# nLab open problems in homotopy type theory

Contents

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.

# Contents

## Metatheory

• “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 $n^{th}$ universe is a model of HoTT with $(n-1)$ 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 $n\gt 0$, an axiom asserting that there are n-types that are not $(n-1)$-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.

## 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 $n$-spheres are $\infty$-truncated.

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

• Define the/a delooping of $S^3$.

• 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:

## Formalization

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

• 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.

## 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.

Last revised on January 26, 2023 at 02:08:56. See the history of this page for a list of all contributions to it.