Introduction to Derived Algebraic Geometry

Derived Algebraic Geometry arises from the interplay between classical algebraic geometry, homotopy theory, and higher category theory. Building on the work of Grothendieck, Verdier, and Illusie, the theory developed through Quillen’s homotopical algebra and reached its modern form in the contributions of Toën–Vezzosi and Lurie. Within this framework, model categories, $\infty$-categories, and Grothendieck topoi combine to provide a language capable of coherently encoding algebraic, geometric, and homotopical information. This thesis introduces the foundations of the subject, emphasizing the role of homotopical algebraic contexts, $\infty$-topoi, and the cotangent complex as central tools in the study of moduli problems and derived structures.