4 Search Results for "Brunerie, Guillaume"

Document

DOI: 10.4230/LIPIcs.TYPES.2023.1

Classification of Covering Spaces and Canonical Change of Basepoint

Authors: Jelle Wemmenhove, Cosmin Manea, and Jim Portegies

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract

Using the language of homotopy type theory (HoTT), we 1) prove a synthetic version of the classification theorem for covering spaces, and 2) explore the existence of canonical change-of-basepoint isomorphisms between homotopy groups. There is some freedom in choosing how to translate concepts from classical algebraic topology into HoTT. The final translations we ended up with are easier to work with than the ones we started with. We discuss some earlier attempts to shed light on this translation process. The proofs are mechanized using the Coq proof assistant and closely follow classical treatments like those by Hatcher [Allen Hatcher, 2002].

Cite as

Jelle Wemmenhove, Cosmin Manea, and Jim Portegies. Classification of Covering Spaces and Canonical Change of Basepoint. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 1:1-1:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{wemmenhove_et_al:LIPIcs.TYPES.2023.1,
  author =	{Wemmenhove, Jelle and Manea, Cosmin and Portegies, Jim},
  title =	{{Classification of Covering Spaces and Canonical Change of Basepoint}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{1:1--1:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.1},
  URN =		{urn:nbn:de:0030-drops-204795},
  doi =		{10.4230/LIPIcs.TYPES.2023.1},
  annote =	{Keywords: Synthetic Homotopy Theory, Homotopy Type Theory, Covering Spaces, Change-of-Basepoint Isomorphism}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.6

Delooping Generated Groups in Homotopy Type Theory

Authors: Camil Champin, Samuel Mimram, and Émile Oleon

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Homotopy type theory is a logical setting based on Martin-Löf type theory in which one can perform geometric constructions and proofs in a synthetic way. Namely, types can be interpreted as spaces (up to continuous deformation) and proofs as homotopy invariant constructions. In this context, loop spaces of pointed connected groupoids provide a natural representation of groups, and any group can be obtained as the loop space of such a type, which is then called a delooping of the group. There are two main methods to construct the delooping of an arbitrary group G. The first one consists in describing it as a pointed higher inductive type, whereas the second one consists in taking the connected component of the principal G-torsor in the type of sets equipped with an action of G. We show here that, when a presentation is known for the group, simpler variants of those constructions can be used to build deloopings. The resulting types are more amenable to computations and lead to simpler meta-theoretic reasoning. We also investigate, in this context, an abstract construction for the Cayley graph of a generated group and show that it encodes the relations of the group. Most of the developments performed in the article have been formalized using the cubical version of the Agda proof assistant.

Cite as

Camil Champin, Samuel Mimram, and Émile Oleon. Delooping Generated Groups in Homotopy Type Theory. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{champin_et_al:LIPIcs.FSCD.2024.6,
  author =	{Champin, Camil and Mimram, Samuel and Oleon, \'{E}mile},
  title =	{{Delooping Generated Groups in Homotopy Type Theory}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{6:1--6:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.6},
  URN =		{urn:nbn:de:0030-drops-203356},
  doi =		{10.4230/LIPIcs.FSCD.2024.6},
  annote =	{Keywords: homotopy type theory, delooping, group, generator, Cayley graph}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.22

Automating Boundary Filling in Cubical Agda

Authors: Maximilian Doré, Evan Cavallo, and Anders Mörtberg

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

When working in a proof assistant, automation is key to discharging routine proof goals such as equations between algebraic expressions. Homotopy Type Theory allows the user to reason about higher structures, such as topological spaces, using higher inductive types (HITs) and univalence. Cubical Agda is an extension of Agda with computational support for HITs and univalence. A difficulty when working in Cubical Agda is dealing with the complex combinatorics of higher structures, an infinite-dimensional generalisation of equational reasoning. To solve these higher-dimensional equations consists in constructing cubes with specified boundaries. We develop a simplified cubical language in which we isolate and study two automation problems: contortion solving, where we attempt to "contort" a cube to fit a given boundary, and the more general Kan solving, where we search for solutions that involve pasting multiple cubes together. Both problems are difficult in the general case - Kan solving is even undecidable - so we focus on heuristics that perform well on practical examples. We provide a solver for the contortion problem using a reformulation of contortions in terms of poset maps, while we solve Kan problems using constraint satisfaction programming. We have implemented our algorithms in an experimental Haskell solver that can be used to automatically solve goals presented by Cubical Agda. We illustrate this with a case study establishing the Eckmann-Hilton theorem using our solver, as well as various benchmarks - providing the ground for further study of proof automation in cubical type theories.

Cite as

Maximilian Doré, Evan Cavallo, and Anders Mörtberg. Automating Boundary Filling in Cubical Agda. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{dore_et_al:LIPIcs.FSCD.2024.22,
  author =	{Dor\'{e}, Maximilian and Cavallo, Evan and M\"{o}rtberg, Anders},
  title =	{{Automating Boundary Filling in Cubical Agda}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{22:1--22:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.22},
  URN =		{urn:nbn:de:0030-drops-203514},
  doi =		{10.4230/LIPIcs.FSCD.2024.22},
  annote =	{Keywords: Cubical Agda, Automated Reasoning, Constraint Satisfaction Programming}
}

Document

DOI: 10.4230/LIPIcs.CSL.2022.11

Synthetic Integral Cohomology in Cubical Agda

Authors: Guillaume Brunerie, Axel Ljungström, and Anders Mörtberg

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract

This paper discusses the formalization of synthetic cohomology theory in a cubical extension of Agda which natively supports univalence and higher inductive types. This enables significant simplifications of many proofs from Homotopy Type Theory and Univalent Foundations as steps that used to require long calculations now hold simply by computation. To this end, we give a new definition of the group structure for cohomology with ℤ-coefficients, optimized for efficient computations. We also invent an optimized definition of the cup product which allows us to give the first complete formalization of the axioms needed to turn the integral cohomology groups into a graded commutative ring. Using this, we characterize the cohomology groups of the spheres, torus, Klein bottle and real/complex projective planes. As all proofs are constructive we can then use Cubical Agda to distinguish between spaces by computation.

Cite as

Guillaume Brunerie, Axel Ljungström, and Anders Mörtberg. Synthetic Integral Cohomology in Cubical Agda. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{brunerie_et_al:LIPIcs.CSL.2022.11,
  author =	{Brunerie, Guillaume and Ljungstr\"{o}m, Axel and M\"{o}rtberg, Anders},
  title =	{{Synthetic Integral Cohomology in Cubical Agda}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.11},
  URN =		{urn:nbn:de:0030-drops-157310},
  doi =		{10.4230/LIPIcs.CSL.2022.11},
  annote =	{Keywords: Synthetic Homotopy Theory, Cohomology Theory, Cubical Agda}
}

Refine by Author
2 Mörtberg, Anders
1 Brunerie, Guillaume
1 Cavallo, Evan
1 Champin, Camil
1 Doré, Maximilian
Show More...

Refine by Classification
3 Theory of computation → Constructive mathematics
3 Theory of computation → Type theory
1 Computing methodologies → Theorem proving algorithms
1 Mathematics of computing → Algebraic topology
1 Theory of computation → Automated reasoning
Show More...

Refine by Keyword
2 Cubical Agda
2 Synthetic Homotopy Theory
1 Automated Reasoning
1 Cayley graph
1 Change-of-Basepoint Isomorphism
Show More...

Refine by Type
4 document

Refine by Publication Year
3 2024
1 2022