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 and groups that can be delooped more than once. A major result is the stabilization theorem, which states that if an n-type can be delooped n + 2 times, then it is an infinite loop type. Most of the results have been formalized in the Lean proof assistant.
Abstract. We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.
We define a variety of notions of cubical sets, based on sites organized using substructural algebraic theories presenting PRO(P)s or Lawvere theories. We prove that all our sites are test categories in the sense of Grothendieck, meaning that the corresponding presheaf categories of cubical sets model classical homotopy theory. We delineate exactly which ones are even strict test categories, meaning that products of cubical sets correspond to products of homotopy types.
Homotopy type theory is a version of Martin-Löf type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of RP n , as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RP n by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take RP −1 to be the empty type. In the inductive step, we take RP n+1 to be the mapping cone of the projection map of the tautological bundle of RP n , and we use its universal property and the univalence axiom to define the tautological bundle on RP n+1 .By showing that the total space of the tautological bundle of RP n is the n-sphere S n , we retrieve the classical description of RP n+1 as RP n with an (n + 1)-disk attached to it. The infinite dimensional real projective space RP ∞ , defined as the sequential colimit of RP n with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z, 1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles.These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.
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