| Title:
|
Homotopy Theory of Ultrametric Spaces (English) |
| Author:
|
Mihara, Tomoki |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
5 |
| Issue:
|
1 |
| Year:
|
2021 |
| Pages:
|
384-400 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Edged Spaces |
| Keyword:
|
Vietoris-Rips Complexes |
| Keyword:
|
Model Categories |
| MSC:
|
18N40 |
| MSC:
|
51F99 |
| idZBL:
|
Zbl 1483.18026 |
| idMR:
|
MR4367225 |
| DOI:
|
10.21136/HS.2021.10 |
| . |
| Date available:
|
2026-03-13T05:39:02Z |
| Last updated:
|
2026-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153442 |
| . |
| Reference:
|
[1] Durov, N. V.: Homotopy Theory of Normed Sets I. Basic Constructions.St. Petersburg Mathematical Journal, Volume 29, Number 6, pp. 887–934 MR 3723811 |
| Reference:
|
[2] Durov, N. V.: Homotopy Theory of Normed Sets II. Model Categories.St. Petersburg Mathematical Journal, Volume 30, Number 1, pp. 25–71 MR 3790744 |
| Reference:
|
[3] Goerss, P. G., Schemmerhorn, K.: Model Categories and Simplicial Methods.Interactions between Homotopy Theory and Algebra, Contemporary Mathematics, Volume 436, pp. 3–49 MR 2355769 |
| Reference:
|
[4] Hausmann, J. C.: On the Vietoris-Rips Complexes and a Cohomology Theory for Metric Spaces.Prospects in Topology, Proceedings of a Conference in Honor of William Browder, Annals of Mathematics Studies, Volume 138, pp. 175–188 |
| Reference:
|
[5] Hovey, M.: Model Categories.Mathematical Surveys and Monographs, Volume 63, Americal Mathematical Society |
| Reference:
|
[6] Lawvere, F. W.: Metric Spaces, Generalized Logic and Closed Categories.Rendiconti del Seminario Matématico e Fisico di Milano, Volume 43, Issue 1, pp. 135–166 |
| Reference:
|
[7] Lurie, J.: Higher Topos Theory.Annals of Mathematics Studies, Volume 170, Princeton University Press MR 2522659 |
| . |
