A Mechanization of the Blakers-Massey Connectivity Theorem in Homotopy Type Theory

Abstract

This paper continues investigations in "synthetic homotopy theory": the use of homotopy type theory to give machine-checked proofs of constructions from homotopy theory.We present a mechanized proof of the Blakers-Massey connectivity theorem, a result relating the higher-dimensional homotopy groups of a pushout type (roughly, a space constructed by gluing two spaces along a shared subspace) to those of the components of the pushout. This theorem gives important information about the pushout type, and has a number of useful corollaries, including the Freudenthal suspension theorem, which has been studied in previous formalizations.The new proof is more elementary than existing ones in abstract homotopy-theoretic settings, and the mechanization is concise and high-level, thanks to novel combinations of ideas from homotopy theory and type theory.

Quotient Inductive-Inductive Types

“…Despite a lot of work making use of concrete HITs [4,[9][10][11]23,26,27], and despite the fact that it is usually -on some intuitive level -clear for the expert how the elimination principle for such a HIT can be derived, giving a general specification and a theoretical foundation for HITs has turned out to be a major difficulty. Several approaches have been proposed [6,18,28,33], and they do indeed give a satisfactory specification of HITs in the sense that they cover all HITs which have been used so far (see related work below).…”

An introduction to univalent foundations for mathematicians

“…In [38] one may find a proof that the fundamental group of S 1 is Z, and in [32] one may find a formalization of the Seifert-van Kampen theorem. In [33] one may find a formalization of the Blakers-Massey connectedness theorem, using higher inductive types to construct homotopy pushouts. A translation of the formal proof into classical homotopy theory was given in [46], with the expectation that it would go through for any ∞-topos.…”

The James Construction and $$\pi _4(\mathbb {S}^{3})$$ in Homotopy Type Theory

“…In this case, however, we can use the James construction to give an explicit definition of the kernel of that map. We will need the Blakers-Massey theorem (see [4]): If f is n-connected and g is m-connected, then h is (n + m)-connected.…”