nLab formal smooth manifold


This entry is about smooth manifolds with infinitesimal thickenings. For “formal spaces” in the sense that their de Rham complex is a formal dg-algebra, see there.


Synthetic differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

singular 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& \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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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.


  • 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

Last revised on November 4, 2018 at 02:15:13. See the history of this page for a list of all contributions to it.