Cartesian monoid

In algebraic geometry and commutative algebra, a ring homomorphism $f:A\to B$ is called formally smooth (from French: Formellement lisse) if it satisfies the following infinitesimal lifting property:

Suppose B is given the structure of an A-algebra via the map f. Given a commutative A-algebra, C, and a nilpotent ideal $N\subseteq C$, any A-algebra homomorphism $B\to C/N$ may be lifted to an A-algebra map $B\to C$. If moreover any such lifting is unique, then f is said to be formally etale.

Formally smooth maps were defined by Alexander Grothendieck in Éléments de géométrie algébrique IV. Among other things, Grothendieck proved that any such map is flat.

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