Takeaway

HoTT interprets types as spaces and equalities as paths; the univalence axiom identifies equivalent types, enabling structural reasoning and machine-checked proofs.

The problem (before → after)

  • Before: Set-theoretic equality can be too rigid for structural mathematics.
  • After: Treat isomorphic structures as equal up to equivalence; computation and proofs coexist in a constructive framework.

Mental model first

An equality proof is a path deforming one point into another; two types are the same when there’s a path (equivalence) connecting them in the “universe” of types.

Just-in-time concepts

  • Types-as-spaces; terms-as-points; paths-as-equalities.
  • Higher inductive types define spaces with points, paths, and higher paths.
  • Univalence: equivalences correspond to equalities of types.

First-pass solution

Develop mathematics inside type theory; use higher inductive types for quotients and tori; reason up to equivalence via univalence; extract programs from proofs.

Iterative refinement

  1. Cubical type theory gives computational content to univalence.
  2. Formalization in Coq/Agda/Lean.
  3. Synthetic homotopy and category theory in HoTT.

Principles, not prescriptions

  • Structure, not representation; proofs are paths.
  • Computation and proof are unified.

Common pitfalls

  • Translating set-based intuition directly.
  • Managing higher path algebra without tooling.

Connections and contrasts

  • See also: category theory, type theory, formal verification.

Quick checks

  1. What is univalence? — Equivalence implies equality of types.
  2. Why higher paths? — Capture equalities between equalities.
  3. Why constructive? — Proofs can compute.

Further reading

  • HoTT Book (source above); Cubical TT papers