Pull-backs and Fibrations in Approximate Pro-categories

In this paper we introduce the category Apro-ANR called the approximate pro-category of ANR’s, whose objects are all systems of ANR’s and whose morphisms are obtained as equivalence classes of system maps for some equivalence relation. We show that any 2-sink X f −→ Z g ←− Y in Apro-ANR admits a weak pull-back and it admits a pull-back if they are systems of compact ANR’s. Moreover, it admits a pull-back if they are objects of pro-ANRU. Here ANRU is the full subcategory of the category Unif of uniform spaces and uniform maps, whose objects are uniform absolute neighborhood retracts (ANRU’s) in the sense of Isbell. We define the approximate homotopy lifting property (AHLP) for morphisms in Apro-ANR and show that the category Apro-ANR with fibration = morphism with the AHLP with respect to paracompact spaces, and weak equivalence = morphism inducing an isomorphisms in pro-H(ANR) satisfies composition and factorization axioms and part of pull-back axiom for fibration category in the sense of Baues. Finally, we show that the limit of the pull-back of any 2-sink X f −→ Z g ←− Y in Apro-ANR consisting of systems of compact ANR’s is a pull-back in the category Top of topological spaces and continuous maps, and conversely every pull-back in the full subcategory CH of Top whose objects are compact Hausdorff spaces admits an expansion which is a pull-back in Apro-ANR.