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A combinatorial approach to coarse geometry
Cencelj, Matija, 1958- ...
S pomočjo analogije s teorijo oblik vložimo grobo kategorijo metričnih prostorov v kategorijo direktnih zaporedij simplicialnih kompleksov s simplicialnimi veznimi preslikavami. Dve direktni ...zaporedji simplicialnih kompleksov sta ekvivalentni, če lahko eno spremenimo v drugo s faktorizacijami do sosednosti in neskončnimi podzaporedji. To vložitev lahko naredimo z Ripsovimi kompleksi ali anti-Čechovimi aproksimacijami prostorov. V tem modelu groba ▫$n$▫-povezanost zaporedja ▫$\mathcal{K} = \{K_1 \to K_2 \to \ldots\}$▫ pomeni, da za vsak ▫$k$▫ obstaja ▫$m > k$▫, da vezna preslikava iz ▫$K_k$▫ v ▫$K_m$▫ inducira trivialni homomorfizem vseh homotopskih grup do vključno ▫$n$▫. Asimptotična dimenzija kvečjemu ▫$n$▫ pomeni, da za vsak ▫$k$▫ obstaja ▫$m > k$▫, da se vezna preslikava iz ▫$K_k$▫ v ▫$K_m$▫ faktorizira (do sosednosti) skozi neki ▫$n$▫-dimenzionalni kompleks. Yujeva lastnost A je ekvivalentna pogoju, da za vsak ▫$k$▫ in vsak ▫$\epsilon > 0$▫ obstaja ▫$m > k$▫, da ima vezna preslikava iz ▫$\vert K_k \vert $▫ v ▫$\vert K_m \vert$▫ sosednostni približek ▫$g\colon \vert K_k \vert \to \vert K_m \vert$▫, ki preslika simplekse od ▫$\vert K_k \vert$▫ v množice s premerom kvečjemu ▫$\epsilon$▫.
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