Constructions with non-recursive higher inductive types

Higher inductive types (HITs) in homotopy type theory are a powerful generalization of inductive types. Not only can they have ordinary constructors to define elements, but also higher constructors to define equalities (paths). We say that a HIT H is non-recursive if its constructors do not quantify over elements or paths in H. The advantage of non-recursive HITs is that their elimination principles are easier to apply than those of general HITs. It is an open question which classes of HITs can be encoded as non-recursive HITs. One result of this paper is the construction of the propositional truncation via a sequence of approximations, yielding a representation as a non-recursive HIT. Compared to a related construction by van Doorn, ours has the advantage that the connectedness level increases in each step, yielding simplified elimination principles into n-types. As the elimination principle of our sequence has strictly lower requirements, we can then prove a similar result for van Doorn’s construction. We further derive general elimination principles of higher truncations (say, k-truncations) into n-types, generalizing a previous result by Capriotti et al. which considered the case n=k+1

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

Impredicative Encodings of (Higher) Inductive Types

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant {\eta}-equalities and consequently do not admit dependent eliminators. To recover {\eta} and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1

Homotopy Type Theory in Lean

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.Comment: 17 pages, accepted for ITP 201

Bisimulation as path type for guarded recursive types

In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming and reasoning about coinductive types is difficult for two reasons: The need for recursive definitions to be productive, and the lack of coincidence of the built-in identity types and the important notion of bisimilarity. Guarded recursion in the sense of Nakano has recently been suggested as a possible approach to dealing with the problem of productivity, allowing this to be encoded in types. Indeed, coinductive types can be encoded using a combination of guarded recursion and universal quantification over clocks. This paper studies the notion of bisimilarity for guarded recursive types in Ticked Cubical Type Theory, an extension of Cubical Type Theory with guarded recursion. We prove that, for any functor, an abstract, category theoretic notion of bisimilarity for the final guarded coalgebra is equivalent (in the sense of homotopy type theory) to path equality (the primitive notion of equality in cubical type theory). As a worked example we study a guarded notion of labelled transition systems, and show that, as a special case of the general theorem, path equality coincides with an adaptation of the usual notion of bisimulation for processes. In particular, this implies that guarded recursion can be used to give simple equational reasoning proofs of bisimilarity. This work should be seen as a step towards obtaining bisimilarity as path equality for coinductive types using the encodings mentioned above

The real projective spaces in homotopy type theory

Homotopy type theory is a version of Martin-L\"of 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, we retrieve the classical description of RP(n+1) as RP(n) with an (n+1)-cell attached to it. The infinite dimensional real projective space, defined as the sequential colimit of the 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.Comment: 8 pages, to appear in proceedings of LICS 201