nLab
formal smooth manifold

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

References

Section I.17 and I.19 of

• Anders Kock, 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