The most general setting for a free object is in category theory, where one defines a functor, the free functor, that is the left adjoint to the forgetful functor.
Consider the category C of algebraic structures; these can be thought of as sets plus operations, obeying some laws. This category has a functor, the forgetful functor, which maps objects and functions in C to Set, the category of sets. The forgetful functor is very simple: it just ignores all of the operations.
The free functor F, when it exists, is the left adjoint to U. That is, takes sets X in Set to their corresponding free objects F(X) in the category C. The set X can be thought of as the set of "generators" of the free object F(X).
For the free functor to be a left adjoint, one must also have a Set-morphism . More explicitly, F is, up to isomorphisms in C, characterized by the following universal property:
- Whenever A is an algebra in C, and g: X→U(A) is a function (a morphism in the category of sets), then there is a unique C-morphism h: F(X)→A such that U(h)oη = g.
Concretely, this sends a set into the free object on that set; it's the "inclusion of a basis". Abusing notation, (this abuses notation because X is a set, while F(X) is an algebra; correctly, it is ).
The natural transformation is called the unit; together with the counit, one may construct a T-algebra, and so a monad. This leads to the next topic: free functors exist when C is a monad over Set.
Read more about this topic: Free Object
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