The Excision Theory for Homology Theory through <math xmlns='http://www.w3.org/1998/Math/MathML'> <mrow> <msub> <mi>L</mi> <mi>∞</mi> </msub> </mrow></math>-Algebra

This paper investigates the homological structures in L ∞ -algebras and their behavior under various conditions, with a focus on the excision theorem, simplicial homology, and bar homology. We introduce the key concepts of excision in the context of L ∞ -algebras and establish how these structures preserve homological relations under certain inclusions. The relationship between simplicial and bar homologies is explored, and we define the homological equivalences between these different structures. We provide results on the equivalence of homological relations and the necessary conditions for maintaining quasi-isomorphisms between the algebras involved. Furthermore, we analyze the conditions under which -unit and other homological properties hold, offering a comprehensive understanding of the interplay between algebraic and homological structures.