Papers › The Hurewicz theorem in Homotopy Type Theory
The Hurewicz theorem in Homotopy Type Theory
J. Daniel Christensen, Luis Scoccola
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We prove the Hurewicz theorem in homotopy type theory, i.e., that for X a pointed, (n-1)-connected type (n ≥1) and A an abelian group, there is a natural isomorphism πₙ(X)ᵃᵇ ⊗A ≅H̃ₙ(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 ∞-toposes, our results can be viewed as extending known results about spaces to all other ∞-toposes.
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