Homotopy Type Theory
HoTT2019 Summer School open problems list (changes)

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Here are the open problems stated at the 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

  • does (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?

  • 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? (Kristina: true?)

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)

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Last revised on August 11, 2019 at 21:21:44. See the history of this page for a list of all contributions to it.