Homotopy Type Theory open problems > history (Rev #11, changes)

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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.

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.

• 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

• 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.

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?

• Is there a meaningful notion of “birthday” for the higher inductive-inductive surreal numbers??

Formalization

• 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

• 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).)