Consider the local behavior of a map by choosing a point and an open neighborhood such that . Now consider the set of all mappings It is possible to put these mappings into categories by introducing an equivalence relation. Given two mappings and , write provided there exists a neighborhood of such that and and the restriction coincides with These equivalence classes are called map germs and members are called representatives of the germ. It follows from this that , hence it is common to write for the germ where .
Consider a map germ , which is an equivalence class of maps agreeing in a small neighborhood of the origin. The group of germs of diffeomorphisms is denoted , whereas the is denoted . These give coordinate changes in the source and target respectively.
Denote the space of all analytic map germs by The group acts on in a standard way. Let , , and . Then .
The orbit of under this action is
If and , then for some and . This is the same as and means the above diagram commutes. In this case, and are said to be -equivalent as map germs.