The Morita Equivalence for the $A_{\infty }$-Algebras

This paper explores the foundational and advanced aspects of $A_{\infty }$ -algebras. We present a comprehensive study of $A_{\infty }$ -algebras and their homology, emphasizing the construction of long exact sequences in simplicial homology and their implications for short exact sequences of $A_{\infty }$ -algebras. Furthermore, we investigate the trace and inclusion maps in matrix $A_{\infty }$ -algebras, proving their mutual invertibility and highlighting their role in preserving homological properties. These results underscore the utility of $A_{\infty }$ -algebras in simplifying complex algebraic systems while maintaining essential structural invariants.