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## Comments

Nice ! Where did you do your PhD?

We're seeing a tiny taste of Grothendieck's 6 operations already in our study of posets: in Lecture 9 we saw that any function \(f : X \to Y\) induces a functor \(f^* : P(Y) \to P(X) \) that has both a left adjoint \(f_{!} : P(X) \to P(Y) \) and a right adjoint \( f_{\ast} : P(X) \to P(Y) \). All of this is a watered-down version of what happens for topoi or model categories. To me it will be very exciting when more and more of these ideas find their way into

appliedmathematics.`Nice ! Where did you do your PhD? We're seeing a tiny taste of Grothendieck's 6 operations already in our study of posets: in [Lecture 9](https://forum.azimuthproject.org/discussion/1931/lecture-9-chapter-1-adjoints-and-the-logic-of-subsets/p1) we saw that any function \\(f : X \to Y\\) induces a functor \\(f^* : P(Y) \to P(X) \\) that has both a left adjoint \\(f_{!} : P(X) \to P(Y) \\) and a right adjoint \\( f_{\ast} : P(X) \to P(Y) \\). All of this is a watered-down version of what happens for topoi or model categories. To me it will be very exciting when more and more of these ideas find their way into _applied_ mathematics.`