Formally smooth map

Summary

A map of commutative rings $R \to A$ is formally smooth when it has a lifting property. Every map from $A$ to a quotient $C/J$ by an ideal $J$ with $J^2 = 0$ must lift to a map to $C$. This is the algebraic form of an idea from geometry. On a smooth space, there is no obstruction to extending a point to a small thickening. Together with finite presentation it gives the notion of a smooth morphism of rings, and with uniqueness of the lift it gives formally étale maps.