[[!redirects HoTT2019 Summer School open problems list]] < [[nlab:HoTT2019 Summer School open problems list]] Here are the open problems stated at the [HoTT 2019 Summer School](https://hott.github.io/HoTT-2019//summer-school/), listed by the instructor who posed the problem: #### Mathieu Anel: Logos Theory#### - Notion of higher theory? (syntactic approach? categorical approach à la Lawvere?) (e.g. CAT^lex, CAT_cc, LOGOS, …) - Explicit distributivity formula between finite limits and colimit in a logos? colim_{something} lim_{something} —> lim_{something} colim_{something} This should lead to a polynomial calculus for logoi (in the sense of polynomial functors) - Explicit description of the symmetric logo functor? Sym: CAT_cc —> LOGOI - Develop internal cat theory in a logos - Define a class of logoi suited for the purpose of logic (and containing all presentable logoi) ####Egbert Rijke: Synthetic Homotopy Theory#### - deloop the 3-sphere: find a pointed connected type X s.th Omega(X) = S^3 - define the Grassmanians (& other interesting CW-complexes) using HITS. - prove the BoTT periodicity theorem ####Jonas Frey: The Coherence Problem#### - define internal operads, semi-simplicial types, (oo,1)-categories in HoTT. ####Anders Mortberg: Cubical Type Theory#### - Efficient evaluation/ computation of cubical programs - have a cubical TT where transport ref x == x. ####Guillaume Brunerie: Computation in Cubical Type Theory#### - find other interesting examples of cubical terms to compute (e.g. cohomology cup products) ####Kristina Sojakova:#### - conservativity of cubical TT over Book HoTT - (suggested by Nicolai Kraus): Does adding a path between fixed points in a type preserve truncation level? Namely, if X is an n-type for n > 0 and a, b : X, is the HIT H generated by [-] : X —> H , p : [a] = [b] , still an n-type? ####Steve Awodey:#### - crossed modules classify 2-types - define a notion of cubical quasi-category - Cubical Homotopy Hypothesis: sSet (w/Kan QMS) ~ cSet (Dedekind cubes w/Sattler QMS) Return to [[open problems]].