Ah, whoops! I hadn't seen this when I posted in lecture 4: "Boolean algebras are an important kind of poset. The power set functor defines an (contravariant) equivalence \\(P: Set^{op}→Bool\\): any function \\(f: X→Y\\) corresponds to the preimage map \\(f^∗:PY→PX\\), which is a monotone map, and also a boolean algebra homomorphism, meaning it preserves meets and joins (the very important fact that preimage preserves set operations)."

We determined that the direct image \\(f_!\\) is the left adjoint to preimage f*. But there is yet another twist to this **puzzle**: the preimage also has a *right* adjoint - what is it? It is certainly less obvious; I'm curious to see the reasoning of someone figuring it out on their own.

This triple adjunction has deep connections to topos theory and logic, which I hope to learn more about.

We determined that the direct image \\(f_!\\) is the left adjoint to preimage f*. But there is yet another twist to this **puzzle**: the preimage also has a *right* adjoint - what is it? It is certainly less obvious; I'm curious to see the reasoning of someone figuring it out on their own.

This triple adjunction has deep connections to topos theory and logic, which I hope to learn more about.