Homotopy Type Theory James construction

Definition

The James construction $J A$ on a pointed type $A$. Is the higher inductive type given by

• $\epsilon_{J A} : J A$
• $\alpha_{J A} : A \to J A \to J A$
• $\delta_{J A} : \prod_{x : J A} x = \alpha_A (*_A, x)$

Properties

We have an equivalence of types $JA \simeq \Omega \Sigma A$ if $A$ is 0-connected.

We can see that $J A$ is simply the free monoid on $A$. The higher inductive type is recursive which can make it difficult to study. This however can be remedied by defining a sequence of types $(J_n A)_{n: \mathbb{N}}$ together with maps $(i_n : J_n A \to J_{n+1} A)_{n:\mathbb{N}}$ such that the type $J_\infty A$ defined as the sequential colimit of $(J_n A)_{n:\mathbb{N}}$ is equivalent to $J A$.

This is useful as we can study the lower homotopy groups of $\Omega \Sigma X$ by studying the lower homotopy groups of the sequential colimit. Which is what allows Brunerie to show $\pi_4(S^3) = \mathbb{Z} / n \mathbb{Z}$.

category: homotopy theory