# Homotopy Type Theory open problems (Rev #15)

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.

## Semantics

• 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)? (Is this a solution?)

• 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.

• Semantics for strict resizing

• Semantics for Higher inductive recursive types.

• Connect univalence and parametricity as suggested here.

## Metatheory

• 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 $n^{th}$ universe is a model of HoTT with $(n-1)$ universes.

• What is the proof theoretic strength of univalent type theory plus HITs? In particular, can they be predicatively justified?

• Is univalence consistent with Induction-Recusion? This would allow us to build a non-univalent universe inside a univalent one.

• 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. Does adding axioms asserting that each universe is univalent increase the logical strength?

## Homotopy theory and algebraic topology

• Prove that the torus? (with the HIT definition involving a 2-dimensional path) is equivalent to a product of two circles. (Kristina has done this. Formalization seems difficult.) Similarly, consider the projective plane, Klein bottle, … as discussed in the book.

• 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$S^3 \to S^7 \to S^4$.

• Define the Toda bracket.

• Prove that $n$-spheres are $\infty$-truncated.

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

• 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.

• 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? For instance, does it satisfy collection or REA?

• 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?

## Formalization

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

• Formalize Semi-simplicial types.

• 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 Higher Inductive Inductive Types in Coq.

## 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.

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

• 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).)

Revision on March 5, 2014 at 09:30:00 by Bas Spitters. See the history of this page for a list of all contributions to it.