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.

1. The CUNY Graduate Center
   365 Fifth Avenue
   New York, NY 10016-4309, USA
   and
   Stony Brook University
   Stony Brook, NY, 11794-3651, USA