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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?

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Why? There are many other nice examples of natural transformations. I don't see a particular reason to single out the Stone functor. – Qiaochu Yuan Oct 10 '12 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. – Hooman Oct 10 '12 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). – Qiaochu Yuan Oct 10 '12 at 18:40
Hmmm... I would have thought the early motivating examples were in homology and homotopy, rather than general topology... – Zhen Lin Oct 10 '12 at 22:05
up vote 3 down vote accepted

(For those who, like me, did not know what the Stone functor was, this is the contravariant functor from topological spaces to Boolean algebras that assigns, to each topological space, the Boolean algebra of its clopen sets with union and intersection operations.)

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 $X \setminus K$).

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