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# Introduction: Gottfried Yang

edited April 2018 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".

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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 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. 
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excuse me for the later answer. I did my PhD at the University Osnabrück(Germany), supervised by Rainer Vogt. Yes, the idea of Grothendieck's 6 operations is to consider the left and right adjoint functors of the pullback-functor f* in the context of monoidal fibered categories

Comment Source:excuse me for the later answer. I did my PhD at the University Osnabrück(Germany), supervised by Rainer Vogt. Yes, the idea of Grothendieck's 6 operations is to consider the left and right adjoint functors of the pullback-functor f* in the context of monoidal fibered categories 
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Topoi can be used as semantics for type theories (e.g. the nLab article https://ncatlab.org/nlab/show/dependent+linear+type+theory about linear dependent type theory, which presumably corresponds to infinity topoi). In the homotopy type theory one notes the classical adjunction f_! -| f ^ * -| f_ * as \ Sigma_f -| f ^ * -| \ Prod_f, because the terms (dependent) sum and product behave so. In the logical interpretation of HoTT, \ Sigma is an existence quanter and \ prod an all-quantor (https://homotopytypetheory.org/book/). Now every additional structure can be encoded on an infinite topos X so that the logic contains more constructs but the topos is still a model. A typical question would be, to an infinite topos X, to ask, which logic T exactly has this topos (up to isomorphism) as model.

Comment Source:Topoi can be used as semantics for type theories (e.g. the nLab article https://ncatlab.org/nlab/show/dependent+linear+type+theory about linear dependent type theory, which presumably corresponds to infinity topoi). In the homotopy type theory one notes the classical adjunction f_! -| f ^ * -| f_ * as \ Sigma_f -| f ^ * -| \ Prod_f, because the terms (dependent) sum and product behave so. In the logical interpretation of HoTT, \ Sigma is an existence quanter and \ prod an all-quantor (https://homotopytypetheory.org/book/). Now every additional structure can be encoded on an infinite topos X so that the logic contains more constructs but the topos is still a model. A typical question would be, to an infinite topos X, to ask, which logic T exactly has this topos (up to isomorphism) as model.