Abstract Lumsdaine

Abstract:  It is widely believed that homotopy type theory should be the internal logic of elementary ∞-toposes — whatever that may mean.  I will present some recent progress (jww Chris Kapulkin) towards a precise statement and proof of this idea.
 Specifically, for various type theories, leading up to “full HoTT”, we assemble their models into an (∞,1)-category, with an ∞-functor to a suitable (∞,1)-category of structured (∞,1)-categories (e.g. LCCC_∞).  One may then conjecture that this ∞-functor is an equivalence — a precise and strong form of the “internal language” conjecture.
  The main technical tool used is the “Reedy span-equivalences” construction.  This turns out to also have applications of a more traditionally logical flavour, for equivalence and conservativity results between different type theories.

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