Cofree functor

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 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.

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:

In this way the free functor that builds the free object A from the set X becomes left adjoint to the forgetful functor.

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