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# Introduction: Gottfried Yang

edited April 15 in Chat

I finished my PhD thesis 2011 about "model structures (à la Thomason) on the diagram categories and Kan-extensions" and am now independent researcher. My interest was stable homotopy theory and arithmetic geometry. Newly I'm also interested in 1) univalent homotopy type theory; 2) fibered categories and stacks, concretely, "GROTHENDIECK'S "6 OPERATIONS" IN CATEGORICAL LOGIC".

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 applied mathematics.
Comment Source: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.