The Hurewicz theorem in Homotopy Type Theory

Abstract

We prove the Hurewicz theorem in homotopy type theory, i.e., that for $X$ a pointed, $(n-1)$-connected type $(n \geq 1)$ and $A$ an abelian group, there is a natural isomorphism $\pi_n(X)^{ab} \otimes A \cong \tilde{H}_n(X; A)$ relating the abelianization of the homotopy groups with the homology. We also compute the connectivity of a smash product of types and express the lowest non-trivial homotopy group as a tensor product. Along the way, we study magmas, loop spaces, connected covers and prespectra, and we use $1$-coherent categories to express naturality and for the Yoneda lemma. As homotopy type theory has models in all $\infty$-toposes, our results can be viewed as extending known results about spaces to all other $\infty$-toposes.