Free object

A free object is an object generated by a set of elements subject only to the axioms of its category. Every function from those generators into the underlying set of another object extends uniquely to a structure-preserving map.

Universal property: A property specifying an object through the unique maps it admits to or from other objects. It characterizes a free object by the unique extension of every map from its generators.
Set: A collection of distinct elements considered without any additional structure. A set provides the generators for the free construction over sets.
Free group: A group generated by a set with no relations beyond the group axioms. It is the standard free object for the forgetful functor from groups to sets.
Presented object: An algebraic object specified by generators together with relations among them. A presentation adds relations that a free object does not impose beyond category axioms.
Monadicity theorem: A theorem characterizing categories that arise as algebras for a monad on another category. It gives conditions under which free objects and their forgetful functor recover a category.
Adjunction: A relationship between functors expressed by natural correspondences between sets of morphisms. The free-object construction is left adjoint to the forgetful functor.
Algebraic signature: A collection of operation symbols, each assigned an arity, used to specify an algebraic structure. The signature determines which formal operations can be applied to generators.
Free monoid: The monoid of finite words over a set, with concatenation and the empty word. It realizes the free construction for monoids using words as elements.
Quotient algebra: An algebra formed by identifying elements according to a congruence relation. Quotients of free algebras impose equations among otherwise independent terms.
Monads in category theory: An endofunctor equipped with unit and multiplication satisfying associativity and identity laws. The monad induced by free and forgetful functors records the resulting algebraic structure.

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Initial and terminal objectsRelated: In many algebraic categories, the initial object is the free object on no generators.