nLab
movable pro-object

Definition

Let Λ\Lambda be a directed set, which we may also view as the associated order category, a directed category. View a pro-object X:Λ opCX:\Lambda^{op}\to C as a family of objects {X λ} λΛ\{X_\lambda\}_{\lambda\in\Lambda} in CC and bonding morphisms p λλ:X λX λp_{\lambda\lambda'}:X_{\lambda'}\to X_\lambda whenever λ>λ\lambda'\gt\lambda.

A pro-object XX is movable if for every λΛ\lambda\in\Lambda there is λ>λ\lambda'\gt\lambda (‘the movability index of λ\lambda’) such that for every λ>λ\lambda''\gt\lambda' there is a morphism r:X λX λr : X_{\lambda'}\to X_{\lambda''} such that p λλr=p λλp_{\lambda\lambda''} \circ r = p_{\lambda\lambda'}.

Relative version

Let CC be a category and C 0CC_0\subset C a full subcategory. (Λ,X,p)(\Lambda, X, p) in pro-CC is movable with respect to C 0C_0 (or C 0C_0-movable) if every λΛ\lambda\in\Lambda admits λ>λ\lambda'\gt\lambda such that for every λ>λ\lambda''\gt\lambda and every morphism h:YX λh:Y\to X_{\lambda'} in CC such that YC 0Y\in C_0 there is a morphism r:YX λr:Y\to X_{\lambda''} in pro-CC such that

p λλr=p λλh.p_{\lambda\lambda''}\circ r = p_{\lambda\lambda'}\circ h.

Movability of topological spaces

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.

Properties

Every retract in pro-CC of a movable pro-object is a movable pro-object.

Let \mathcal{F} be a full subcategory of the category Grp of groups whose objects are the free groups. Then a pro-group is \mathcal{F}-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.

Literature

  • Ch. II, Sec. 2 in S. Mardešić, J. Segal, Shape theory, North Holland 1982

Last revised on September 16, 2017 at 13:33:42. See the history of this page for a list of all contributions to it.