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In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It 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 category theory, although this is in yet more abstract terms. Examples include free groups, tensor algebras, or free lattices. Informally, a free object over a set A can be thought 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. ==Definition== Free objects are the direct generalization to categories 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''. Conversely, a function ''u : E1 → E2'' defined on a basis of ''E1'' can be uniquely extended to a linear function. The following definition translates this to any category. Let ''(C,F)'' be a concrete category (i.e. ''F: C → Set'' is a faithful functor), let ''X'' be a set (called ''basis''), ''A'' ∈ ''C'' an object, and ''i: X → F(A)'' a map between sets (called ''canonical injection''). We say that ''A'' is the free object on ''X'' (with respect to ''i'') if and only if they satisfy this universal property: :for any object ''B'' and any map between sets ''f: X → F(B)'', there exists a unique morphism ''g: A → B'' such that ''f = F(g) o i''. That is, the following diagram commutes: : In this way the free functor that builds the free object ''A'' from the set ''X'' becomes left adjoint to the forgetful functor. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「free object」の詳細全文を読む スポンサード リンク
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