Completions and Simple Homotopy
Résumé
We propose an extension of simple homotopy by considering homotopic pairs. Intuitively, a homotopic pair is a couple of objects (X,Y) such that X is included in Y and (X,Y) may be transformed to a trivial couple by simple homotopic deformations that keep X inside Y. Thus, these objects are linked by a “relative homotopy relation”.
We formalize these notions by means of completions, which are inductive properties expressed in a declarative way. In a previous work, through the notion of a dyad, we showed that completions were able to handle couples of objects that are linked by a certain “relative homology relation”.
The main result of the paper is a theorem that makes clear the link between homotopic pairs and dyads. Thus, we prove that, in the unified framework of completions, it is possible to handle notions relative to both homotopy and homology.