hott

I am sure it doesn't actually take HoTT, but I'm indoctrinated there right now mostly due to the accessibility of the HoTT book.

You can check Voevodsky's github page[1], and most of the other HoTT collaborators are active members as well.

That this also applies to identity types in HoTT is a sign that something is going right.

This is the same Voevodsky that came up with the univalence axiom[1] that is foundational in HoTT[2] (homotopy type theory)?

The novelty of HoTT, and only extension over intensional Martin-Loef type-theory, is the univalence axiom.

It turns out that proofs in HoTT are formally similar to certain aspects of geometry/topology.

Much of current research on HoTT is about the consequences of the univalence axiom and the similarity between logic and geometry/topology.

Kind of like what you might get if you took the intro to HoTT and lengthened it and made it more philosophical.

A top level language providing suitable abstractions is needed and HoTT is one proposal.

You can say that HoTT derives non-constructive mathematics on top of constructive foundations.

The key hope in all this is that HoTT will streamline formalised mathematics.

In the end, the current implementation of HoTT might not be the one standing the test of time; But the central idea behind the programme is here to stay: A refactoring of mathematics.

In HoTT, one can define (and reason about) the basic concepts of homotopy theory straight from "day one".

This "computational interpretation" is not unique to HoTT; in fact, this is what the linked nLab article is about.

Secondly, HoTT is the first foundation with which computer-formalized proofs of nontrivial theorems in a subject called "homotopy theory" are possible.

Thirdly, one intriguing aspect of HoTT is that it might be possible to "run" proofs (this is the objective of current research).