26 research outputs found
CANONICITY AND HOMOTOPY CANONICITY FOR CUBICAL TYPE THEORY
Cubical type theory provides a constructive justification of homotopy type theory. A crucial ingredient of cubical type theory is a path lifting operation which is explained computationally by induction on the type involving several non-canonical choices. We present in this article two canonicity results, both proved by a sconing argument: a homotopy canonicity result, every natural number is path equal to a numeral, even if we take away the equations defining the lifting operation on the type structure, and a canonicity result, which uses these equations in a crucial way. Both proofs are done internally in a presheaf model
On the -topos semantics of homotopy type theory
Many introductions to homotopy type theory and the univalence axiom gloss
over the semantics of this new formal system in traditional set-based
foundations. This expository article, written as lecture notes to accompany a
3-part mini course delivered at the Logic and Higher Structures workshop at
CIRM-Luminy, attempt to survey the state of the art, first presenting
Voevodsky's simplicial model of univalent foundations and then touring
Shulman's vast generalization, which provides an interpretation of homotopy
type theory with strict univalent universes in any -topos. As we will
explain, this achievement was the product of a community effort to abstract and
streamline the original arguments as well as develop new lines of reasoning.Comment: These lecture notes were written to accompany a 4.5 hour mini-course
delivered at the workshop Logique et structures sup\'erieures held at CIRM -
Luminy from 21-25 February 2022. Video is available at
https://www.carmin.tv/en/collections/logic-and-higher-structures-logique-et-structures-superieure
Categorical Structures for Type Theory in Univalent Foundations
In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in the setting of univalent foundations, where the relationships between them can be stated more transparently. Specifically, we construct maps between the different structures and show that these maps are equivalences under suitable assumptions.
We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce relative universes, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure.
We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library
The Univalence Principle
The Univalence Principle is the statement that equivalent mathematical
structures are indistinguishable. We prove a general version of this principle
that applies to all set-based, categorical, and higher-categorical structures
defined in a non-algebraic and space-based style, as well as models of
higher-order theories such as topological spaces. In particular, we formulate a
general definition of indiscernibility for objects of any such structure, and a
corresponding univalence condition that generalizes Rezk's completeness
condition for Segal spaces and ensures that all equivalences of structures are
levelwise equivalences.
Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is
expressed in Voevodsky's Univalent Foundations (UF), extending previous work on
the Structure Identity Principle and univalent categories in UF. This enables
indistinguishability to be expressed simply as identification, and yields a
formal theory that is interpretable in classical homotopy theory, but also in
other higher topos models. It follows that Univalent Foundations is a fully
equivalence-invariant foundation for higher-categorical mathematics, as
intended by Voevodsky.Comment: A short version of this book is available as arXiv:2004.06572. v2:
added references and some details on morphisms of premonoidal categorie
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
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
Domain Theory in Constructive and Predicative Univalent Foundations
We develop domain theory in constructive and predicative univalent
foundations (also known as homotopy type theory). That we work predicatively
means that we do not assume Voevodsky's propositional resizing axioms. Our work
is constructive in the sense that we do not rely on excluded middle or the
axiom of (countable) choice. Domain theory studies so-called directed complete
posets (dcpos) and Scott continuous maps between them and has applications in
programming language semantics, higher-type computability and topology. A
common approach to deal with size issues in a predicative foundation is to work
with information systems, abstract bases or formal topologies rather than
dcpos, and approximable relations rather than Scott continuous functions. In
our type-theoretic approach, we instead accept that dcpos may be large and work
with type universes to account for this. A priori one might expect that complex
constructions of dcpos result in a need for ever-increasing universes and are
predicatively impossible. We show that such constructions can be carried out in
a predicative setting. We illustrate the development with applications in the
semantics of programming languages: the soundness and computational adequacy of
the Scott model of PCF and Scott's model of the untyped
-calculus. We also give a predicative account of continuous and
algebraic dcpos, and of the related notions of a small basis and its rounded
ideal completion. The fact that nontrivial dcpos have large carriers is in fact
unavoidable and characteristic of our predicative setting, as we explain in a
complementary chapter on the constructive and predicative limitations of
univalent foundations. Our account of domain theory in univalent foundations is
fully formalised with only a few minor exceptions. The ability of the proof
assistant Agda to infer universe levels has been invaluable for our purposes.Comment: PhD thesis, extended abstract in the pdf. v5: Fixed minor typos in
6.2.18, 6.2.19 and 6.4.