# A nice example of a functor naturally ismorphic to Stone-functor.

I want to explain the natural transformation with an example involving the Stone functor, but, I can't think of any non-trivial one. Does any one have one?

• Why? There are many other nice examples of natural transformations. I don't see a particular reason to single out the Stone functor. Commented Oct 10, 2012 at 18:33
• I was just curious, since I have heard that in the early years of the category theory, Stone functor was among the beloved motivating examples of functors. Commented Oct 10, 2012 at 18:40
• It's a good example of a functor, but that doesn't imply that it leads to interesting examples of natural transformations (although I could of course be wrong about this). Commented Oct 10, 2012 at 18:40
• Hmmm... I would have thought the early motivating examples were in homology and homotopy, rather than general topology... Commented Oct 10, 2012 at 22:05

There is a contravariant functor from topological spaces to commutative unital rings given by $X \mapsto C(X)$ that sends a topological space to its ring of continuous real-valued (or complex-valued, if you prefer) functions.
There is a covariant functor from commutative unital rings to Boolean algebras, which sends a ring $R$ to the set of its idempotents, using the following operations on two idempotents $e, f \in R$:
• $e \vee f = e + f - ef$
• $e \wedge f = ef$
• The "bottom" and "top" elements are $0$ and $1$, respectively.
Now an exercise for you: prove that the Stone functor is naturally isomorphic to the composite $\mathbf{Top}^{op} \to \mathbf{cRing} \to \mathbf{Boolean}$ of the two functors defined above. A healthy hint for you: the isomorphism between these two Boolean algebras should send a clopen subset $K \subseteq X$ with the characteristic function $\chi_K$ (whose value is $1$ on $K$ and $0$ on $X \setminus K$).