  We interpret mathematically the pair (master equation, solution of master
equation) up to equivalence, as the pair (a presentation of a free triangular
dga T over a combination operad O, dga map of T into C, a dga over O) up to
homotopy equivalence of dgOa maps, see Definition 1. We sketch two general
applications: I to the theory of the definition and homotopy theory of infinity
versions of general algebraic structures including noncompact frobenius
algebras and Lie bialgebras. Here the target C would be the total Hom complex
between various tensor products of another chain complex B, C = HomB, O
describes combinations of operations like composition and tensor product
sufficient to describe the algebraic structure and one says that B has the
algebraic structure in question. II to geometric systems of moduli spaces up to
deformation like the moduli of J holomorphic curves. Here C is some geometric
chain complex containing the fundamental classes of the moduli spaces of the
geometric problem. We also discuss analogues of homotopy groups and Postnikov
systems for maps and impediments to using them related to linear terms in the
master equation called anomalies.
