44 research outputs found
Unifying Cubical Models of Univalent Type Theory
We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure
Abstract and concrete type theories
In this thesis, we study abstract and concrete type theories. We introduce an abstract notion of a type theory to obtain general results in the semantics of type theories, but we also provide a syntactic way of presenting a type theory to allow us a further investigation into a concrete type theory to obtain consistency and independence results
Strict Rezk completions of models of HoTT and homotopy canonicity
We give a new constructive proof of homotopy canonicity for homotopy type
theory (HoTT). Canonicity proofs typically involve gluing constructions over
the syntax of type theory. We instead use a gluing construction over a "strict
Rezk completion" of the syntax of HoTT. The strict Rezk completion is specified
and constructed in the topos of cartesian cubical sets. It completes a model of
HoTT to an equivalent model satisfying a saturation condition, providing an
equivalence between terms of identity types and cubical paths between terms.
This generalizes the ordinary Rezk completion of a 1-category
Set Theory and its Place in the Foundations of Mathematics:a new look at an old question
This paper reviews the claims of several main-stream candidates to be the foundations of mathematics, including set theory. The review concludes that at this level of mathematical knowledge it would be very unreasonable to settle with any one of these foundations and that the only reasonable choice is a pluralist one
Code Generation for Higher Inductive Types
Higher inductive types are inductive types that include nontrivial
higher-dimensional structure, represented as identifications that are not
reflexivity. While work proceeds on type theories with a computational
interpretation of univalence and higher inductive types, it is convenient to
encode these structures in more traditional type theories with mature
implementations. However, these encodings involve a great deal of error-prone
additional syntax. We present a library that uses Agda's metaprogramming
facilities to automate this process, allowing higher inductive types to be
specified with minimal additional syntax.Comment: 16 pages, Accepted for presentation in WFLP 201
Transpension: The Right Adjoint to the Pi-type
Presheaf models of dependent type theory have been successfully applied to
model HoTT, parametricity, and directed, guarded and nominal type theory. There
has been considerable interest in internalizing aspects of these presheaf
models, either to make the resulting language more expressive, or in order to
carry out further reasoning internally, allowing greater abstraction and
sometimes automated verification. While the constructions of presheaf models
largely follow a common pattern, approaches towards internalization do not.
Throughout the literature, various internal presheaf operators (,
, , , ,
, the strictness axiom and locally fresh names) can be found and
little is known about their relative expressivenes. Moreover, some of these
require that variables whose type is a shape (representable presheaf, e.g. an
interval) be used affinely.
We propose a novel type former, the transpension type, which is right adjoint
to universal quantification over a shape. Its structure resembles a dependent
version of the suspension type in HoTT. We give general typing rules and a
presheaf semantics in terms of base category functors dubbed multipliers.
Structural rules for shape variables and certain aspects of the transpension
type depend on characteristics of the multiplier. We demonstrate how the
transpension type and the strictness axiom can be combined to implement all and
improve some of the aforementioned internalization operators (without formal
claim in the case of locally fresh names)
Twisted Cubes and their Applications in Type Theory
This thesis captures the ongoing development of twisted cubes, which is a
modification of cubes (in a topological sense) where its homotopy type theory
does not require paths or higher paths to be invertible. My original motivation
to develop the twisted cubes was to resolve the incompatibility between cubical
type theory and directed type theory. The development of twisted cubes is still
in the early stages and the intermediate goal, for now, is to define a twisted
cube category and its twisted cubical sets that can be used to construct a
potential definition of (infinity, n)-categories. The intermediate goal above
leads me to discover a novel framework that uses graph theory to transform
convex polytopes, such as simplices and (standard) cubes, into base categories.
Intuitively, an n-dimensional polytope is transformed into a directed graph
consists 0-faces (extreme points) of the polytope as its nodes and 1-faces of
the polytope as its edges. Then, we define the base category as the full
subcategory of the graph category induced by the family of these graphs from
all n-dimensional cases. With this framework, the modification from cubes to
twisted cubes can formally be done by reversing some edges of cube graphs.
Equivalently, the twisted n-cube graph is the result of a certain endofunctor
being applied n times to the singleton graph; this endofunctor (called twisted
prism functor) duplicates the input, reverses all edges in the first copy, and
then pairwisely links nodes from the first copy to the second copy. The core
feature of a twisted graph is its unique Hamiltonian path, which is useful to
prove many properties of twisted cubes. In particular, the reflexive transitive
closure of a twisted graph is isomorphic to the simplex graph counterpart,
which remarkably suggests that twisted cubes not only relate to (standard)
cubes but also simplices.Comment: PhD thesis (accepted at the University of Nottingham), 162 page