Coquand’s cubical set model for homotopy type theory provides the basis for a computational interpretation of the univalence axiom and some higher inductive types, as implemented in the cubical proof assistant. This paper contributes to the understanding of this model. We make three contributions: 1. Johnstone’s topological topos was created to present the geometric realization of simplicial se...

Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky’s univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some...

The goal of this work is to establish a new algorithm for computing the cohomology ring of cubical complexes. The cubical structure enables an explicit recurrence formula for the cup product. We derive this formula and, next, show how to extend the Mrozek and Batko [7] homology coreduction algorithm to the cohomology ring structure. The implementation of the algorithm is a work in progress. Thi...

In this paper we describe a model of concurrency enjoying an algebraic structure reflecting the parallel composition. For the sake of simplicity we restrict to linear concurrent programs i.e. the ones with neither loops nor branchings. Such programs are given a semantics using cubical areas that we call geometric. The collection of all cubical areas admits a structure of tensor product in the c...

Betti numbers are topological invariants that count the number of holes of each dimension in a space. Cubical complexes are a class of CW complex whose cells are cubes of different dimensions such as points, segments, squares, cubes, etc. They are particularly useful for modeling structured data such as binary volumes. We introduce a fast and simple method for computing the Betti numbers of a t...

This is the third in a series of papers extending Martin-Löf’s meaning explanations of dependent type theory to a Cartesian cubical realizability framework that accounts for higherdimensional types. We extend this framework to include a cumulative hierarchy of univalent Kan universes of Kan types; exact equality and other pretypes lacking Kan structure; and a cumulative hierarchy of pretype uni...

In this paper we present an efficient framework for computation of persistent homology of cubical data in arbitrary dimensions. An existing algorithm using simplicial complexes is adapted to the setting of cubical complexes. The proposed approach enables efficient application of persistent homology in domains where the data is naturally given in a cubical form. By avoiding triangulation of the ...

This paper displays a closed model structure for the category of cubical sets and shows that the resulting homotopy category is equivalent to the ordinary homotopy category for topological spaces. The main results are Theorem 19, which gives the model structure, and Theorem 29 and Corollary 30 which together imply the equivalence of homotopy categories. The cofibrations and weak equivalences fo...

Algorithms for many geometric and physical algorithms rely on a decomposition of 3-D space. The cubical decomposition that is typically used can lead to costly communication overheads when implemented on multicomputers since each cubical cell is adjacent to, and may interact with, as many as 26 neighbouring cells. We explore an alternate decomposition based on truncated octahedra that, along wi...