freeness

Freeness is a mathematical notion describing the absence of imposed relations on a generating set or the existence of a universal property that determines maps from a given structure uniquely. It is a context-dependent idea that appears in algebra, topology, and category theory.

In group theory, a free group on a set S is a group F(S) together with an inclusion of S into F(S) such that every function from S to any group G extends uniquely to a group homomorphism from F(S) to G. Elements of F(S) can be represented as reduced words formed from elements of S and their formal inverses, with no relations other than those required by group axioms. The rank of a free group equals the cardinality of S.

In module theory, a free module over a ring R is a module isomorphic to a direct sum of copies of R. A basis of a free module is a set of elements that generate the module and are linearly independent, and the number of basis elements is the module’s rank. Freeness of modules generalizes the notion of a basis from vector spaces to rings that may not be fields.

In the context of group actions, an action of a group G on a set X is free if no nonidentity element of G fixes a point of X. Free actions have quotient structures with favorable properties; for example, in topology a free action by a group on a space can yield a covering space of the quotient.

In category theory, freeness describes free objects relative to a forgetful functor, such as the free group functor from sets to groups or the free module functor from sets to R-modules, characterized by a universal mapping property.