We will introduce the notion of a differential graded algebra that is not strictly associative, but associative up to chain homotopy. Such objects are called A-infinity algebras or strong homotopy associative algebras. The goal is to state and prove the Homotopy Transfer Theorem, which states that any chain complex that is chain homotopic to an A-infinity algebra has an A-infinity structure and the maps in the chain homotopy can be lifted to a map of A-infinity algebras.

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