Let be a directed set, which we may also view as the associated order category, a directed category. View a pro-object as a family of objects in and bonding morphisms whenever .
A pro-object is movable if for every there is (‘the movability index of ’) such that for every there is a morphism such that .
Let be a category and a full subcategory. in pro- is movable with respect to (or -movable) if every admits such that for every and every morphism in such that there is a morphism in pro- such that
Term movability comes from topology where it denotes a property studied by Borsuk and which generalizes the class of topological spaces having the shape of an absolute neighborhood retract. This can be restated using expansions by pro-objects. There are versions for topological spaces and for pointed topological spaces. As pointed space is a special case of a pair of topological spaces it is natural to define the movability of pairs by requiring that there exist an expansion of the pair by the pro-object in the category of pairs in the homotopy category of polyhedra which is movable as an abstract pro-object.
Every retract in pro- of a movable pro-object is a movable pro-object.
Let be a full subcategory of the category Grp of groups whose objects are the free groups. Then a pro-group is -movable iff the underlying pro-object in the category of pointed sets is movable. A pointed pro-set is movable iff it satisfies the Mittag-Leffler property. This may be used to show that every retract of a pro-object satisfying Mittag-Leffler property is also Mittag-Leffler.
Last revised on September 16, 2017 at 17:33:42. See the history of this page for a list of all contributions to it.