We prove the Hurewicz theorem in homotopy type theory, i.e., that for $X$ a pointed, $(n-1)$-connected $(n \geq 1)$ and $A$ an abelian group, there is natural isomorphism $\pi_n(X)^{ab} \otimes A \cong \tilde{H}_n(X; A)$ relating abelianization of groups with homology. also compute connectivity smash product types express lowest non-trivial group as tensor product. Along way, we study magmas, l...

We continue the investigation of CW homotopy type of spaces of continuous functions between two CW complexes begun by J. Milnor in 1959 and P. Kahn in 1984. Viewing function spaces as particular cases of inverse limits we also study certain inverse systems of fibrations between CW homotopy type spaces. If the limit space Z∞ of an inverse sequence {Zi} of fibrations between CW type spaces has CW...

The cubical sets model of Homotopy Type Theory was introduced by Bezem, Coquand and Huber using a particular category of presheaves. We show that this category is equivalent to a category of sets whose elements have a finite support property with respect to an action of a monoid of name substitutions; and that this is isomorphic to a category of nominal sets equipped with source and target maps...

An exposition of some proofs of the Freudenthal suspension theorem and the BlakersMassey theorem. These are meant to be reverse engineered versions of proofs in homotopy type theory due to Lumsdaine, Finster, and Licata. The proof of Blakers-Massey given here is based on a formalization given by Favonia.

These lecture notes are based on and partly contain material from the HoTT book and are licensed under Creative Commons Attribution-ShareAlike 3.0.

We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this viewpoint, as well as higher dimensional groups...