Is a typedef or using declaration necessary in this case?
Generally speaking, the only time that I've been known to use either is when dealing with template traits and policy classes (**typically, to make my own types inter-op with the standard library containers and algorithms).
If you're trying to come up with a naming convention for std::map<Type1, Type2>, maybe try something like:
using MapT1T2 std::map<Type1, Type2>;
In this case, it would be: using MapPosObj = ....;
or maybe you want: Map_PosObj;
Whatever you do, be consistent. I've noticed that your class types (Position, Object) start with a capitol, so this should start with a capitol letter as well. It's fairly common for people to precede a name with a short description of the type like: strHello (string), fHello (float), vStuff (vector).
I like to use capitols for my types, and lower-case for variables. I use m_ prefix for members, p_ prefix for parameters, g_ prefix for globals and no prefix for locals.