For the convenience of the reader we summarize in the first section of this appendix some basic notions and elementary results in the homotopy theory of differential modules. We are actually interested in situations where some additional structure is given, e.g., ℤ- or ℤ2-graded differential modules over a (possibly) graded ring, but the fundamental ideas can be most easily understood in the general framework of differential modules, and the reader, who has become familiar with this situation, should have no trouble with the minor modifications, which are necessary in the different more special cases. A general reference dealing with chain complexes over arbitrary rings (i.e. ℤ-graded differential modules, such that the differential has degree −1) is [Dold, 1960], which contains much more than we are going to discuss here. An exposition for chain complexes over the integers ℤ is given in [Dold, 1980]. See also, e.g., [Brown, K.S. 1982], Chapter I which contains a summary of relevant material from homological algebra, discussing in particular complexes over arbitrary rings.

The general principles in developing the homotopy theory of differential graded algebras (see [Quillen, 1967], [Sullivan, 1977], [Halperin, 1977(1983)], [Bousfield, Gugenheim, 1976]) are very similar to those in the case of differential (graded) modules, but the technical apparatus is much more involved.