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Keywords:
Edged Spaces; Vietoris-Rips Complexes; Model Categories
Edged Spaces; Vietoris-Rips Complexes; Model Categories
Summary:
We introduce the notion of an edged space as an extension of that of a metric space, and study two model structures on edged spaces transferred through Quillen adjunctions given by Vietoris–Rips complexes. We show that a metric space is a fibrant-cofibrant object with respect to one of the model structures if and only if it is an ultrametric space. The two model categories give a new foundation of homotopy theories of ultrametric spaces and edged spaces.
References:
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