Cohomology theory through infinity algebras

This research explores the foundational structures and theoretical framework of A_∞-algebras and L_∞-algebras within the wider setting of homological algebra and higher category theory. These "infinity" algebras extend classical associative and Lie algebras by encoding operations that satisfy generalized coherence relations up to homotopy. Their flexible and homotopically rich structure provides a unifying language for dealing with complex algebraic phenomena that are not accessible through traditional means.The study begins by reviewing some definitions and algebraic properties of A_∞- and L_∞-algebras, emphasizing their realization as differential graded structures governed by an infinite sequence of multilinear operations. This hierarchy of operations is structured via higher associativity or higher Jacobi-type identities, which hold up to coherent homotopies. Operads are introduced as essential organizing tools that capture and formalize these intricate patterns of relations.The homological dimensions of these algebras are developed through constructions such as the bar complexes, as well as the theory of Maurer–Cartan elements, which serve as central objects in encoding deformations. A special emphasis is placed on Hochschild and Chevalley–Eilenberg cohomology, which classify extensions and control deformation theory in these settings.Furthermore, the notion of homotopy equivalence between infinity morphisms is investigated to understand equivalence classes of algebraic structures. This provides a natural framework for studying moduli problems and organizing higher algebraic invariants in a coherent way.