Internal Language, Univalence, and the HoTT¶
Job: develop the internal language of \(\infty\)-topoi, the univalent universe / object classifier, the synthetic homotopy, and the cubical / computational — with the univalent \(\mathcal{U}\) / HoTT as the worked example.
Core¶
The internal language of ∞-topoi¶
| Internal language / HoTT / ∞-topos | Description |
|---|---|
| Internal language | the HoTT the internal language of the \(\infty\)-topos (the HoTT) |
| HoTT / ∞-topos | the HoTT, the \(\infty\)-topos the internal |
The univalence and the object classifier¶
| Univalence / \(\mathcal{U}\) / object classifier | Description |
|---|---|
| Univalence | \(\mathcal{U}\) the univalent universe: \((A\simeq B)\to(\mathcal{U}(A)=\mathcal{U}(B))\) (the HoTT) |
| Object classifier | \(\mathcal{U}\to\mathcal{U}\) the universal bundle, maps \(X\to\mathcal{U}\) classify bundles \(E\to X\) (the foundations) |
The synthetic homotopy and the ∞-topos¶
| Synthetic / HoTT / ∞-topos | Description |
|---|---|
| Synthetic homotopy | the HoTT the synthetic (the HoTT) |
| Synthetic / ∞-topos | the HoTT, the \(\infty\)-topos the synthetic |
The cubical and the computational¶
| Cubical / computational / HoTT | Description |
|---|---|
| Cubical type theory | the computational (the HoTT) |
| Cubical / ∞-topos | the HoTT, the \(\infty\)-topos the cubical |
See it¶
Univalent universe \(\mathcal{U}\) as the object classifier and the HoTT internal language. The univalence: the \(\mathcal{U}\) the univalent universe (the \((A\simeq B)\to(\mathcal{U}(A)=\mathcal{U}(B))\) the univalence axiom, the HoTT). The object classifier: the \(\mathcal{U}\to\mathcal{U}\) the universal bundle (the maps \(X\to\mathcal{U}\) classify the bundles \(E\to X\), the foundations). The internal language: the HoTT the internal language of the \(\infty\)-topos (the HoTT). The key point: the \(\mathcal{U}\) the univalent, the object classifier, the HoTT the internal language.
Check¶
Explain the univalence axiom and the object classifier.
The univalence: the \(\mathcal{U}\) the univalent universe (the \((A\simeq B)\to(\mathrm{Id}_\mathcal{U}(A,B))\) the univalence axiom, the Voevodsky, the HoTT). The object classifier: the \(\mathcal{U}\to\mathcal{U}\) the universal bundle (the maps \(X\to\mathcal{U}\) classify the bundles \(E\to X\) with small fibers, the foundations). The \(\infty\)-topos: the \(\mathcal{U}\) the object classifier (the foundations, the HoTT). So the univalence the \(\mathcal{U}\), the object classifier the \(\mathcal{U}\to\mathcal{U}\).
Explain the HoTT as the internal language and the cubical type theory.
The HoTT: the internal language of the \(\infty\)-topos (the types = objects, the terms = morphisms, the identity types = path spaces, the univalence the \(\mathcal{U}\), the HoTT). The cubical: the computational (the cubical type theory, the Kan composition, the HoTT). The synthetic: the HoTT the synthetic homotopy (the \(\pi_n\) the identity types, the HoTT). The Shulman: the \(\infty\)-topos the univalent (the HoTT). So the HoTT the internal language, the cubical the computational.
Pitfalls¶
- The univalence the Voevodsky (\(\mathcal{U}\), the HoTT) — the univalence the \(\simeq\to=\), the Voevodsky.
- The object classifier the [foundations] (\(\mathcal{U}\to\mathcal{U}\), the foundations) — the object classifier the bundles, the \(\mathcal{U}\).
- The HoTT the internal language (the HoTT) — the HoTT the \(\infty\)-topos, the synthetic.
- The cubical the computational (the HoTT) — the cubical the Kan, the computational.
Refs¶
- Voevodsky, "Univalent foundations" (2009–); the univalence
- Shulman, "All (∞,1)-toposes have strict univalent universes" (2019); the internal language
- Cohen–Coquand–Huber–Mörtberg, "Cubical type theory" (2016); the cubical (the HoTT)
- The HoTT Book (2013); the HoTT
Next¶
∞-topoi index · Foundations · Descent & Stacks · Shape & Cosmoi · Cohesive & Differential · HoTT · Higher categories