Homotopy Type Theory Formalized Homotopy Theory > history (Rev #11)

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, Evan Cavallo

In progress

Cohomology

• 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.

At least one proof has been formalized

Whitehead’s theorem

• Dan’s proof for n-types.

K(G,n)

• Dan’s construction.
• $K(G,n)$ is a spectrum, formalized

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.

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.

Homotopy limits

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

Van Kampen

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

Covering spaces

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

Blakers-Massey

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