Homotopy Type Theory
open problems (Rev #7)

A list of (believed to be) open problems in homotopy type theory. To add more detail about a problem (such as why it is hard or interesting, or what ideas have been tried), make a link to a new page.


  • Is there a Quillen model structure on semi-simplicial sets? (with semi-simplicial Kan fibrations) making it Quillen equivalent to simplicial sets (with Kan fibrations)?

  • Construct a model of type theory in an (infinity,1)-topos in general.

  • Construct an (infinity,1)-topos? out of (the syntactic category of) a system of type theory.

  • Define a notion of elementary (infinity,1)-topos and show that any such admits a model of homotopy type theory.


  • Give a computational interpretation? of univalence and HITs.

  • Is there a cubical type theory? that describes more exactly what happens in the model of type theory in cubical sets??

  • Show in HoTT that the n thn^{th} universe is a model of HoTT with (n1)(n-1) universes.

  • Consider the type theory with a sequence of universes plus for each n>0n\gt 0, an axiom asserting that there are n-types? that are not (n1)(n-1)-types. Does adding axioms asserting that each universe is univalent increase the logical strength?

Homotopy theory

  • Prove that the torus? (with the HIT definition involving a 2-dimensional path) is equivalent to a product of two circles. (Has Kristina done this?)

  • What is the loop space of a wedge of circles indexed by a set without decidable equality?

  • Calculate more homotopy groups of spheres.

  • Show that the homotopy groups of spheres are all finitely generated, and are finite with the same exceptions as classically.

  • Construct the “super Hopf fibration” S 3S 7S 4S^3 \to S^7 \to S^4.

  • Define the Whitehead product.

  • Define the Toda bracket.

Higher algebra and higher category theory

  • Define semi-simplicial types? in type theory.

  • Define a weak omega-category in type theory?.

Other mathematics

  • What axioms of set theory are satisfied by the HIT model of ZFC? constructed in chapter 10 of the book?

  • Are any of the stronger forms of the axiom of choice? mentioned in the book consistent with univalence?

  • Do the higher inductive-inductive real numbers? from the book coincide with the Escardo-Simpson reals?


  • Formalize the construction of models of type theory using contextual categories.

  • Formalize what remains to be done from chapters 8, 10, and 11 of the book.

Closed problems

How about keeping a running list of solutions like this:?

Maybe only things not in the HoTT book? Else the list of solved problems gets very long!

  • Construct the Hopf fibration. (Lumsdaine gave the construction in 2012 and Brunerie proved it was correct in 2013.)

  • Prove the Siefert–van Kampen Theorem. (Shulman did it in 2013.)

  • Construct Eilenberg–MacLane spaces and use them to define cohomology. (Licata and Finster did it in 2013, written up in this paper (pdf).)

Revision on February 20, 2014 at 18:12:37 by Steve Awodey?. See the history of this page for a list of all contributions to it.