Completions, Perforations and Fillings
Résumé
We introduce new local operations, perforations and fillings, which transform couples of objects (X, Y) such that X is a subset of Y. We complement these two operations by collapses and anti-collapses of both X and Y. This set of operations may be seen as a generalization of transformations based solely on collapses, thus a generalization of simple homotopy. Though these transformations do not, in general, preserve homotopy, we make a link between perforations, fillings, and simple homotopy. We give a full characterization of the collection of couples of objects that is closed under the above set of operations. Also, we provide a full characterization of the collection of couples of objects that is closed under the single operation of collapse.
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