Codimension of a singularity

Summary

The codimension of a singularity of a smooth function counts how many independent conditions a function must satisfy to have that singularity. Equivalently, it is the number of parameters a family of functions needs before the singularity occurs in it in a persistent way. Persistent means that small perturbations of the family do not remove it. It is computed from the Jacobian ideal of the germ. It is also the least number of parameters of a versal unfolding, not counting the constant term. In catastrophe theory it sorts the elementary catastrophes: the fold has codimension $1$ and the cusp has codimension $2$.