Related papers: Inductive types in homotopy type theory
Initial Semantics aims at characterizing the syntax associated to a signature as the initial object of some category. We present an initial semantics result for typed higher-order syntax together with its formalization in the Coq proof…
In this paper we examine the natural interpretation of a ramified type hierarchy into Martin-L\"of type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of…
Algebraic theories with dependency between sorts form the structural core of Martin-L\"of type theory and similar systems. Their denotational semantics are typically studied using categorical techniques; many different categorical…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
In this paper we combine the principled approach to modalities from multimodal type theory (MTT) with the computationally well-behaved realization of identity types from cubical type theory (CTT). The result -- cubical modal type theory…
The goal of this dissertation is to present synthetic homotopy theory in the setting of homotopy type theory. We will present various results in this framework, most notably the construction of the Atiyah-Hirzebruch and Serre spectral…
This paper proposes a way of doing type theory informally, assuming a cubical style of reasoning. It can thus be viewed as a first step toward a cubical alternative to the program of informalization of type theory carried out in the…
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…
In this work we study the induction (induced and coinduced)theory for Hopf group coalgebra. We define a substructure B of a Hopf group coalgebra $H$, called subHopf group coalgebra. Also, we have introduced the definition of Hopf group…
Native type systems are those in which type constructors are derived from term constructors, as well as the constructors of predicate logic and intuitionistic type theory. We present a method to construct native type systems for a broad…
Covering spaces are a fundamental tool in algebraic topology because of the close relationship they bear with the fundamental groups of spaces. Indeed, they are in correspondence with the subgroups of the fundamental group: this is known as…
The ability to cast values between related types is a leitmotiv of many flavors of dependent type theory, such as observational type theories, subtyping, or cast calculi for gradual typing. These casts all exhibit a common structural…
In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by…
When working in a proof assistant, automation is key to discharging routine proof goals such as equations between algebraic expressions. Homotopy type theory allows the user to reason about higher structures, such as topological spaces,…
In order to avoid well-know paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type$_0$ : Type$_1$ :…
We give a new solution of the "homotopy periods" problem, as highlighted by Sullivan, which places explicit geometrically meaningful formulae first dating back to Whitehead in the context of Quillen's formalism for rational homotopy theory…
We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode…
The aim of this paper is to extend the definition of motivic homotopy theory from schemes to a large class of algebraic stacks and establish a six functor formalism. The class of algebraic stacks that we consider includes many interesting…
A reflective subuniverse in homotopy type theory is an internal version of the notion of a localization in topology or in the theory of $\infty$-categories. Working in homotopy type theory, we give new characterizations of the following…
Hammers are tools that provide general purpose automation for formal proof assistants. Despite the gaining popularity of the more advanced versions of type theory, there are no hammers for such systems. We present an extension of the…