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  • 4 - The Functor Ext and the Homology Sequences

  • Félix Cabello Sánchez, Universidad de Extremadura, Spain, Jesús M. F. Castillo, Universidad de Extremadura, Spain
  • Book:
    Homological Methods in Banach Space Theory
    Published online:
    19 January 2023
    Print publication:
    26 January 2023, pp 197-242

  • Summary

    This chapter lights from a categorical perspective many of the results treated in previous chapters. Contrary to its notorious reputation, category theory helps in understanding concrete constructions, leads to the right questions and, oftentimes, suggests answers. Categories are used in an elementary way but without sacrificing rigour. The topics covered include the functor $\operatorname{Ext}$, the natural equivalence between $\operatorname{Ext}$ and the spaces of quasilinear maps studied in Chapter 3 (including the categorical meaning of `natural’) and the form in which all the pieces fit together in longer exact sequences and their uses, adjoint and derived functors, the topological structure of the spaces $\operatorname{Ext}(X,Y)$ and its connection with the geometry of the spaces.