I'm reading introductory notes on category theory. While discussing the notion of opposite category, the author makes a remark that if we take opposite category of SET, that is $SET^{op}$ category, we will get the (isomorphic) category of Boolean algebra. But the book didn't explain it beyond that. I would like to know why this is so? Taking the opposite category simply change the direction of arrow, how can this changing the arrows direction results in a Boolean algebra? Any answer to build intuition or any suggestion of reference that does it in elementary way--that is without using advanced theorems in category theory?

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    $\begingroup$ It's not the category of Boolean algebras but the category of complete atomic Boolean algebras. There's also a natural Boolean algebra structure associated with a set and that assignment conveniently extends to a contravariant functor on $\mathbf{Set}$. You may attempt to derive the result yourself from this by trying to explicitly construct the equivalence. $\endgroup$ – Derek Elkins Feb 23 at 6:38

This isn't quite true. But the idea is that the opposite of the category of sets may be thought of as the category of powersets and morphisms preserving intersections and unions, since to every function $f:S\to T$ there uniquely corresponds such a function $f^{-1}:P(T)\to P(S)$. Furthermore, $f^{-1}$ is a Boolean algebra homomorphism. This sets up an isomorphism between the opposite of sets and a certain full subcategory of Boolean algebras, namely that whose objects are powersets.

It's true that the opposite category is just some formal construction involving arrows. The point of this example is that in certain (rare) examples, the opposite of a category is isomorphic to some familiar category. Most other examples of this kind are categories which are isomorphic to their own opposite, such as the category of sets and relations, and the category of finite-dimensional vector spaces.


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