Elementary Lifting Operations for Realm-Like Structures

Florian's thoughts about lifting operations:

Many data structures admit a lifting operation that allows to map a function over the values "stored" in that data structure. Examples:

1. Lists
2. Tuples
3. Functions (by applying another function to the values of the first function)

-> Any type in Haskell in the Functor class (via fmap :: (a -> b) -> f a -> f b) should admit such a data structure

Some non-Functor-like examples:

1. a\mapsto\{a\}
2. \{a\}\mapsto a
3. list \mapsto bag \mapsto set
4. partial function f\mapsto total function over the domain of definition of f
5. total function \mapsto partial functon over a larger domain
6. algebraic structure A\mapsto carrier set of A
7. list/vector/... \mapsto its i-th entry
8. function defined on a generating set (e.g. basis of a vector space) \mapsto function on the whole structure (e.g. whole vector space)
9. function defined on the constants of a theory \mapsto theory morphism
10. a\mapsto[a] (modding out an equivalence relation on a set)