Realization of Chain Maps

Abstract

In this chapter we consider fundamental properties of the chain functor C* which carries a T-complex X _≥1 to a chain complex A = C*X _≥1. Then X_≥1 is termed a realization of the chain complex A. We introduce partial realizations of a chain complex which are termed “twisted homotopy systems”; this generalizes the notion of a twisted chain complex in chapter II. Using twisted homotopy systems we study partial realizations of chain maps. This leads to an obstruction theory both for the realization of a chain complex and for the realization of chain maps. To discuss these properties we introduce some useful language on “linear extensions of categories”, “exact sequences for functors” and “towers of categories”; see § 5. The homological tower of categories in § 6 is a first main result which is needed to prove the homological Whitehead theorem in § 7 and the “model lifting property” of the twisted chain functor in § 8. The model lifting property is a key point in the proof of the Hurewicz theorem in § 10 and in the proof of the finiteness obstruction theorem in the next chapter VII.