In this paper, we first introduce the notion of a completion. Completions are inductive properties which may be expressed in a declarative way and which may be combined. In the sequel of the paper, we show that completions may be used for describing structures or transformations which appear in combinatorial topology. We present two completions in order to define, in an axiomatic way, a remarkable collection of acyclic complexes. We give some basic properties of this collection. Then, we present a theorem which shows the equivalence between this collection and the collection made of all complexes that are acyclic in the sense of homology theory.
