The Archimedean place is a blurred interval at infinity

Abstract

Classically, the places of ℚ are often regarded as a one-point compactification of Spec(ℤ), with the real place corresponding to a formal ``prime'' added at infinity. Re-examining this picture from a topos-theoretic perspective reveals a subtler geometry: while the non-Archimedean places are identified with singletons indexed by the non-zero prime ideals of ℤ, the Archimedean place is represented by the space of upper reals overleftarrow[0,1], which may be informally thought of as the unit interval equipped with a non-Hausdorff topology. On a technical level, our analysis brings together geometric logic and descent techniques from topos theory, distinguishing standard descent from lax descent toposes both at the level of sheaves and of the geometric theories they classify. More broadly, this paper brings into conversation two parallel distinctions: on the number-theoretic side, between Archimedean and non-Archimedean phenomena, and on the topos-theoretic side, between standard and lax descent. Looked at from a high level, these perspectives begin to converge on a common theme: how should the connected and the disconnected interact?

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BibTeX RIS

BibTeX

@misc{ng2026archimedean,
  title = {{The Archimedean place is a blurred interval at infinity}},
  author = {Ming Ng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09117},
  eprint = {2609.09117},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09117},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - The Archimedean place is a blurred interval at infinity
AU  - Ming Ng
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09117
UR  - https://arxiv.org/abs/2609.09117
AB  - Classically, the places of ℚ are often regarded as a one-point compactification of Spec(ℤ), with the real place corresponding to a formal ``prime'' added at infinity. Re-examining this picture from a topos-theoretic perspective reveals a subtler geometry: while the non-Archimedean places are identified with singletons indexed by the non-zero prime ideals of ℤ, the Archimedean place is represented by the space of upper reals overleftarrow[0,1], which may be informally thought of as the unit interval equipped with a non-Hausdorff topology. On a technical level, our analysis brings together geometric logic and descent techniques from topos theory, distinguishing standard descent from lax descent toposes both at the level of sheaves and of the geometric theories they classify. More broadly, this paper brings into conversation two parallel distinctions: on the number-theoretic side, between Archimedean and non-Archimedean phenomena, and on the topos-theoretic side, between standard and lax descent. Looked at from a high level, these perspectives begin to converge on a common theme: how should the connected and the disconnected interact?
ER  -

Abstract

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