Homotopical Decompositions of Simplicial and Vietoris Rips Complexes

Explore homotopical decompositions of simplicial and Vietoris Rips complexes in this 31-minute lecture from the Applied Algebraic Topology Network. Delve into the study of decompositions of simplicial complexes induced by coverings of their vertices, with a focus on their applications in Topological Data Analysis (TDA). Learn about obstruction complexes and their role in measuring how decompositions approximate the original complex, including their use in providing conditions for Mayer-Vietoris sequences to calculate homology. Discover the advantages of working with small categories rather than directly with simplicial complexes. Examine the main result applicable to arbitrary simplicial complexes, and investigate its specialization to Vietoris-Rips complexes, emphasizing the importance of the triangular inequality.