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.
Installation details are explained in the file INSTALL.txt. You will need to compile a
custom version of Coq which supports the -relevant-equality and
-warn-universe-inconsistency command-line options. We hope to have these options pushed
into standard Coq.
If you are looking for an older version of HoTT which works with standard Coq, have a look
at the one tagged as pure-coq-8.3. Note however that we do not support the old
library anymore.
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 o 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.
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!