set, etc.) which may satisfy some axioms. For these objects, a commonly considered forgetful functor is as follows. Let be any category based on sets, e.g. groups—sets of elements—or topological spaces—sets of 'points'. As usual, write for the objects of and write for the morphisms of the same. Consider the rule: For all in the underlying set of For all in the morphism, , as a map of sets. The functor is then the forgetful functor from to Set, the category of sets. Forgetful functors are almost always faithful. Concrete categories have forgetful functors to the category of sets—indeed they may be defined as those categories that admit a faithful functor to that category. Forgetful functors that only forget axioms are always fully faithful, since every morphism that respects the structure between objects that satisfy the axioms automatically also respects the axioms. Forgetful functors that forget structures need not be full; some morphisms don't respect the structure. These functors are still faithful however because distinct morphisms that do respect the structure are still distinct when the structure is forgotten. Functors that forget the extra sets need not be faithful, since distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set. In the language of formal logic, a functor of the first kind removes axioms, a functor of the second kind removes predicates, and a functor of the third kind remove types. An example of the first kind is the forgetful functor Ab → Grp. One of the second kind is the forgetful functor Ab → Set. A functor of the third kind is the functor Mod → Ab, where Mod is the fibred category of all modules over arbitrary rings. To see this, just choose a ring homomorphism between the underlying rings that does not change the ring action. Under the forgetful functor, this morphism yields the identity. Note that an object in Mod is a tuple, which includes a ring and an abelian group, so which to forget is a matter of taste. Left adjoints of forgetful functors Forgetful functors tend to have left adjoints, which are 'free' constructions. For example: free module: the forgetful functor from (the category of -modules) to has left adjoint , with , the free -module with basis . free group free lattice tensor algebra free category, adjoint to the forgetful functor from categories to quivers universal enveloping