

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Recordings of talks available at www.youtube.com/channel/UCLrmyGpqxyeVpTcA1b5HcMw/videos