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.

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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 $\mathbb{R}^n \times D$, where $\mathbb{R}^n$ is a Cartesian space and $D$ is an infinitesimally thickened point. Here $\mathbb{R}^n$ is the underlying reduced manifold.

Related concepts

• jet bundle

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 $X \times D$ in the category of smooth loci for $X$ an ordinary smooth manifold and $D$ 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