323 lines
22 KiB
BibTeX
323 lines
22 KiB
BibTeX
|
||
@incollection{hatcher_spectral_2004,
|
||
title = {Spectral Sequences},
|
||
url = {https://pi.math.cornell.edu/~hatcher/AT/SSpage.html},
|
||
booktitle = {Algebraic Topology},
|
||
author = {Hatcher, Allen},
|
||
urldate = {2024-09-25},
|
||
date = {2004},
|
||
file = {ATch5.pdf:/Users/michael/Zotero/storage/EGAC34WS/ATch5.pdf:application/pdf},
|
||
}
|
||
|
||
@book{hatcher_algebraic_2001,
|
||
title = {Algebraic Topology},
|
||
url = {https://pi.math.cornell.edu/~hatcher/AT/ATpage.html},
|
||
author = {Hatcher, Allen},
|
||
urldate = {2024-09-25},
|
||
date = {2001},
|
||
file = {AT+.pdf:/Users/michael/Zotero/storage/89RKS7DH/AT+.pdf:application/pdf},
|
||
}
|
||
|
||
@misc{van_doorn_formalization_2018,
|
||
title = {On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory},
|
||
url = {http://arxiv.org/abs/1808.10690},
|
||
doi = {10.48550/arXiv.1808.10690},
|
||
abstract = {The goal of this dissertation is to present synthetic homotopy theory in the setting of homotopy type theory. We will present various results in this framework, most notably the construction of the Atiyah-Hirzebruch and Serre spectral sequences for cohomology, which have been fully formalized in the Lean proof assistant.},
|
||
number = {{arXiv}:1808.10690},
|
||
publisher = {{arXiv}},
|
||
author = {van Doorn, Floris},
|
||
urldate = {2024-09-25},
|
||
date = {2018-08-31},
|
||
eprinttype = {arxiv},
|
||
eprint = {1808.10690 [cs, math]},
|
||
keywords = {55T05 (Primary) 55T10, 55T25, 55P20, 03B15, 55U35 (Secondary), Computer Science - Logic in Computer Science, F.4.1, Mathematics - Algebraic Topology, Mathematics - Logic},
|
||
file = {arXiv Fulltext PDF:/Users/michael/Zotero/storage/3NZN52L9/van Doorn - 2018 - On the Formalization of Higher Inductive Types and.pdf:application/pdf;arXiv.org Snapshot:/Users/michael/Zotero/storage/G3XTTNC6/1808.html:text/html},
|
||
}
|
||
|
||
@misc{cohen_cubical_2016,
|
||
title = {Cubical Type Theory: a constructive interpretation of the univalence axiom},
|
||
url = {http://arxiv.org/abs/1611.02108},
|
||
doi = {10.48550/arXiv.1611.02108},
|
||
shorttitle = {Cubical Type Theory},
|
||
abstract = {This paper presents a type theory in which it is possible to directly manipulate \$n\$-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.},
|
||
number = {{arXiv}:1611.02108},
|
||
publisher = {{arXiv}},
|
||
author = {Cohen, Cyril and Coquand, Thierry and Huber, Simon and Mörtberg, Anders},
|
||
urldate = {2024-09-25},
|
||
date = {2016-11-07},
|
||
eprinttype = {arxiv},
|
||
eprint = {1611.02108 [cs, math]},
|
||
keywords = {Computer Science - Logic in Computer Science, F.4.1, Mathematics - Logic, F.3.2},
|
||
file = {arXiv Fulltext PDF:/Users/michael/Zotero/storage/PIFTAIIS/Cohen et al. - 2016 - Cubical Type Theory a constructive interpretation.pdf:application/pdf;arXiv.org Snapshot:/Users/michael/Zotero/storage/6JNE6NJS/1611.html:text/html},
|
||
}
|
||
|
||
@online{noauthor_cubical_nodate,
|
||
title = {Cubical — Agda 2.8.0 documentation},
|
||
url = {https://agda.readthedocs.io/en/latest/language/cubical.html},
|
||
urldate = {2024-09-25},
|
||
file = {Cubical — Agda 2.8.0 documentation:/Users/michael/Zotero/storage/R7TGJ3L9/cubical.html:text/html},
|
||
}
|
||
|
||
@article{vezzosi_cubical_2021,
|
||
title = {Cubical Agda: A dependently typed programming language with univalence and higher inductive types},
|
||
volume = {31},
|
||
issn = {0956-7968, 1469-7653},
|
||
url = {https://www.cambridge.org/core/product/identifier/S0956796821000034/type/journal_article},
|
||
doi = {10.1017/S0956796821000034},
|
||
shorttitle = {Cubical Agda},
|
||
abstract = {Abstract
|
||
Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types ({HITs}). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of {HITs}. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.},
|
||
pages = {e8},
|
||
journaltitle = {Journal of Functional Programming},
|
||
shortjournal = {J. Funct. Prog.},
|
||
author = {Vezzosi, Andrea and Mörtberg, Anders and Abel, Andreas},
|
||
urldate = {2024-09-25},
|
||
date = {2021},
|
||
langid = {english},
|
||
file = {Full Text:/Users/michael/Zotero/storage/7MTLCPWG/Vezzosi et al. - 2021 - Cubical Agda A dependently typed programming lang.pdf:application/pdf},
|
||
}
|
||
|
||
@book{the_univalent_foundations_program_homotopy_2013,
|
||
title = {Homotopy Type Theory: Univalent Foundations of Mathematics},
|
||
url = {https://homotopytypetheory.org/book},
|
||
author = {{The \{Univalent Foundations Program\}}},
|
||
date = {2013},
|
||
}
|
||
|
||
@video{hottest_floris_2018,
|
||
title = {Floris van Doorn, Towards spectral sequences for homology},
|
||
url = {https://www.youtube.com/watch?v=Q3zaqeKhUKg},
|
||
abstract = {Homotopy Type Theory Electronic Seminar Talks, 2018-11-22
|
||
|
||
Spectral sequences form a powerful tool which can be used to compute homotopy, homology and cohomology groups of a wide variety of spaces. We have constructed two important spectral sequences in homotopy type theory, the Atiyah-Hirzebruch and Serre spectral sequences for cohomology. These spectral sequences have analogues for homology, but they have not been constructed in {HoTT} yet. However, many parts of our construction could be reused to construct the corresponding spectral sequences for homology.
|
||
|
||
In this talk I will introduce spectral sequences and review the spectral sequences we have constructed and some of their applications. Furthermore, I will describe what parts are still missing to construct the Atiyah-Hirzebruch and Serre spectral sequences for homology.
|
||
|
||
The construction of the spectral sequences for cohomology is joint work with Jeremy Avigad, Steve Awodey, Ulrik Buchholtz, Egbert Rijke and Mike Shulman.},
|
||
author = {{HoTTEST}},
|
||
urldate = {2024-10-03},
|
||
date = {2018-11-23},
|
||
}
|
||
|
||
@article{lyons_elementary_nodate,
|
||
title = {An Elementary Introduction to the Hopf Fibration},
|
||
author = {Lyons, David W},
|
||
langid = {english},
|
||
file = {PDF:/Users/michael/Zotero/storage/MKUPPFCK/Lyons - An Elementary Introduction to the Hopf Fibration.pdf:application/pdf},
|
||
}
|
||
|
||
@misc{buchholtz_cayley-dickson_2016,
|
||
title = {The Cayley-Dickson Construction in Homotopy Type Theory},
|
||
url = {http://arxiv.org/abs/1610.01134},
|
||
abstract = {We define in the setting of homotopy type theory an H-space structure on \${\textbackslash}mathbb S{\textasciicircum}3\$. Hence we obtain a description of the quaternionic Hopf fibration \${\textbackslash}mathbb S{\textasciicircum}3{\textbackslash}hookrightarrow{\textbackslash}mathbb S{\textasciicircum}7{\textbackslash}twoheadrightarrow{\textbackslash}mathbb S{\textasciicircum}4\$, using only homotopy invariant tools.},
|
||
number = {{arXiv}:1610.01134},
|
||
publisher = {{arXiv}},
|
||
author = {Buchholtz, Ulrik and Rijke, Egbert},
|
||
urldate = {2024-10-17},
|
||
date = {2016-10-04},
|
||
eprinttype = {arxiv},
|
||
eprint = {1610.01134},
|
||
keywords = {Computer Science - Logic in Computer Science, Mathematics - Algebraic Topology, Mathematics - Logic, Mathematics - Category Theory},
|
||
file = {Preprint PDF:/Users/michael/Zotero/storage/Y39DL75F/Buchholtz and Rijke - 2016 - The Cayley-Dickson Construction in Homotopy Type T.pdf:application/pdf;Snapshot:/Users/michael/Zotero/storage/Q9HVDU7R/1610.html:text/html},
|
||
}
|
||
|
||
@online{kovacs_answer_2023,
|
||
title = {Answer to "What are the complex induction patterns supported by Agda?"},
|
||
url = {https://proofassistants.stackexchange.com/a/2002},
|
||
shorttitle = {Answer to "What are the complex induction patterns supported by Agda?},
|
||
titleaddon = {Proof Assistants Stack Exchange},
|
||
author = {Kovács, András},
|
||
urldate = {2024-10-16},
|
||
date = {2023-02-20},
|
||
file = {Snapshot:/Users/michael/Zotero/storage/RC9J6FMA/what-are-the-complex-induction-patterns-supported-by-agda.html:text/html},
|
||
}
|
||
|
||
@online{wood_answer_2024,
|
||
title = {Answer to "What are the principal differences between Agda's core type theory and Coq's?"},
|
||
url = {https://proofassistants.stackexchange.com/a/2740},
|
||
shorttitle = {Answer to "What are the principal differences between Agda's core type theory and Coq's?},
|
||
titleaddon = {Proof Assistants Stack Exchange},
|
||
author = {Wood, James},
|
||
urldate = {2024-10-16},
|
||
date = {2024-02-14},
|
||
file = {Snapshot:/Users/michael/Zotero/storage/FXDZCGPH/what-are-the-principal-differences-between-agdas-core-type-theory-and-coqs.html:text/html},
|
||
}
|
||
|
||
@online{noauthor_calculus_nodate,
|
||
title = {Calculus of Inductive Constructions — Coq 8.9.1 documentation},
|
||
url = {https://coq.inria.fr/doc/v8.9/refman/language/cic.html},
|
||
urldate = {2024-10-16},
|
||
file = {Calculus of Inductive Constructions — Coq 8.9.1 documentation:/Users/michael/Zotero/storage/C6LMRM5W/cic.html:text/html},
|
||
}
|
||
|
||
@incollection{goos_polymorphism_1984,
|
||
location = {Berlin, Heidelberg},
|
||
title = {Polymorphism is not set-theoretic},
|
||
volume = {173},
|
||
isbn = {978-3-540-13346-9 978-3-540-38891-3},
|
||
url = {http://link.springer.com/10.1007/3-540-13346-1_7},
|
||
pages = {145--156},
|
||
booktitle = {Semantics of Data Types},
|
||
publisher = {Springer Berlin Heidelberg},
|
||
author = {Reynolds, John C.},
|
||
editor = {Kahn, Gilles and {MacQueen}, David B. and Plotkin, Gordon},
|
||
editorb = {Goos, G. and Hartmanis, J. and Barstow, D. and Brauer, W. and Brinch Hansen, P. and Gries, D. and Luckham, D. and Moler, C. and Pnueli, A. and Seegmüller, G. and Stoer, J. and Wirth, N.},
|
||
editorbtype = {redactor},
|
||
urldate = {2024-10-16},
|
||
date = {1984},
|
||
doi = {10.1007/3-540-13346-1_7},
|
||
note = {Series Title: Lecture Notes in Computer Science},
|
||
file = {Reynolds - 1984 - Polymorphism is not set-theoretic.pdf:/Users/michael/Zotero/storage/72S4JJZM/Reynolds - 1984 - Polymorphism is not set-theoretic.pdf:application/pdf},
|
||
}
|
||
|
||
@article{chow_you_2006,
|
||
title = {You Could Have Invented Spectral Sequences},
|
||
volume = {53},
|
||
number = {1},
|
||
author = {Chow, Timothy Y},
|
||
date = {2006},
|
||
langid = {english},
|
||
file = {Chow - 2006 - You Could Have Invented Spectral Sequences.pdf:/Users/michael/Zotero/storage/89SGKENP/Chow - 2006 - You Could Have Invented Spectral Sequences.pdf:application/pdf},
|
||
}
|
||
|
||
@article{strong_introduction_nodate,
|
||
title = {Introduction to Spectral Sequences},
|
||
author = {Strong, Kimball},
|
||
langid = {english},
|
||
file = {Strong - Introduction to Spectral Sequences.pdf:/Users/michael/Zotero/storage/VPL9V4UW/Strong - Introduction to Spectral Sequences.pdf:application/pdf},
|
||
}
|
||
|
||
@online{noauthor_penrose_nodate,
|
||
title = {Penrose},
|
||
url = {https://penrose.cs.cmu.edu/},
|
||
urldate = {2024-11-05},
|
||
file = {Penrose:/Users/michael/Zotero/storage/67ZBXZU7/penrose.cs.cmu.edu.html:text/html},
|
||
}
|
||
|
||
@article{buchsbaum_exact_nodate,
|
||
title = {{EXACT} {CATEGORIES} {AND} {DUALITY}},
|
||
author = {Buchsbaum, D A},
|
||
langid = {english},
|
||
file = {PDF:/Users/michael/Zotero/storage/BIADJWY4/Buchsbaum - EXACT CATEGORIES AND DUALITY.pdf:application/pdf},
|
||
}
|
||
|
||
@inproceedings{licata_eilenberg-maclane_2014,
|
||
location = {Vienna Austria},
|
||
title = {Eilenberg-{MacLane} spaces in homotopy type theory},
|
||
isbn = {978-1-4503-2886-9},
|
||
url = {https://dl.acm.org/doi/10.1145/2603088.2603153},
|
||
doi = {10.1145/2603088.2603153},
|
||
abstract = {Homotopy type theory is an extension of Martin-Löf type theory with principles inspired by category theory and homotopy theory. With these extensions, type theory can be used to construct proofs of homotopy-theoretic theorems, in a way that is very amenable to computer-checked proofs in proof assistants such as Coq and Agda. In this paper, we give a computer-checked construction of Eilenberg-{MacLane} spaces. For an abelian group G, an {EilenbergMacLane} space K(G, n) is a space (type) whose nth homotopy group is G, and whose homotopy groups are trivial otherwise. These spaces are a basic tool in algebraic topology; for example, they can be used to build spaces with specified homotopy groups, and to define the notion of cohomology with coefficients in G. Their construction in type theory is an illustrative example, which ties together many of the constructions and methods that have been used in homotopy type theory so far.},
|
||
eventtitle = {{CSL}-{LICS} '14: {JOINT} {MEETING} {OF} the Twenty-Third {EACSL} Annual Conference on {COMPUTER} {SCIENCE} {LOGIC}},
|
||
pages = {1--9},
|
||
booktitle = {Proceedings of the Joint Meeting of the Twenty-Third {EACSL} Annual Conference on Computer Science Logic ({CSL}) and the Twenty-Ninth Annual {ACM}/{IEEE} Symposium on Logic in Computer Science ({LICS})},
|
||
publisher = {{ACM}},
|
||
author = {Licata, Daniel R. and Finster, Eric},
|
||
urldate = {2024-11-20},
|
||
date = {2014-07-14},
|
||
langid = {english},
|
||
file = {PDF:/Users/michael/Zotero/storage/W3YYYBQW/Licata and Finster - 2014 - Eilenberg-MacLane spaces in homotopy type theory.pdf:application/pdf},
|
||
}
|
||
|
||
@software{licata_dlicata335hott-agda_2024,
|
||
title = {dlicata335/hott-agda},
|
||
url = {https://github.com/dlicata335/hott-agda},
|
||
author = {Licata, Dan},
|
||
urldate = {2024-11-20},
|
||
date = {2024-07-08},
|
||
note = {original-date: 2012-02-29T23:25:09Z},
|
||
}
|
||
|
||
@online{noauthor_spectral_nodate,
|
||
title = {spectral sequences in homotopy type theory in {nLab}},
|
||
url = {https://ncatlab.org/nlab/show/spectral+sequences+in+homotopy+type+theory},
|
||
urldate = {2024-11-26},
|
||
file = {spectral sequences in homotopy type theory in nLab:/Users/michael/Zotero/storage/EDEHNJYF/spectral+sequences+in+homotopy+type+theory.html:text/html},
|
||
}
|
||
|
||
@article{yang_homotopy_nodate,
|
||
title = {Homotopy Groups of Spheres in Homotopy Type Theory},
|
||
author = {Yang, Jinghui},
|
||
langid = {english},
|
||
file = {PDF:/Users/michael/Zotero/storage/TWWEY6E9/Yang - Homotopy Groups of Spheres in Homotopy Type Theory.pdf:application/pdf},
|
||
}
|
||
|
||
@misc{ljungstrom_computational_2024,
|
||
title = {Computational Synthetic Cohomology Theory in Homotopy Type Theory},
|
||
url = {http://arxiv.org/abs/2401.16336},
|
||
doi = {10.48550/arXiv.2401.16336},
|
||
abstract = {This paper discusses the development of synthetic cohomology in Homotopy Type Theory ({HoTT}), as well as its computer formalisation. The objectives of this paper are (1) to generalise previous work on integral cohomology in {HoTT} by the current authors and Brunerie (2022) to cohomology with arbitrary coefficients and (2) to provide the mathematical details of, as well as extend, results underpinning the computer formalisation of cohomology rings by the current authors and Lamiaux (2023). With respect to objective (1), we provide new direct definitions of the cohomology group operations and of the cup product, which, just as in (Brunerie et al., 2022), enable significant simplifications of many earlier proofs in synthetic cohomology theory. In particular, the new definition of the cup product allows us to give the first complete formalisation of the axioms needed to turn the cohomology groups into a graded commutative ring. We also establish that this cohomology theory satisfies the {HoTT} formulation of the Eilenberg-Steenrod axioms for cohomology and study the classical Mayer-Vietoris and Gysin sequences. With respect to objective (2), we characterise the cohomology groups and rings of various spaces, including the spheres, torus, Klein bottle, real/complex projective planes, and infinite real projective space. All results have been formalised in Cubical Agda and we obtain multiple new numbers, similar to the famous `Brunerie number', which can be used as benchmarks for computational implementations of {HoTT}. Some of these numbers are infeasible to compute in Cubical Agda and hence provide new computational challenges and open problems which are much easier to define than the original Brunerie number.},
|
||
number = {{arXiv}:2401.16336},
|
||
publisher = {{arXiv}},
|
||
author = {Ljungström, Axel and Mörtberg, Anders},
|
||
urldate = {2025-01-08},
|
||
date = {2024-03-25},
|
||
eprinttype = {arxiv},
|
||
eprint = {2401.16336 [math]},
|
||
keywords = {Computer Science - Logic in Computer Science, Mathematics - Algebraic Topology},
|
||
file = {Preprint PDF:/Users/michael/Zotero/storage/ITW5YQ7P/Ljungström and Mörtberg - 2024 - Computational Synthetic Cohomology Theory in Homotopy Type Theory.pdf:application/pdf;Snapshot:/Users/michael/Zotero/storage/QX3LMQPF/2401.html:text/html},
|
||
}
|
||
|
||
@misc{ljungstrom_formalising_2024,
|
||
title = {Formalising and Computing the Fourth Homotopy Group of the \$3\$-Sphere in Cubical Agda},
|
||
url = {http://arxiv.org/abs/2302.00151},
|
||
doi = {10.48550/arXiv.2302.00151},
|
||
abstract = {Brunerie's 2016 {PhD} thesis contains the first synthetic proof in Homotopy Type Theory ({HoTT}) of the classical result that the fourth homotopy group of the 3-sphere is \${\textbackslash}mathbb\{Z\}/2{\textbackslash}mathbb\{Z\}\$. The proof is one of the most impressive pieces of synthetic homotopy theory to date and uses a lot of advanced classical algebraic topology rephrased synthetically. Furthermore, the proof is fully constructive and the main result can be reduced to the question of whether a particular "Brunerie number" \${\textbackslash}beta\$ can be normalised to \${\textbackslash}pm 2\$. The question of whether Brunerie's proof could be formalised in a proof assistant, either by computing this number or by formalising the pen-and-paper proof, has since remained open. In this paper, we present a complete formalisation in Cubical Agda. We do this by modifying Brunerie's proof so that a key technical result, whose proof Brunerie only sketched in his thesis, can be avoided. We also present a formalisation of a new and much simpler proof that \${\textbackslash}beta\$ is \${\textbackslash}pm 2\$. This formalisation provides us with a sequence of simpler Brunerie numbers, one of which normalises very quickly to \$-2\$ in Cubical Agda, resulting in a fully formalised computer-assisted proof that \${\textbackslash}pi\_4({\textbackslash}mathbb\{S\}{\textasciicircum}3) {\textbackslash}cong {\textbackslash}mathbb\{Z\}/2{\textbackslash}mathbb\{Z\}\$.},
|
||
number = {{arXiv}:2302.00151},
|
||
publisher = {{arXiv}},
|
||
author = {Ljungström, Axel and Mörtberg, Anders},
|
||
urldate = {2025-01-08},
|
||
date = {2024-04-30},
|
||
eprinttype = {arxiv},
|
||
eprint = {2302.00151 [math]},
|
||
keywords = {Computer Science - Logic in Computer Science, Mathematics - Algebraic Topology},
|
||
file = {Preprint PDF:/Users/michael/Zotero/storage/VGLRRI2M/Ljungström and Mörtberg - 2024 - Formalising and Computing the Fourth Homotopy Group of the \$3\$-Sphere in Cubical Agda.pdf:application/pdf;Snapshot:/Users/michael/Zotero/storage/4ECF3ZJD/2302.html:text/html},
|
||
}
|
||
|
||
@inproceedings{ljungstrom_formalizing_2023,
|
||
title = {Formalizing π4(S3) ≅Z/2Z and Computing a Brunerie Number in Cubical Agda},
|
||
url = {https://ieeexplore.ieee.org/document/10175833/?arnumber=10175833},
|
||
doi = {10.1109/LICS56636.2023.10175833},
|
||
abstract = {Brunerie’s 2016 {PhD} thesis contains the first synthetic proof in Homotopy Type Theory ({HoTT}) of the classical result that the fourth homotopy group of the 3-sphere is ℤ/2ℤ. The proof is one of the most impressive pieces of synthetic homotopy theory to date and uses a lot of advanced classical algebraic topology rephrased synthetically. Furthermore, the proof is fully constructive and the main result can be reduced to the question of whether a particular "Brunerie number" β can be normalized to ±2. The question of whether Brunerie’s proof could be formalized in a proof assistant, either by computing this number or by formalizing the pen-and-paper proof, has since remained open. In this paper, we present a complete formalization in Cubical Agda. We do this by modifying Brunerie’s proof so that a key technical result, whose proof Brunerie only sketched in his thesis, can be avoided. We also present a formalization of a new and much simpler proof that β is ±2. This formalization provides us with a sequence of simpler Brunerie numbers, one of which normalizes very quickly to −2 in Cubical Agda, resulting in a fully formalized computer-assisted proof that {\textbackslash}pi \_4({\textbackslash}{mathbbS}{\textasciicircum}3) {\textbackslash}cong {\textbackslash}{mathbbZ}/2{\textbackslash}{mathbbZ}.},
|
||
eventtitle = {2023 38th Annual {ACM}/{IEEE} Symposium on Logic in Computer Science ({LICS})},
|
||
pages = {1--13},
|
||
booktitle = {2023 38th Annual {ACM}/{IEEE} Symposium on Logic in Computer Science ({LICS})},
|
||
author = {Ljungström, Axel and Mörtberg, Anders},
|
||
urldate = {2025-01-09},
|
||
date = {2023-06},
|
||
keywords = {Algebra, Codes, Computer science, Libraries, Machinery, Stress, Topology},
|
||
file = {Full Text PDF:/Users/michael/Zotero/storage/VU3UKR5E/Ljungström and Mörtberg - 2023 - Formalizing π4(S3) ≅Z2Z and Computing a Brunerie Number in Cubical Agda.pdf:application/pdf;IEEE Xplore Abstract Record:/Users/michael/Zotero/storage/TE6RX488/10175833.html:text/html},
|
||
}
|
||
|
||
@software{the_agda_community_cubical_2024,
|
||
title = {Cubical Agda Library},
|
||
url = {https://github.com/agda/cubical},
|
||
abstract = {An experimental library for Cubical Agda},
|
||
version = {0.7},
|
||
author = {{The Agda Community}},
|
||
urldate = {2025-01-09},
|
||
date = {2024-02},
|
||
note = {original-date: 2018-10-15T09:28:28Z},
|
||
}
|
||
|
||
@software{agda_developers_agda_2025,
|
||
title = {Agda},
|
||
url = {https://agda.readthedocs.io/},
|
||
abstract = {Agda is a dependently typed programming language / interactive theorem prover.},
|
||
version = {2.8.0},
|
||
author = {{Agda Developers}},
|
||
urldate = {2025-01-14},
|
||
date = {2025-01-13},
|
||
note = {original-date: 2015-08-08T17:51:48Z},
|
||
}
|
||
|
||
@incollection{martin-lof_intuitionistic_1975,
|
||
title = {An Intuitionistic Theory of Types: Predicative Part},
|
||
shorttitle = {An Intuitionistic Theory of Types},
|
||
booktitle = {? Iterose1975},
|
||
publisher = {North Holland},
|
||
author = {Martin-Löf, Per},
|
||
editor = {Martin-Löf, Per},
|
||
date = {1975},
|
||
file = {An-Intuitionistic-Theory-of-Types-1972.pdf:/Users/michael/Zotero/storage/ICVD458G/An-Intuitionistic-Theory-of-Types-1972.pdf:application/pdf;Snapshot:/Users/michael/Zotero/storage/JLETE5WU/MARAIT-27.html:text/html},
|
||
}
|