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Concept
Free object
Summary
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices.
The concept is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a formulation in terms of , although this is in yet more abstract terms.
Free objects are the direct generalization to of the notion of basis in a vector space. A linear function u : E1 → E2 between vector spaces is entirely determined by its values on a basis of the vector space E1. The following definition translates this to any category.
A is a category that is equipped with a faithful functor to Set, the . Let C be a concrete category with a faithful functor U : C → Set. Let X be a set (that is, an object in Set), which will be the basis of the free object to be defined. A free object on X is a pair consisting of an object in C and an injection (called the canonical injection), that satisfies the following universal property:
For any object B in C and any map between sets , there exists a unique morphism in C such that . That is, the following diagram commutes:
If free objects exist in C, the universal property implies every map between two sets induces a unique morphism between the free objects built on them, and this defines a functor . It follows that, if free objects exist in C, the functor F, called the free functor is a left adjoint to the forgetful functor U; that is, there is a bijection
The creation of free objects proceeds in two steps. For algebras that conform to the associative law, the first step is to consider the collection of all possible words formed from an alphabet. Then one imposes a set of equivalence relations upon the words, where the relations are the defining relations of the algebraic object at hand.
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