Categorías de modelos en espacios topológicos

Abstract

The monograph delves into the concept of weak equivalences, highlighting their fundamental role in homotopy theory. Within this theoretical framework, morphisms with closure properties similar to isomorphism are examined; however, they allow for a more subtle interaction. Homotopical categories are investigated, such as topological spaces and chain complexes, along with some homotopic functors like homotopy groups and homology groups. Furthermore, the importance of model categories is emphasized, especially the H and Q model categories, which unify concepts such as fibrations, cofibrations, and weak equivalences in a coherent context. Emphasis is placed on the work of Quillen and his contributions to bifibrant replacements, which bring order and coherence to homotopy theory. This monograph explores connections between model categories in topological spaces and chain complexes, enriching the understanding of algebraic topology and homotopy theory.