Properties of odd tangent bundle
1 Canonical odd vector field
1.1 General definition of 
-parameter families of diffeomorphisms
In the language of functors of points, a
-parameter group of diffeomorphisms of a supermanifold
is
defined as a map, for any supermanifold
:
consistent with morphisms between different
’s.
1.2 Case of odd tangent bundle
For any supermanifold
, the
has a canonically defined
-parameter group of
diffeomorphisms. It is defined as follows. For any
define a “shift”:
Then define the map:
2 Odd tangent bundle of a group
If 
is a supergroup, then 
is again a supergroup.
