Higher Homotopies in a Hierarchy of Univalent Universes

For Martin-Lof type theory with a hierarchy U(0): U(1): U(2): ... of univalent universes, we show that U(n) is not an n-type. Our construction also solves the problem of finding a type that strictly has some high truncation level without using higher inductive types. In particular, U(n) is such a type if we restrict it to n-types. We have fully formalized and verified our results within the dependently typed language and proof assistant Agda.Comment: v1: 30 pages, main results and a connectedness construction; v2: 14 pages, only main results, improved presentation, final journal version, ancillary files with electronic appendix; v3: content unchanged, different documentclass reduced the number of pages to 1

Modalities in homotopy type theory

Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the theory of factorization systems, reflective subuniverses, and modalities in homotopy type theory, including their construction using a "localization" higher inductive type. This produces in particular the ($n$-connected, $n$-truncated) factorization system as well as internal presentations of subtoposes, through lex modalities. We also develop the semantics of these constructions

Partial Univalence in n-truncated Type Theory

It is well known that univalence is incompatible with uniqueness of identity proofs (UIP), the axiom that all types are h-sets. This is due to finite h-sets having non-trivial automorphisms as soon as they are not h-propositions. A natural question is then whether univalence restricted to h-propositions is compatible with UIP. We answer this affirmatively by constructing a model where types are elements of a closed universe defined as a higher inductive type in homotopy type theory. This universe has a path constructor for simultaneous "partial" univalent completion, i.e., restricted to h-propositions. More generally, we show that univalence restricted to $(n-1)$-types is consistent with the assumption that all types are $n$-truncated. Moreover we parametrize our construction by a suitably well-behaved container, to abstract from a concrete choice of type formers for the universe.Comment: 21 pages, long version of paper accepted at LICS 202

Formalizing two-level type theory with cofibrant exo-nat

This study provides some results about two-level type-theoretic notions in a way that the proofs are fully formalizable in a proof assistant implementing two-level type theory such as Agda. The difference from prior works is that these proofs do not assume any abuse of notation, providing us with more direct formalization. Moreover, some new notions, such as function extensionality for cofibrant types, are introduced. The necessity of such notions arises during the task of formalization. In addition, we provide some novel results about inductive types using cofibrant exo-nat, the natural number type at the non-fibrant level. While emphasizing the necessity of this axiom by citing new applications as justifications, we also touch upon the semantic aspect of the theory by presenting various models that satisfy this axiom

Homotopy-initial algebras in type theory

We investigate inductive types in type theory, using the insights provided by homotopy type theory and univalent foundations of mathematics. We do so by introducing the new notion of a homotopy-initial algebra. This notion is defined by a purely type-theoretic contractibility condition which replaces the standard, category-theoretic universal property involving the existence and uniqueness of appropriate morphisms. Our main result characterises the types that are equivalent to W-types as homotopy-initial algebras