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 such that . That is, the following diagram commutes:
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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