A generalization of Artin-Mazur completion and rigid completion functors in homotopy theory

AbstractA generalization of the definition of the pro-category Pro-C for a category C is introduced, by taking Grothendieck's ‘double limit’ formula as the definition of morphisms in Pro-C even where the index categories are no longer cofiltered. Using this, a generalized Artin-Mazur completion is defined starting with an arbitrary functor K: C →D. It is shown that the shape category of K can be identified with a full subcategory of Pro-C. Finally, a ‘rigid’ Artin-Mazur completion is obtained by taking D (resp. C) to be the category of pointed CW complexes (resp. pointed CW complexes which have finite homotopy groups) and pointed continuous maps