At the suggestion of Yemon, I have moved my comment to an answer. The <Gelfand representation>[Gelfand representation][1]
[1]: http://en.wikipedia.org/wiki/Gelfand_representation

gives an equivalence between the category of commutative, unital $C^*$-algebras and the opposite category of compact Hausdorff spaces. 

Breifly, let $A$ be a $C^*$-algebra, and let $\Sigma$ be the collection of nonzero homomorphisms $A \rightarrow \mathbb{C}$. Then $\Sigma$ sits inside $A^*$, 
the dual of $A$. Thus we can endow it with the weak-$^*$ topology. With this topology, if we consider $C(\Sigma)$, the algebra of continuous functions $\Sigma \rightarrow \mathbb{C}$, it turns out that we obtain a canonical isometric $*$-isomorphism $A \rightarrow C(\Sigma)$. 

The functor $CommC^*Alg \rightarrow CptHdTop^{op}$ given by $A \mapsto \Sigma$ defined above is an equivalence of categories. A great reference for the details is Conway's book on functional analysis (but he doesn't mention categories or functors).