Categorical characterisations of quasi-isometric embeddings

2024/11/13 by Tang, Robert · 2 citations
#20F65 #51F30 #Category Theory (math.CT) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2411.08501

Abstract

We characterise the (closeness classes of) quasi-isometric embeddings as the regular monomorphisms in the coarsely Lipschitz category, formalising the notion that they are isomorphisms onto their image. Furthermore, we prove that the coarsely Lipschitz category is coregular, and hence admits an (Epi, RegMono)--orthogonal factorisation system. Consequently, quasi-isometric embeddings are equivalently characterised as the effective, strong, or extremal monomorphisms. Finally, we prove that the coarsely Lipschitz category is not coexact in the sense of Barr.

Cited by

• Epimorphism classes and relatively maximal metrics in large-scale geometry
• The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spaces

Related

• Quasi-isometric embedding between *-algebras
• Quasi-isometric embeddings inapproximable by Anosov representations
• Quasi-isometric embeddings of symmetric spaces and lattices: reducible case
• Categorical approach to rigidity of Roe-like algebras of coarse spaces
• Embedding mapping class groups into finite products of trees