https://github.com/ayberkt/formal-topology-in-uf
Formal Topology in Univalent Foundations (WIP).
https://github.com/ayberkt/formal-topology-in-uf
constructive-topology formal-topologies homotopy-type-theory locale-theory pointless-topology topology univalent-foundations univalent-mathematics
Last synced: about 2 months ago
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Formal Topology in Univalent Foundations (WIP).
- Host: GitHub
- URL: https://github.com/ayberkt/formal-topology-in-uf
- Owner: ayberkt
- License: gpl-3.0
- Created: 2020-05-13T11:52:39.000Z (about 5 years ago)
- Default Branch: master
- Last Pushed: 2022-07-29T15:42:37.000Z (almost 3 years ago)
- Last Synced: 2023-04-06T15:37:26.891Z (about 2 years ago)
- Topics: constructive-topology, formal-topologies, homotopy-type-theory, locale-theory, pointless-topology, topology, univalent-foundations, univalent-mathematics
- Language: CSS
- Homepage:
- Size: 10.5 MB
- Stars: 29
- Watchers: 4
- Forks: 2
- Open Issues: 5
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# Formal Topology in Univalent Foundations
[**Link to thesis**](https://hdl.handle.net/20.500.12380/301098).
This is the Agda development accompanying my (upcoming) master's thesis at Chalmers
University of Technology to be titled _Formal Topology in Univalent Foundations_.
**Note**: This library is not actively maintained. The dependency `cubical`
needs to be checked out to commit `09cc7134082573cf82436f5d317405812856f7f6`.
For an actively developed version of the locale theory development here, see
the
[`Locales`](https://github.com/martinescardo/TypeTopology/tree/master/source/Locales)
in [`TypeTopology`](https://github.com/martinescardo/TypeTopology).
The approach to formal topology implemented here follows an idea of Thierry Coquand [0] to
define formal topologies as posets endowed with [interaction systems][2]. The main novelty
in this development is the definition of covering as an HIT. This seems to be necessary in
the context of univalent type theory to avoid using a form of the axiom of choice.
The version of the code presented in the thesis will be archived whereas this repository
(which is, as of now, mostly the same) will be maintained and developed further.
## Question: what is formal topology?
Here is an answer by Giovanni Sambin [1]:
> What is formal topology? A good approximation to the correct answer is: formal topology
> is topology as developed in (Martin-Löf's) type theory.
Also, [this blog post][6] by Mike Shulman contains interesting remarks about predicative
mathematics and formal topology.
## Overview
The main development comprises nine modules. If you are interested in reading the code, I
suggest the following order:
1. `Basis`. Basic definitions of univalent type theory, many of which are imported from
`agda/cubical` and some of which are adapted from Martín Escardó's
[introduction to HoTT/UF][4].
2. `Poset`.
3. `Frame`. A rudimentary development of [frames][5].
4. `Nucleus`. The notion of a [nucleus][3] on a frame.
5. `FormalTopology`. Definition of a formal topology as an interaction system.
6. `Cover`. The cover relation induced by the structure of a formal topology.
7. `CoverFormsNucleus`. Contains the proof that the cover relation of a formal topology is
a nucleus on the frame of downwards-closed subsets of its underlying poset.
8. `UniversalProperty`. Contains the proof that formal topologies present frames as
expected.
9. `CantorSpace`. The definition of the formal Cantor topology along with a proof that it
is compact.
10. `Sierpinski`. The formal topology of the Sierpinski space.
## Clickable HTML
A rendering of the code in glorious clickable HTML can be [accessed here][7].
## Credits
This work was carried out under the supervision of Thierry Coquand.
## References
0. Coquand, T. 1996. Formal Topology with Posets. http://www.cse.chalmers.se/~coquand/formal.html
1. Sambin, G. 2000. Formal Topology and Domains. Electronic Notes in Theoretical Computer Science. 35, (Jan. 2000), 177–190. DOI:https://doi.org/10.1016/S1571-0661(05)80742-X.
[2]: http://www.dcs.ed.ac.uk/home/pgh/interactive_systems.html
[3]: https://ncatlab.org/nlab/show/nucleus
[4]: https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html
[5]: https://ncatlab.org/nlab/show/frame
[6]: https://golem.ph.utexas.edu/category/2011/01/topos_theory_can_make_you_a_pr.html
[7]: https://ayberkt.github.io/formal-topology-in-UF/Main.html