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A Cubical Approach to Synthetic Homotopy Theory

Daniel R. Licata, Guillaume Brunerie
2015 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science  
In this paper, we describe a cubical approach to developing homotopy theory within type theory.  ...  Homotopy theory can be developed synthetically in homotopy type theory, using types to describe spaces, the identity type to describe paths in a space, and iterated identity types to describe higher-dimensional  ...  Acknowledgments We thank Joseph Lee, who attempted to prove the torus-circles equivalence with the first author in summer 2012.  ... 
doi:10.1109/lics.2015.19 dblp:conf/lics/LicataB15 fatcat:vqpgtlx3wbdsvaodmvivex2e7a

Homotopy Type Theory in Lean [chapter]

Floris van Doorn, Jakob von Raumer, Ulrik Buchholtz
2017 Lecture Notes in Computer Science  
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.  ...  Lastly, we want to thank all contributors to the HoTT library and the Spectral repository, most notably Egbert Rijke and Mike Shulman.  ... 
doi:10.1007/978-3-319-66107-0_30 fatcat:vhywj5fvy5hgzowlzj4ja6ym3y

The RedPRL Proof Assistant (Invited Paper)

Carlo Angiuli, Evan Cavallo, Kuen-Bang Hou (Favonia), Robert Harper, Jonathan Sterling
2018 Electronic Proceedings in Theoretical Computer Science  
RedPRL is an experimental proof assistant based on Cartesian cubical computational type theory, a new type theory for higher-dimensional constructions inspired by homotopy type theory.  ...  Notably, RedPRL implements a two-level type theory, allowing an extensional, proof-irrelevant notion of exact equality to coexist with a higher-dimensional proof-relevant notion of paths.  ...  Acknowledgements We would like to thank everyone who has contributed to the RedPRL project and its predecessors, including Eugene Akentyev, Tim Baumann, David Thrane Christiansen, Daniel Gratzer, Darin  ... 
doi:10.4204/eptcs.274.1 fatcat:bycs5desrzfatki2mfj66ff5wu

Cartesian Cubical Computational Type Theory: Constructive Reasoning with Paths and Equalities

Carlo Angiuli, Hou (Favonia), Kuen-Bang, Robert Harper, Michael Wagner
2018 Annual Conference for Computer Science Logic  
Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate  ...  We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types.  ...  Acknowledgements We are greatly indebted to Steve Awodey, Marc Bezem, Evan Cavallo, Daniel Gratzer, Simon Huber, Dan Licata, Ed Morehouse, Anders Mörtberg, Andrew Pitts, Jonathan Sterling, and Todd Wilson  ... 
doi:10.4230/lipics.csl.2018.6 dblp:conf/csl/AngiuliF018 fatcat:56hfopefhfghdk6ty2zyy54hq4

Higher Structures in Homotopy Type Theory [article]

Ulrik Buchholtz
2018 arXiv   pre-print
The intended model of the homotopy type theories used in Univalent Foundations is the infinity-category of homotopy types, also known as infinity-groupoids.  ...  theories in Univalent Foundations.  ...  Homotopy type theories are synthetic theories of ∞-groupoids.  ... 
arXiv:1807.02177v1 fatcat:lgrt5uxqpvhanjj6pmw7ey7umm

Pregeometric Spaces from Wolfram Model Rewriting Systems as Homotopy Types [article]

Xerxes D. Arsiwalla, Jonathan Gorard
2021 arXiv   pre-print
A key issue we have addressed here is to relate abstract non-deterministic rewriting systems to higher homotopy spaces.  ...  A consequence of constructing spaces and geometry synthetically is that it eliminates ad hoc assumptions about geometric attributes of a model such as an a priori background or pre-assigned geometric data  ...  Thanks also to Amar Hadzihasanovic, Nils A Baas, Nicolas Behr and Hatem Elshatlawy for useful feedback.  ... 
arXiv:2111.03460v2 fatcat:ckmayfudijgmvil4tua6hfjpmi

Homotopy Type Theory in Isabelle [article]

Joshua Chen
2021 arXiv   pre-print
The infrastructure developed is then used to formalize foundational results from the Homotopy Type Theory book.  ...  This paper introduces Isabelle/HoTT, the first development of homotopy type theory in the Isabelle proof assistant.  ...  In addition, although the logic presented here is formulated in the axiomatic style of the HoTT book, one could use the same approach to develop two-level type theory [1, 3] and cubical type theory  ... 
arXiv:2002.09282v2 fatcat:cgaeh2o2jrhlrb37ofqayl4pre

Varieties of Cubical Sets [article]

Ulrik Buchholtz, Edward Morehouse
2017 arXiv   pre-print
We define a variety of notions of cubical sets, based on sites organized using substructural algebraic theories presenting PRO(P)s or Lawvere theories.  ...  We delineate exactly which ones are even strict test categories, meaning that products of cubical sets correspond to products of homotopy types.  ...  to a given class of models, decidability of equality of terms, existence of canonical forms) and its homotopy-theoretic characteristics, most importantly, that the synthetic homotopy theory to which it  ... 
arXiv:1701.08189v2 fatcat:eln74dav6fanvdkj4t6fcbhhby

The HoTT Library: A formalization of homotopy type theory in Coq [article]

Andrej Bauer, Jason Gross, Peter LeFanu Lumsdaine, Mike Shulman, Matthieu Sozeau, Bas Spitters
2016 arXiv   pre-print
It formalizes most of basic homotopy type theory, including univalence, higher inductive types, and significant amounts of synthetic homotopy theory, as well as category theory and modalities.  ...  We report on the development of the HoTT library, a formalization of homotopy type theory in the Coq proof assistant.  ...  Synthetic Homotopy theory The library also contains a variety of other de nitions and results, many relevant to synthetic homotopy theory or higher category theory.  ... 
arXiv:1610.04591v2 fatcat:hej2265co5dndciufqrpa43ltq

Cubical Assemblies and the Independence of the Propositional Resizing Axiom [article]

Taichi Uemura
2018 arXiv   pre-print
We construct a model of cubical type theory with a univalent and impredicative universe in a category of cubical assemblies.  ...  We show that the cubical assembly model does not satisfy the propositional resizing axiom.  ...  This work is part of the research programme "The Computational Content of Homotopy Type Theory" with project number 613.001.602, which is financed by the Netherlands Organisation for Scientific Research  ... 
arXiv:1803.06649v2 fatcat:pij6ffctlbgk5p446nad3itecy

IN MEMORIAM: VLADIMIR VOEVODSKY 1966–2017

Daniel R. Grayson
2018 Bulletin of Symbolic Logic  
A new field, synthetic homotopy theory, proves analogues of theorems of homotopy theory and algebraic topology in this landscape, including the computation of some homotopy groups of spheres, such as n  ...  Eventually formalizing his work in motivic homotopy theory would be a fitting memorial to Vladimir.  ... 
doi:10.1017/bsl.2018.20 fatcat:cp5qo6vdzrac3a4goybp247esm

Computational higher-dimensional type theory

Carlo Angiuli, Robert Harper, Todd Wilson
2017 SIGPLAN notices  
From a computer science perspective, interest in type theory arises from its applications to programming languages.  ...  Formal constructive type theory has proved to be an effective language for mechanized proof.  ...  Cubical methods in type theory have since been used in applications ranging from synthetic homotopy theory to guarded recursion [9, 10, 24, 25] .  ... 
doi:10.1145/3093333.3009861 fatcat:xgwzynahwzguxofci4s43wmz3y

Computational higher-dimensional type theory

Carlo Angiuli, Robert Harper, Todd Wilson
2017 Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages - POPL 2017  
From a computer science perspective, interest in type theory arises from its applications to programming languages.  ...  Formal constructive type theory has proved to be an effective language for mechanized proof.  ...  Cubical methods in type theory have since been used in applications ranging from synthetic homotopy theory to guarded recursion [9, 10, 24, 25] .  ... 
doi:10.1145/3009837.3009861 fatcat:mczi2nwjrvcabhbqxs5rbnbfbe

The real projective spaces in homotopy type theory [article]

Ulrik Buchholtz, Egbert Rijke
2017 arXiv   pre-print
Homotopy type theory is a version of Martin-L\"of type theory taking advantage of its homotopical models.  ...  The classical definition of RP(n), as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory.  ...  Homotopy type theory allows us to reason synthetically about the objects of algebraic topology (spaces, paths, homotopies, etc.) analogously to how the setting of Euclidean geometry allows us to reason  ... 
arXiv:1704.05770v1 fatcat:3koprjbypfbv7dg4gjqldnxzhu

Internal Universes in Models of Homotopy Type Theory [article]

Daniel R. Licata, Ian Orton, Andrew M. Pitts, Bas Spitters
2018 arXiv   pre-print
This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory.  ...  is tiny - a property that the interval in cubical sets does indeed have.  ...  One approach to validating these universe axioms would be to check them directly in a cubical set model; but we can in fact do more work using crisp type theory as the internal language and reduce the  ... 
arXiv:1801.07664v3 fatcat:eolkydqwfjedjloq3rtephble4
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