New Structures Based on Completions

International audienceWe propose new axioms relative to combinatorial topology. These axioms are settled in the framework of completions which are inductive properties expressed in a declarative way, and that may be combined. We introduce several completions for describing dyads. A dyad is a pair of complexes which are, in a certain sense, linked by a "relative topology". We first give some basic properties of dyads, then we introduce a second set of axioms for relative dendrites. This allows us to establish a theorem which provides a link between dyads and dendrites, a dendrite is an acyclic complex which may be also described by completions. Thanks to a previous result, this result makes clear the relation between dyads, relative dendrites, and complexes which are acyclic in the sense of homology