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Abstract

Using the algebraic theory of homotopies between maps of dga's we obtain a homotopy theory for algebraic structures defined by collections of multiplications and comultiplications. This is done by expressing these structures and resolved versions of them in terms of dga maps. This same homotopy theory of dga maps applies to extract invariants beyond homological periods from systems of moduli spaces that determine systems of chains that satisfy master equations like $dX + X*X = 0$. Minimal models of these objects resemble Postnikov decompositions in the homotopy theory of spaces and maps.