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 $n^{th}$ universe is a model of HoTT with $(n-1)$ universes.
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?
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” $S^3 \to S^7 \to S^4$.
Define the Whitehead product.
Define the Toda bracket.
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.
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).)