nLab synthetic algebraic geometry

Idea

Synthetic algebraic geometry is concerned with synthetic formulations of algebraic geometry, hence with systems of axioms akin to those for synthetic differential geometry but whose natural/intended categorical semantics is specifically in topoi from algebraic geometry (cf. Hakim 1972), such as notably the Zariski topos.

Concretely, in the formulation of Cherubini, Coquand & Hutzler 2024, axioms are added to homotopy type theory which are intended to have HoTT-like categorical semantics in the $(\infty,1)$-topos version of the Zariski topos. (After a previous formulation, in Wellen/Cherubini 2017-2024, of synthetic differential Cartan geometry in a fragment of differentially cohesive homotopy type theory.)

Related concepts

• algebraic geometry

• synthetic mathematics

  • synthetic geometry

  • synthetic topology

  • synthetic differential geometry

  • synthetic differential topology

  • synthetic homotopy theory

  • synthetic $(\infty,1)$-category theory

  • synthetic probability theory

References

• Anders Kock, §I.12 in: Synthetic Differential Geometry, Cambridge University Press (1981, 2006) [pdf, doi:10.1017/CBO9780511550812]

• Ingo Blechschmidt: Using the internal language of toposes in algebraic geometry, PhD thesis (2017) [original:pdf, arXiv:2111.03685]

• Felix Cherubini, Thierry Coquand, Matthias Hutzler: A foundation for synthetic algebraic geometry, Mathematical Structures in Computer Science 34 Special Issue 9: Advances in Homotopy type theory (2024) 1008-1053 [doi:10.1017/S0960129524000239, arXiv:2307.00073]

• Felix Cherubini, A Foundation for Synthetic Algebraic Geometry, talk at Homotopy Type Theory Electronic Seminar Talks (Oct 2023) [slides:pdf, video:YT]

• Synthetic Algebraic Geometry in the Zariski-Topos (Github)

On projective spaces:

• Felix Cherubini, Thierry Coquand, Matthias Hutzler, David Wärn: Projective Space in Synthetic Algebraic Geometry, 2024, [arxiv:2405.13916]

On Châtelet’s theorem:

• Thierry Coquand, Hugo Moeneclaey, Chatelet’s Theorem in Synthetic Algebraic Geometry [arXiv:2504.08326]