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https://github.com/HoTT/HoTT

A Coq library for Homotopy Type Theory
https://github.com/HoTT/HoTT

homotopy-type-theory type-theory univalent-foundations

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A Coq library for Homotopy Type Theory

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README 

        
[![Github Actions CI][1]][2]

[![HoTT Zulip chat][3]][4]



[Homotopy Type Theory][5] is an interpretation of

Martin-Löf’s intensional type theory into abstract homotopy theory.

Propositional equality is interpreted as homotopy and type isomorphism as

homotopy equivalence. Logical constructions in type theory then correspond to

homotopy-invariant constructions on spaces, while theorems and even proofs in

the logical system inherit a homotopical meaning. As the natural logic of

homotopy, type theory is also related to higher category theory as it is used

e.g. in the notion of a higher topos.



The HoTT library is a development of homotopy-theoretic ideas in the Coq proof

assistant. It draws many ideas from Vladimir Voevodsky's [Foundations][6]

library (which has since been incorporated into the [UniMath][7] library) and

also cross-pollinates with the [HoTT-Agda][8] library. See also: [HoTT in

Lean2][9], [Spectral Sequences in Lean2][10], and [Cubical Agda][11].



More information about this library can be found in:



- _The HoTT Library: A formalization of homotopy type theory in Coq_, Andrej

  Bauer, Jason Gross, Peter LeFanu Lumsdaine, Mike Shulman, Matthieu Sozeau, Bas

  Spitters, 2016 [arXiv][12] [CPP17][13]



Other publications related to the library can be found

[here][14].



# Installation



The HoTT library is part of the [Coq

Platform][15] and can be installed using

the installation instructions there.



More detailed installation instructions are provided in the file

[INSTALL.md](/INSTALL.md).



# Usage



The HoTT library can be used like any other Coq library. If you wish to use the

HoTT library in your own project, make sure to put the following arguments in

your `_CoqProject` file:



```

-arg -noinit

-arg -indices-matter

```



For more advanced use such as contribution see [INSTALL.md](/INSTALL.md).



We recommend the following text editors:



 * [Emacs][16] together with [Proof General][17].

 * [CoqIDE][18] part of the [Coq Proof Assistant][19].

 * [Visual Studio Code][20] together with [VSCoq][21].



Other methods of developing in `coq` will work as long as the correct arguments

are passed.



# Contributing



Contributions to the HoTT library are very welcome! For style guidelines and

further information, see the file [STYLE.md](/STYLE.md).



# Licensing



The library is released under the permissive BSD 2-clause license, see the file

[LICENSE.txt](/LICENSE.txt) for further information. In brief, this means you

can do whatever you like with it, as long as you preserve the Copyright

messages. And of course, no warranty!



# Wiki



More information can be found in the [Wiki][22].



[1]: https://github.com/HoTT/HoTT/workflows/CI/badge.svg?branch=master

[2]: https://github.com/HoTT/HoTT/actions?query=workflow%3ACI+branch%3Amaster

[3]: https://img.shields.io/badge/zulip-join_chat-brightgreen.svg

[4]: https://hott.zulipchat.com/



[5]: http://homotopytypetheory.org/



[6]: https://github.com/vladimirias/Foundations

[7]: https://github.com/UniMath/UniMath

[8]: https://github.com/HoTT/HoTT-Agda

[9]: https://github.com/leanprover/lean2/tree/master/hott

[10]: https://github.com/cmu-phil/Spectral

[11]: https://agda.readthedocs.io/en/v2.6.0.1/language/cubical.html



[12]: https://arxiv.org/abs/1610.04591

[13]: http://cpp2017.mpi-sws.org/

[14]: https://github.com/HoTT/HoTT/wiki/Publications-based-on-the-HoTT-library

[15]: https://github.com/coq/platform/releases



[16]: http://www.gnu.org/software/emacs/

[17]: http://proofgeneral.inf.ed.ac.uk

[18]: https://coq.inria.fr/refman/practical-tools/coqide.html

[19]: https://github.com/coq/coq

[20]: https://code.visualstudio.com/

[21]: https://github.com/coq-community/vscoq



[22]: https://github.com/HoTT/HoTT/wiki