Homotopy type theory may be seen as an internal language for the ∞-category of weak ∞-groupoids. Moreover, weak ∞-groupoids model the univalence axiom. Voevodsky proposes this (language for) weak ∞-groupoids as a new foundation for Mathematics called the univalent foundations. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of) the traditional set theoretical foundations as a special case. We thus wonder whether those 'discrete' groupoids do in fact form a (predicative) topos. More generally, homotopy type theory is conjectured to be the internal language of 'elementary' of ∞-toposes. We prove that sets in homotopy type theory form a ΠW-pretopos. This is similar to the fact that the 0-truncation of an ∞-topos is a topos. We show that both a subobject classifier and a 0-object classifier are available for the type theoretical universe of sets. However, both of these are large and moreover the 0-object classifier for sets is a function between 1-types (i.e. groupoids) rather than between sets. Assuming an impredicative propositional resizing rule we may render the subobject classifier small and then we actually obtain a topos of sets. IntroductionA preliminary version of this paper was ready when the standard reference book on homotopy type theory (The Univalent Foundations Program 2013) was produced. In fact, many of the results of this paper can now also be found in chapter 10 of that book. Conversely, the collaborative writing of that chapter helped us to clarify the presentation of the present article. The paper is also meant to give a readable account of some computer proofs which have meanwhile found their way to https://github.com/HoTT/HoTT/.Homotopy type theory (Awodey 2012) extends the Curry-Howard correspondence between simply typed λ-calculus, Cartesian closed categories and minimal logic, via extensional dependent type theory, locally Cartesian closed categories and predicate logic (Lambek and Scott 1988; Jacobs 1999) to Martin-Löf type theory with identity types and certain homotopical models. The univalent foundations program (Kapulkin 2012; Pelayo and Warren 2012; The Univalent Foundations Program 2013) extends homotopy type theory with the so-called univalence axiom, thus providing a language for ∞-groupoids. Voevodsky's insight was that this can serve as a new foundation for Mathematics. The ∞-groupoids form the prototypical higher topos (Lurie 2009;Rezk 2010). In fact, homotopy type theory and the univalence axiom can be interpreted in † The research leading to these results has received funding from the European Union's 7th Framework Programme under grant agreement nr. 243847 (ForMath).at https://www.cambridge.org/core/terms. https://doi

We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type X, the natural map X → X (p) induces algebraic localizations on all homotopy groups. In order to prove this, we further develop the theory of reflective subuniverses. In particular, we show that for any reflective subuniverse L, the subuniverse of L-separated types is again a reflective subuniverse, which we call L ′ . Furthermore, we prove results establishing that L ′ is almost left exact. We next focus on localization with respect to a map, giving results on preservation of coproducts and connectivity. We also study how such localizations interact with other reflective subuniverses and orthogonal factorization systems. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an Eilenberg-Mac Lane space K(G, n) with G abelian. We also include a partial converse to the main theorem.

In homotopy type theory we can define the join of maps as a binary operation on maps with a common codomain. This operation is commutative, associative, and the unique map from the empty type into the common codomain is a neutral element. Moreover, we show that the idempotents of the join of maps are precisely the embeddings, and we prove the 'join connectivity theorem', which states that the connectivity of the join of maps equals the join of the connectivities of the individual maps.We define the image of a map f : A → X in U via the join construction, as the colimit of the finite join powers of f . The join powers therefore provide approximations of the image inclusion, and the join connectivity theorem implies that the approximating maps into the image increase in connectivity.A modified version of the join construction can be used to show that for any map f : A → X in which X is only assumed to be locally small, the image is a small type. We use the modified join construction to give an alternative construction of set-quotients, the Rezk completion of a precategory, and we define the n-truncation for any n : N. Thus we see that each of these are definable operations on a univalent universe for Martin-Löf type theory with a natural numbers object, that is moreover closed under homotopy coequalizers.

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.

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.