FROM DG-CATEGORIES OVER K TO K-LINEAR STABLE INFINITY-CATEGORIES

The purpose of this thesis is to prove that “idempotent- complete pretriangulated dg-categories over k are stable k-linear idempotent com- plete ∞-categories”. This result is a meeting point between algebraic topology and algebraic geometry. In 2013 L. Cohn proved it in a preprint, and in 2016, he updated the preprint to its latest version. Since then the ∞-category theory and the spectral algebraic geometry theory have made progress: therefore, we think that a reproof can be useful in make this result clearer for more mathematicians. The first four chapters are devoted to laying the groundwork for the next chapter where linear category theory can finally be presented. In the fifth chapter, we tackle for the first time the problem of defining the theory of k-linearization using enriched ∞-category theory (Lurie has previously addressed this topic but with a different setting [1, Spectral Algebraic Geoemetry]). This chapter comprises three parts: the first two run almost in parallel, while the third compares them. In the first part, we delve into the theory of dg-categories over a fixed ring k, which serves as the algebraic- geometric setting for k-linearization. One significant result is the demonstration that M odHk-enriched categories are left Hk-module objects in CatSp ∞ . This result elucidates why the two parallel sections converge, prompting an additional section to present it in a categorical setting. This finding enables k-linear Morita theory to be explored using nonlinear theory. In the second part, we investigate linearization from the viewpoint of spectral algebraic geometry (or algebraic topology). We provide a conceptual overview of linearization and define Morita theory for k-linear ∞-categories. In this part, we will come to realize that there exists an incredible number of equivalent definitions of k-linear ∞-categories, and the main outcome of this section is to demonstrate their equality. In the final part of the fifth chapter, we achieve our goal and prove the main results. The thesis concludes with a brief chapter outlining possible directions for future research following this work.