nLab open problems in homotopy type theory


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.




  • “Writing all inductive definitions in terms of Sigma-types and W-types is theoretically possible, but extremely tedious … it’s unknown whether any similar reduction for HITs is possible.” (source)

  • Give a computational interpretation? of univalence and HITs. The cubical set model makes progress on this. PDF

  • Is there a cubical type theory that describes more exactly what happens in the model of type theory in cubical sets?? Partial solutions: the cubical language, this and this

  • Show in HoTT that the n thn^{th} universe is a model of HoTT with (n1)(n-1) universes. Part of this question is: what is an internal model of HoTT.

  • What is the proof theoretic strength of univalent type theory plus HITs? In particular, can they be predicatively justified? One could start by considering simple classes of HITs; e.g. here before continuing to, say, pushouts, the cumulative sets, the Cauchy reals and the surreals.

  • Is univalence consistent with Induction-Recusion? This would allow us to build a non-univalent universe inside a univalent one. Related to this, higher inductive types can be used to define a univalent universe.

  • 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; see here. Does adding axioms asserting that each universe is univalent increase the logical strength? Of course, univalence gives funext.

  • Prove or disprove the conjecture that every fibrant type in HTS is equivalent to one definable in MLTT. A potential method of disproof would be to solve the model invariance problem positively for MLTT but negatively for fibrant types in HTS.

  • Define higher inductive types in higher observational type theory.

Homotopy theory and algebraic topology

  • Homotopy theory in HoTT?

  • Similarly to the torus, consider the projective plane, Klein bottle, … as discussed in the book (sec 6.6). Show that the Klein bottle is not orientable. (This requires defining “orientable”.)

    • This also requires defining what a surface is.
  • 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.

    • The classical proof requires Hurewicz, and now that spectral sequences are around should be possible.
  • Define the Hurewicz map and prove the Hurewicz theorem

  • Define the Toda bracket.

  • Prove that nn-spheres are \infty-truncated.

  • Prove that S 2S^2 is not an nn-type.

  • Define the/a delooping of S 3S^3.

  • Can we verify computational algebraic topology using HoTT?

  • Bott periodicity

  • Develop synthetic stable homotopy theory

Higher algebra and higher category theory

  • Define semi-simplicial types in type theory, or show that this is not possible. Define Segal space complete Segal space.

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

  • Is it possible to have limits be judgmentally the same as opposite colimits, and simultaneously have (co)limits be judgmentally the same as particular Kan extensions. As partially detailed on an issue on the HoTT/HoTT repo, part of the problem is that the functor ᵒᵖ : (D → C)ᵒᵖ → (Dᵒᵖ → Cᵒᵖ) has a judgmental inverse (the composition is judgmentally the identity functor on objects and morphisms), and, assuming univalence, the precategories (D → C)ᵒᵖ and (Dᵒᵖ → Cᵒᵖ) are propositionally but not judgmentally equal.

  • Eric Finster, Towards Higher Universal Algebra in Type Theory

  • Construct the first 8 \mathbb{N}-indexed families of \infty-groups in the Whitehead tower of the orthogonal groups.

Other mathematics

  • What axioms of set theory are satisfied by the HIT model of ZFC? constructed in chapter 10 of the book? For instance, does it satisfy collection or REA?

  • Are any of the stronger forms of the axiom of choice mentioned in the book (ex 7.8) consistent with univalence?

  • Do the higher inductive-inductive real numbers from the book coincide with the Escardo-Simpson reals, here and here? Solved by Auke Booij using resizing/impredicativity. Can a predicative treatment be obtained from hSets as a predicative topos (by Rijke/Spitters) and the predicative formalization of Cauchy reals by Gilbert.

  • Can multiplication be defined for the higher inductive-inductive surreal numbers from the book?

  • Can Rezk complete categories remove the non-constructivity from the applications of Freyd's adjoint functor theorem?

  • Is there a predicative and constructive abstract definition of the category of real Hilbert spaces, in a similar fashion as what Chris Heunen and Andre Kornell did in classical mathematics in their article (Axioms for the category of Hilbert spaces)? Solér’s theorem, which is used in their proof, is only valid in classical mathematics.

In cohesive homotopy type theory

Problems in cohesive homotopy type theory:

See also the commented list of problems at:

In cubical type theory


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

  • Formalize semi-simplicial types in homotopy type theory.

  • Formalize \infty-groupoids, \infty-categories within HoTT.

  • Formalize what remains to be done from chapters 8, 10, and 11 of the book. In particular, develop the Higher Inductive-Inductive real numbers in some language, such as Coq (the basics of the surreal numbers have been done).

    In general, this file contains a Coq outline of the book. Instructions for how to contribute are here.

Other lists of open problems

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!

OK, here’s the rule: if it was stated here (or on the UF-wiki) as an open problem, then it gets recorded here once it’s solved.

  • There is a model structure on semi-simplicial sets.

  • Prove that the torus (with the HIT definition involving a 2-dimensional path) is equivalent to a product of two circles. See Sojakova’s proof: torus.pdf. A shorter formalized proof is here

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

  • Construct the “super Hopf fibration” (AKA quaternionic Hopf fibration) S 3S 7S 4S^3 \to S^7 \to S^4. (Buchholtz and Rijke solved this early 2016 through a homotopy version of the Cayley-Dickson construction.)

  • Prove the Seifert-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).)

  • Define the Whitehead product. Guillaume Brunerie did this in 2017, written up in this paper. He also gives a constructive proof of π 4(S 3)=Z/nZ\pi_4( S^3)= Z/n Z for some nn and a construction of the reduced James product as a HIT and a homotopy colimit. It is also proven that ΩΣXJ(X)\Omega \Sigma X \simeq J(X) for some pointed connected type XX. All of these constructions can be found in detail in his thesis.

Last revised on August 3, 2022 at 12:03:02. See the history of this page for a list of all contributions to it.