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https://github.com/martinescardo/TypeTopology

Logical manifestations of topological concepts, and other things, via the univalent point of view.
https://github.com/martinescardo/TypeTopology

agda compact-type constructive-mathematics homotopy-type-theory injective-type ordinal searchable-set totally-separated-type type-theory univalent-foundations univalent-mathematics

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Logical manifestations of topological concepts, and other things, via the univalent point of view.

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# Various new theorems in constructive univalent mathematics written in Agda



This development was started by Martin Escardo in 2010 as an `svn` project, and

transferred to `github` Monday 5th February 2018.



If you contribute, please add your full (legal or adopted) name and date

at the place of contribution.



An [html rendering of the Agda

code](http://www.cs.bham.ac.uk/~mhe/TypeTopology/index.html) is hosted at

[Martin Escardo](https://www.cs.bham.ac.uk/~mhe/index.html)'s institutional web

page.



## How to cite



You can use the following BibTeX entry to cite `TypeTopology`:



```bibtex

@misc{type-topology,

  title    = {{TypeTopology}},

  author   = {Escard\'{o}, Mart\'{i}n H. and {contributors}},

  url      = {https://github.com/martinescardo/TypeTopology},

  note     = {{Agda} development},

}

```



If you are citing only your own files, then create a different bibtex file with

only your name as author.



## Root of the development



 * [source/index.lagda](source/index.lagda) (only `--safe` modules).

 * [source/AllModulesIndex.lagda](source/AllModulesIndex.lagda) (including

   "unsafe" ones).

 * Each subdirectory in [source/](source/) has its own index file.



## Current contributors in alphabetical order of first name



Please add yourself the first time you contribute.



* Alice Laroche

* Andrew Sneap

* Andrew Swan

* Ayberk Tosun

* Brendan Hart

* Bruno Paiva

* Chuangjie Xu

* Cory Knapp

* Ettore Aldrovandi

* Fredrik Nordvall Forsberg

* Ian Ray

* Igor Arrieta (ii)

* Jon Sterling

* Kelton OBrien

* Keri D'Angelo

* Lane Biocini

* Marc Bezem

* Martin Escardo

* Nicolai Kraus

* Ohad Kammar

* Paul Levy (i)

* Paulo Oliva

* Peter Dybjer

* Simcha van Collem

* Thierry Coquand

* Todd Waugh Ambridge

* Tom de Jong

* Vincent Rahli



(i) These authors didn't write any single line of Agda code here, but

they contributed to constructions, theorems and proofs via the hands

of Martin Escardo.



(ii) These authors didn't write single line of Agda code here, but they

contributed to constructions, theorems and proofs via the hands of Ayberk

Tosun.



## Publications resulting from [`TypeTopology`]()



1. Martín H. Escardó. *Infinite sets that satisfy the principle of

   omniscience in any variety of constructive mathematics.* [The

   Journal of Symbolic

   Logic](https://www.cambridge.org/core/journals/journal-of-symbolic-logic),

   Volume 78 , Issue 3 , 2013 , pp. 764 - 784.



   https://doi.org/10.2178/jsl.7803040



1. Martín H. Escardó. *Continuity of Gödel's system T functionals via

   effectful forcing.* [Electronic Notes in Theoretical Computer

   Science](https://www.sciencedirect.com/journal/electronic-notes-in-theoretical-computer-science),

   Volume 298, 2013, Pages 119-141. [MFPS XXIX](https://www.cs.cornell.edu/Conferences/MFPS29/)



   https://doi.org/10.1016/j.entcs.2013.09.010



1. Nicolai Kraus, Martín H. Escardó, T. Coquand,

   T. Altenkirch. *Generalizations of Hedberg's Theorem.* In: Hasegawa,

   M. (eds) Typed Lambda Calculi and Applications. [TLCA

   2013](https://www.kurims.kyoto-u.ac.jp/tlca2013/). Lecture Notes in

   Computer Science, vol 7941. Springer.



   https://doi.org/10.1007/978-3-642-38946-7_14



1. Martín H. Escardó. *Constructive decidability of classical continuity.*

   [Mathematical Structures in Computer Science][MSCS], Volume 25, Special

   Issue 7: Computing with Infinite Data: Topological and Logical Foundations

   Part 1, October 2015, pp. 1578 - 1589 DOI:



   https://doi.org/10.1017/S096012951300042X



1. Martín H. Escardó and Chuangjie Xu. *The inconsistency of a

   Brouwerian continuity principle with the Curry-Howard

   interpretation.* [13th International Conference on Typed Lambda

   Calculi and Applications (TLCA 2015)](https://drops.dagstuhl.de/opus/portals/lipics/index.php?semnr=15006).



   https://doi.org/10.4230/LIPIcs.TLCA.2015.153



1. Martín H. Escardó and T. Streicher. *The intrinsic topology of

   Martin-Löf universes.* [Annals of Pure and Applied

   Logic](https://www.sciencedirect.com/journal/annals-of-pure-and-applied-logic),

   Volume 167, Issue 9, 2016, Pages 794-805.



   https://doi.org/10.1016/j.apal.2016.04.010



1. Martín H. Escardó and Cory Knapp. *Partial elements and recursion via

   dominances in univalent type theory.* [Leibniz International Proceedings in

   Informatics

   (LIPIcs)][LIPICS],

   Proceedings of [CSL

   2017](https://www.math-stockholm.se/konferenser-och-akti/logic-in-stockholm-2/26th-eacsl-annual-co/computer-science-logic-2017-august-20-24-1.717663).



   https://doi.org/10.4230/LIPIcs.CSL.2017.21



1. Nicolai Kraus, Martín H. Escardó, T. Coquand, T. Altenkirch. *Notions of

   Anonymous Existence in Martin-Löf Type Theory.*  [Logical Methods in

   Computer Science][LMCS], March 24, 2017, Volume 13, Issue 1.



   https://doi.org/10.23638/LMCS-13(1:15)2017



1. Tom de Jong. *The Scott model of PCF in univalent type

   theory.* [Mathematical Structures in Computer

   Science](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science),

   Volume 31, Issue 10 - Homotopy Type Theory 2019, July 2021.



   https://doi.org/10.1017/S0960129521000153



1. Martín H. Escardó. *The Cantor-Schröder-Bernstein Theorem for

   ∞-groupoids.*  [Journal of Homotopy and Related

   Structures](https://tcms.org.ge/Journals/JHRS/), 16(3), 363-366,

   2021.



   https://doi.org/10.1007/s40062-021-00284-6



1. Martín H. Escardó.  *Injective types in univalent

   mathematics.* [Mathematical Structures in Computer

   Science](https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science),

   Volume 31 , Issue 1 , 2021 , pp. 89 - 111.



   https://doi.org/10.1017/S0960129520000225



1. Tom de Jong and Martín H. Escardó. *Domain Theory in Constructive

   and Predicative Univalent Foundations.* [Leibniz International

   Proceedings in Informatics

   (LIPIcs)](https://www.dagstuhl.de/en/publishing/series/details/LIPIcs),

   Volume 183 - Proceedings of [CSL 2021][CSL21], January

   2021.



   https://doi.org/10.4230/LIPIcs.CSL.2021.28



1. Dan R. Ghica and Todd Waugh Ambridge. *Global Optimisation with

   Constructive Reals.*

   [Logic in Computer Science (LICS)](https://dl.acm.org/conference/lics),

   Proceedings of [LICS 2021][LICS21], June 2021.



   https://doi.org/10.1109/LICS52264.2021.9470549



1. Tom de Jong and Martín H. Escardó. *Predicative Aspects of Order

   Theory in Univalent Foundations.* [Leibniz International Proceedings

   in Informatics

   (LIPIcs)](https://www.dagstuhl.de/en/publishing/series/details/LIPIcs),

   Volume 195 - Proceedings of [FSCD 2021][FSCD21], July 2021.



   https://doi.org/10.4230/LIPIcs.FSCD.2021.8



1. Tom de Jong. *Domain Theory in Constructive and Predicative Univalent

   Foundations*. PhD thesis. School of Computer Science, University of

   Birmingham, UK. Submitted: 30 September 2022; accepted: 1 February 2023.



   https://etheses.bham.ac.uk/id/eprint/13401/ \

   Updated versions: \

   https://arxiv.org/abs/2301.12405 \

   https://tdejong.com/writings/phd-thesis.pdf



1. Ayberk Tosun and Martín H. Escardó. *Patch Locale of a Spectral

   Locale in Univalent Type Theory.* [Electronic Notes in Theoretical

   Informatics and Computer Science](https://entics.episciences.org/),

   Volume 1 - Proceedings of [MFPS XXXVIII][MFPS38], February

   2023.



   https://doi.org/10.46298/entics.10808



1. Tom de Jong, Nicolai Kraus, Fredrik Nordvall Forsberg and Chuangjie

   Xu. *Set-Theoretic and Type-Theoretic Ordinals Coincide.*

   [Logic in Computer Science (LICS)](https://dl.acm.org/conference/lics),

   Proceedings of [LICS 2023][LICS23]. June 2023.



   https://doi.org/10.1109/LICS56636.2023.10175762



   Publicly available at https://arxiv.org/abs/2301.10696.



1. Tom de Jong and Martín H. Escardó. *On Small Types in Univalent

   Foundations.* [Logical Methods in Computer

   Science](https://lmcs.episciences.org/), Volume 19, Issue 2, May

   2023.



   https://doi.org/10.46298/lmcs-19(2:8)2023



1. Martín H. Escardó and Paulo Oliva. *Higher-order Games with

   Dependent Types*.



   [Theoretical Computer

   Science](https://www.sciencedirect.com/journal/theoretical-computer-science),

   Special issue "Continuity, Computability, Constructivity: From

   Logic to Algorithms", dedicated to Ulrich Berger's 65th birthday,

   volume 974, 29 September 2023, available online 2 August 2023.



   https://doi.org/10.1016/j.tcs.2023.114111



1. Todd Waugh Ambridge. *Exact Real Search: Formalised Optimisation and

   Regression in Constructive Univalent Mathematics.* January 2024. University

   of Birmingham. PhD thesis.



   https://doi.org/10.48550/arXiv.2401.09270



1. Igor Arrieta, Martín H. Escardó and Ayberk Tosun. *The Patch Topology in

   Univalent Foundations*.



   ArXiv preprint. Available online 5 February 2024.



   https://arxiv.org/abs/2402.03134



1. Tom de Jong. *Domain theory in univalent foundations I: Directed complete

   posets and Scott's D∞*. July 2024.



   https://doi.org/10.48550/arxiv.2407.06952



1. Tom de Jong and Martín H. Escardó. *Domain theory in univalent

   foundations II: Continuous and algebraic domains*. July 2024.



   https://doi.org/10.48550/arxiv.2407.06956



[CSL21]:  https://csl2021.fmf.uni-lj.si/

[FSCD21]: https://fscd2021.dc.uba.ar/

[LICS21]: https://easyconferences.eu/lics2021/

[LICS23]: https://lics.siglog.org/lics23/

[MFPS38]: https://www.cs.cornell.edu/mfps-2022/

[LMCS]: https://lmcs.episciences.org/

[MSCS]: https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science

[LIPICS]: https://www.dagstuhl.de/en/publishing/series/details/LIPIcs