The Morita Equivalence for the <math xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msub> <mi>A</mi> <mi>∞</mi> </msub> </mrow></math>-Algebras

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