Homotopy Type Theory (HoTT): The Mathematics of Paths and Types

Back when the GenAI hype was just getting started, I ran an experiment: an entire video script written by ChatGPT, covering one of the most challenging areas of modern mathematics — homotopy type theory. The conclusion was predictable — generative AI still struggles to explain things like this. The result was too shallow and abstract, and most viewers didn’t connect with it. Still, the experiment was worth running. And a huge thanks to Wild Mathing for the ending!

Someday I’ll have to write that script the old-fashioned way. In the meantime, here’s a reminder about this important theory — one that sits at the foundation of the mathematics of the future.

What Is Homotopy Type Theory?

Homotopy type theory (HoTT) is a framework that unifies mathematics and computer science. In HoTT, mathematical objects are treated as “shapes” connected by paths that can be “traversed” in different ways. It goes beyond the simple equality of numbers: the idea of choosing a path emerges — much like taking different routes on a map.

A Brief History of Homotopy Type Theory

The idea that types could be treated as homotopic structures — shapes with paths — first emerged in the mid-twentieth century. In the 1990s, researchers began formulating how groupoids (models that account for multiple distinct paths) could be applied within type theory. The real breakthrough came in the early 2000s, when scientists including Vladimir Voevodsky started applying HoTT to concrete problems.

The pivotal development was the introduction of the Univalence Axiom, which states that “equivalent types can be considered identical.” This became a cornerstone for an entirely new wave of mathematical research.

How Homotopy Type Theory Is Applied in Practice

Imagine writing a program to prove a theorem. In traditional mathematics, you would need to spell out every step explicitly, and any deviation would require building new proofs from scratch. In HoTT, you can “travel” along different paths, simplifying the proof process — and a computer program can understand those paths and verify proofs automatically.

This makes mathematics more intuitive and opens up new possibilities for tackling complex theorems that would otherwise be intractable.

Why Homotopy Type Theory Matters

HoTT makes mathematics more flexible and enables computers to participate meaningfully in verifying proofs. It opens the door to building systems where precision and reliability are critical — and brings us closer to a future where mathematics and technology work together more seamlessly than ever before.