On the homotopy types - Alexandre, Thiago (2022)
This dissertation is concerned with the foundations of homotopy theory following the ideas of the manuscripts Les Derivateurs and Pursuing Stacks of Grothendieck. In particular, we discuss how the formalism of derivators allows us to think about homotopy types intrinsically, or, even as a primitive concept for mathematics, for which sets are a particular case. We show how category theory is naturally extended to homotopical algebra, understood here as the formalism of derivators. Then, we proof in details a theorem of Heller and Cisinski, characterizing the category of homotopy types with a suitable universal property in the language of derivators, which extends the Yoneda universal property of the category of sets with respect to the cocomplete categories. From this result, we propose a synthetic re-denition of the category of homotopy types. This establishes a mathematical conceptual explanation for the the links between homotopy type theory, 1-categories and homotopical algebra, and also for the recent program of re-foundations of mathematics via homotopy type theory envisioned by Voevodsky. In this sense, the research on foundations of homotopy theory re ects in a discussion about the re-foundations of mathematics. We also expose the theory of Grothendieck-Maltsiniotis 1-groupoids and the famous Homotopy Hypothesis conjectured by Grothendieck, which arms the (homotopical) equivalence between spaces and 1-groupoids. This conjectured, if proved, provides a strictly algebraic picture of spaces.