Here are some links to proofs about homotopy theory, formalized in homotopy type theory. Please add!

Cast of characters so far: Jeremy Avigad, Guillaume Brunerie, Favonia, Eric Finster, Chris Kapulkin, Dan Licata, Peter Lumsdaine, Mike Shulman.

In progress


  • Guillaume proved that there is some n such that π4(S3) is Z_n. With a computational interpretation, we could run this proof and check that it's 2.


  • Peter's construction of the Hopf fibration as a dependent type. Lots of people around know the construction, but I don't know anywhere it's written up. Here's some Agda code with it in it.
  • Guillaume's proof that the total space of the Hopf fibration is S3, together with π_n(Sn), imply this by a long-exact-sequence argument, but this hasn't been formalized.
  • Prove that K(G,n) is a spectrum (Eric?)
  • To do cohomology with finite coefficients, all we need is the boring work of defining Z/pZ as an explicit group.
  • Calculate some cohomology groups

At least one proof has been formalized

Whitehead's theorem


Freudenthal Suspension Theorem

Implies π_k(Sn) = π_k+1(Sn+1) whenever k <= 2n - 2
  • Peter's encode/decode-style proof, formalized by Dan, using a clever lemma about maps out of 2 n-connected types. This is the "95%" version, which shows that the relevant map is an equivalence on truncations.
  • The full "100%" version, formalized by Dan, which shows that the relevant map is 2n-connected.


π_k(Sn) for k < n


  • Guillaume's proof
  • Dan's encode/decode-style proof


  • Mike's proof by contractibility of total space of universal cover (HoTT blog).
  • Dan's encode/decode-style proof (HoTT blog).
    A paper mostly about the encode/decode-style proof, but also describing the relationship between the two.
  • Guillaume's proof using the flattening lemma.

Homotopy limits

  • Chris/Peter/Jeremy's development (link?)

Van Kampen

Covering spaces

  • Favionia's proofs (link in flux due to library rewrite).


  • Favonia/Peter/Guillaume/Dan's formalization of Peter/Eric/Dan's proof (link in flux due to library rewrite).