nLab
formal smooth manifold

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

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & 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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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)

Contents

Idea

A formal smooth manifold is a smooth manifold equipped possibly with infinitesimal extension.

In the differential cohesion of synthetic differential infinity-groupoids these are the spaces locally isomorphic to n×D\mathbb{R}^n \times D, where n\mathbb{R}^n is a Cartesian space and DD is an infinitesimally thickened point. Here n\mathbb{R}^n is the underlying reduced manifold.

References

  • Anders Kock, Formal manifolds and synthetic theory of jet bundles, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1980) Volume: 21, Issue: 3 (Numdam)

  • Anders Kock, section I.17 and I.19 of Synthetic Differential Geometry, (pdf)

Formal smooth manifolds of the simple product form X×DX \times D in the category of smooth loci for XX an ordinary smooth manifold and DD and infinitesimal space have been considered in section 4 of

  • Eduardo Dubuc, Sur les modeles de la geometrie differentielle synthetique Cahiers de Topologie et Géométrie Différentielle Catégoriques, 20 no. 3 (1979), p. 231-279 (numdam).

For more on this see Cahiers topos

Revised on May 13, 2015 12:02:30 by Urs Schreiber (195.113.30.252)