202 research outputs found
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
A type theory for synthetic -categories
We propose foundations for a synthetic theory of -categories
within homotopy type theory. We axiomatize a directed interval type, then
define higher simplices from it and use them to probe the internal categorical
structures of arbitrary types. We define Segal types, in which binary
composites exist uniquely up to homotopy; this automatically ensures
composition is coherently associative and unital at all dimensions. We define
Rezk types, in which the categorical isomorphisms are additionally equivalent
to the type-theoretic identities - a "local univalence" condition. And we
define covariant fibrations, which are type families varying functorially over
a Segal type, and prove a "dependent Yoneda lemma" that can be viewed as a
directed form of the usual elimination rule for identity types. We conclude by
studying homotopically correct adjunctions between Segal types, and showing
that for a functor between Rezk types to have an adjoint is a mere proposition.
To make the bookkeeping in such proofs manageable, we use a three-layered
type theory with shapes, whose contexts are extended by polytopes within
directed cubes, which can be abstracted over using "extension types" that
generalize the path-types of cubical type theory. In an appendix, we describe
the motivating semantics in the Reedy model structure on bisimplicial sets, in
which our Segal and Rezk types correspond to Segal spaces and complete Segal
spaces.Comment: 78 pages; v2 has minor updates inspired by discussions at the
Mathematics Research Community on Homotopy Type Theory; v3 incorporates many
expository improvements suggested by the referee; v4 is the final journal
version to appear in Higher Structures with a more precise syntax for our
type theory with shape
Homological Region Adjacency Tree for a 3D Binary Digital Image via HSF Model
Given a 3D binary digital image I, we define and compute
an edge-weighted tree, called Homological Region Tree (or Hom-Tree,
for short). It coincides, as unweighted graph, with the classical Region
Adjacency Tree of black 6-connected components (CCs) and white 26-
connected components of I. In addition, we define the weight of an edge
(R, S) as the number of tunnels that the CCs R and S “share”. The
Hom-Tree structure is still an isotopic invariant of I. Thus, it provides
information about how the different homology groups interact between
them, while preserving the duality of black and white CCs.
An experimentation with a set of synthetic images showing different
shapes and different complexity of connected component nesting is performed
for numerically validating the method.Ministerio de Economía y Competitividad MTM2016-81030-
Internal Parametricity for Cubical Type Theory
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we call relativity. We demonstrate the use of the theory by analyzing polymorphic functions between higher inductive types, and we give an account of the identity extension lemma for internal parametricity
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
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