nLab homotopy theory in homotopy type theory -- references

Homotopy theory formalized in homotopy type theory

The following is a list of topics and results of homotopy theory that have been formalized in homotopy type theory.

On homotopy groups of spheres:

• homotopy groups of spheres in HoTT

On Whitehead's theorem:

• Dan’s proof for n-types.

On Eilenberg-MacLane spaces:

• Dan’s construction.

• $K(G,n)$ is a spectrum, formalized

On ordinary cohomology:

• Deriving cohomology theories from spectra

• Mayer-Vietoris sequence, with exposition

• Cohomology of the n-torus, which could be easily extended to any finite product of spheres.

• to do:

  • cohomology with finite coefficients, all we need is the boring work of defining $\mathbb{Z}/p\mathbb{Z}$ as an explicit group.

  • Calculate some more cohomology groups.

  • Compute the loop space of this construction and use it to define spectra.

On the Freudenthal suspension theorem:

• Implies $\pi_k(S^n) = \pi_{k+1}(S^{n+1})$ whenever $k \leq 2n - 2$

• 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.

On homotopy limits:

• Jeremy Avigad, Chris Kapulkin and Peter Lumsdaine arXiv Coq code

On the van Kampen theorem:

• Mike’s proofs are in the book and Favonia formalized it. {{deadlink}}

On covering spaces:

• Favonia’s proofs (link in flux due to library rewrite).

On Blakers-Massey theorem:

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