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We interpret mathematically the pair (master equation, solution of masterequation) up to equivalence, as the pair (a presentation of a free triangulardga T over a combination operad O, dga map of T into C, a dga over O) up tohomotopy equivalence of dgOa maps, see Definition 1. We sketch two generalapplications: I to the theory of the definition and homotopy theory of infinityversions of general algebraic structures including noncompact frobeniusalgebras and Lie bialgebras. Here the target C would be the total Hom complexbetween various tensor products of another chain complex B, C = HomB, Odescribes combinations of operations like composition and tensor productsufficient to describe the algebraic structure and one says that B has thealgebraic structure in question. II to geometric systems of moduli spaces up todeformation like the moduli of J holomorphic curves. Here C is some geometricchain complex containing the fundamental classes of the moduli spaces of thegeometric problem. We also discuss analogues of homotopy groups and Postnikovsystems for maps and impediments to using them related to linear terms in themaster equation called anomalies.

Homotopy theory of the master equation package applied to algebra and geometry: a sketch of two interlocking programs