# Category theoretic proof of Stone's representation theorem

Can Stone's representation theorem about Boolean algebras that every Boolean algebra $$B$$ is isomorphic to the algebra of clopen subsets of its Stone space $$S(B)$$, be proven categorically using Yoneda lemma ?

• Is there some reason you would expect there to be such a proof? I'm sure you could find a way to phrase the proof so that it uses Yoneda's lemma somehow, but ultimately it's a theorem about the actual algebraic structure of Boolean algebras and is not going to have a proof that is purely abstract nonsense. – Eric Wofsey Nov 23 '18 at 0:08

$$(\text{Ind}(\text{FinBool}))^{op} \cong \text{Pro}(\text{FinBool}^{op}) \cong \text{Pro}(\text{FinSet})$$