Seminar

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One of the main aims of the transference problem in homological algebra is to transfer the dg algebra, A∞-algebra or more generally A∞-category structures to cochain complexes under some conditions which provide some solutions to the problem. One of the solutions to the transference problem is provided by ”Homological Perturbation Lemma (HPL)”. In this talk, we state HPL in the language of non-unital A∞-categories and as an application, we show that any quasi-isomorphism between two non-unital A∞-categories has an inverse up to homotopy. However, this is not necessarily true in the dg category of cochain complexes. Hence, in this case, we should consider a zigzag of quasi-isomorphisms. HPL also suggests us to consider a minimal model of a non-unital A∞-category and formal dg algebras. We examine some of the relationship between the formality of a dg algebra and the higher order Massey products. Finally, we present some examples of formal dg algebras, and non-formal dg algebras like Borromean rings and more generally Brunnian links. For HPL and the result, a good reference that we follow is Seidel's book "Fukaya Categories and Picard-Lefschetz Theory".