nLab Topos Theoretic Methods in Geometry

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Context

Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id ⊣ id ∨ ∨ fermionic ⇉ ⊣ ⇝ bosonic ⊥ ⊥ bosonic ⇝ ⊣ Rh rheonomic ∨ ∨ reduced ℜ ⊣ ℑ infinitesimal ⊥ ⊥ infinitesimal ℑ ⊣ & étale ∨ ∨ cohesive ʃ ⊣ ♭ discrete ⊥ ⊥ discrete ♭ ⊣ ♯ continuous ∨ ∨ ∅ ⊣ * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

This page is to record the reference:

  • Anders Kock (ed.):


    Topos Theoretic Methods in Geometry


    Various Publications Series 30

    Aarhus University (1979)

    pdf, pdf

compiling early discussions of synthetic differential geometry.

Contents:

  • F. William Lawvere:

    Categorical dynamics

  • Marie-Françoise Coste, Michel Coste:

    The generic model of an ϵ\epsilon-stable geometric extension of the theory of rings is of line type

  • Marie-Françoise Coste, Michel Coste:

    Topologies for real algebraic geometry

  • E. J. Dubuc, G. E. Reyes:

    Subtoposes of the ring classifier

  • Anders Kock:

    Formally real local rings, and infinitesimal stability

  • Peter T. Johnstone:

    Extracts from two letters to Anders Kock, March 1978

  • Anders Kock:

    On the synthetic theory of vector fields

  • A. Kock, G. E. Reyes:

    Connections in formal differential geometry

  • Christiane Rousseau:

    Complex structures on topoi

  • G. C. Wraith:

    Complexification of variable fields

  • Report on the Open House on Topos Theoretic Methods in Geometry and Analysis, Aarhus, May 10–24, 1978.

category: reference

Last revised on December 22, 2025 at 15:06:59. See the history of this page for a list of all contributions to it.