Category-theoretic Definition
In the language of category theory, free Boolean algebras can be defined simply in terms of an adjunction between the category of sets and functions, Set, and the category of Boolean algebras and Boolean algebra homomorphisms, BA. In fact, this approach generalizes to any algebraic structure definable in the framework of universal algebra.
Above, we said that a free Boolean algebra is a Boolean algebra with a set of generators that behave a certain way; alternatively, one might start with a set and ask which algebra it generates. Every set X generates a free Boolean algebra FX defined as the algebra such that for every algebra B and function f : X → B, there is a unique Boolean algebra homomorphism f′ : FX → B that extends f. Diagrammatically,
where iX is the inclusion, and the dashed arrow denotes uniqueness. The idea is that once one chooses where to send the elements of X, the laws for Boolean algebra homomorphisms determine where to send everything else in the free algebra FX. If FX contained elements inexpressible as combinations of elements of X, then f′ wouldn't be unique, and if the elements of X weren't sufficiently independent, then f′ wouldn't be well defined! It's easily shown that FX is unique (up to isomorphism), so this definition makes sense. It's also easily shown that a free Boolean algebra with generating set X, as defined originally, is isomorphic to FX, so the two definitions agree.
One shortcoming of the above definition is that the diagram doesn't capture that f′ is a homomorphism; since it's a diagram in Set each arrow denotes a mere function. We can fix this by separating it into two diagrams, one in BA and one in Set. To relate the two, we introduce a functor U : BA → Set that "forgets" the algebraic structure, mapping algebras and homomorphisms to their underlying sets and functions.
If we interpret the top arrow as a diagram in BA and the bottom triangle as a diagram in Set, then this diagram properly expresses that every function f : X → B extends to a unique Boolean algebra homomorphism f′ : FX → B. The functor U can be thought of as a device to pull the homomorphism f′ back into Set so it can be related to f.
The remarkable aspect of this is that the latter diagram is one of the various (equivalent) definitions of when two functors are adjoint. Our F easily extends to a functor Set → BA, and our definition of X generating a free Boolean algebra FX is precisely that U has a left adjoint F.
Read more about this topic: Free Boolean Algebra
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