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.
