Homotopy type theory
Coq Shell OCaml Makefile CSS M4 Other
Latest commit 378c6b1 Oct 17, 2016 @spitters spitters committed with JasonGross Updating the readme with lean, cubical and our paper (#835)
* updating readme with lean, cubical and our paper

* updating readme with lean, cubical and our paper
Failed to load latest commit information.
contrib Updating HoTTBook.v (through section 6.12) Jul 13, 2016
coq-HoTT @ e1661dc Bumped coq-HoTT submodule to 8.5pl2, pulling in a few bugfixes. Aug 15, 2016
coq Strip -no-native-compiler Apr 19, 2016
etc theories/{hit => HIT,categories => Categories} Jul 13, 2016
theories Merge pull request #824 from SkySkimmer/build-precat-typo Sep 16, 2016
.gitignore add more coq/ symlinks to .gitignore Sep 14, 2014
.gitmodules Move to trunk Jul 23, 2014
.mailmap Update .mailmap Jul 13, 2016
.travis.yml Also build HoTT with the tip of v8.5 and trunk Mar 2, 2015
CREDITS.txt Update CREDITS.txt Aug 26, 2013
INSTALL.md Update INSTALL.md to 8.5pl1 Apr 19, 2016
LICENSE.txt Added legaleze Oct 3, 2012
Makefile.am theories/{hit => HIT,categories => Categories} Jul 13, 2016
README.md Updating the readme with lean, cubical and our paper (#835) Oct 17, 2016
STYLE.md theories/{hit => HIT,categories => Categories} Jul 13, 2016
UNICODE.txt Add unicode instructions Aug 26, 2014
_CoqProject theories/{hit => HIT,categories => Categories} Jul 13, 2016
autogen.sh Added warning in [autogen.sh] that fallback to git may fail for old l… Aug 5, 2015
configure.ac Make checking for symlink support more robust Jun 23, 2015
hoq-config.in Strip -no-native-compiler Apr 19, 2016
hoqc Strip -no-native-compiler Apr 19, 2016
hoqdep Strip -no-native-compiler Apr 19, 2016
hoqide Strip -no-native-compiler Apr 19, 2016
hoqtop Strip -no-native-compiler Apr 19, 2016
hoqtop.byte Strip -no-native-compiler Apr 19, 2016


Build Status

Homotopy Type Theory 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 library (which has since been incorporated into the UniMath library) and also cross-pollinates with the HoTT-Agda library. Recently, there are also the Lean library and the cubical type checker.

More information about this libary 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


Installation details are explained in the file INSTALL.md.


It is possible to use the HoTT library directly on the command line with the hoqtop script, but who does that?

It is probably better to use Proof General and Emacs. When Proof General asks you where to find the coqtop executable, just point it to the hoqtop script. If Emacs runs a coqtop without asking, you should probably customize set the variable proof-prog-name-ask to nil (in Emacs type C-h v proof-prog-name-ask RET to see what this is about).

At the moment there is no hoqide equivalent of coqide, but getting one is high on our to-do list.


Contributions to the HoTT library are very welcome! For style guidelines and further information, see the file STYLE.md.


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