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Preliminaries

  • Jun Zhang
Chapter
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Part of the CRM Short Courses book series (CRMSC)

Abstract

In this chapter, we give detailed descriptions of the concepts of derived category, persistence k-module, and singular support. These serve as preparations for the topics treated in later chapters. Sections 2.1 to 2.5 are devoted to derived category and derived functors, as well as their applications in the category of sheaves. These are basic ingredients for Tamarkin categories. Sections 2.6 and 2.7 are devoted to the theory of persistence k-module theory, which figures the interleaving distance and barcodes. Two main theorems highlight this theory: Normal Form Theorem and Isometry Theorem. Sections 2.8 and 2.9 are devoted to the definition of the singular support, its various functorial properties, and an important result, the microlocal Morse lemma, which generalizes the classical Morse lemma to a microlocal formulation. This lemma is essential in the constructions of Tamarkin categories.

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Jun Zhang
    • 1
  1. 1.Département de Mathématiques et StatistiqueUniversity of MontrealMontréalCanada

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