Pull-backs and fibrations in approximate pro-categories

Takahisa Miyata
2008 Glasnik Matematicki - Serija III  
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 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
more » ... 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. 2000 Mathematics Subject Classification. 54C56, 54C55, 55U35.
doi:10.3336/gm.43.2.15 fatcat:oddcr6x63baprerd4btcalsjly