Models, Semantics, and the ∞-Topos¶
Job: develop the simplicial model (Voevodsky), the ∞-topos semantics (Shulman), the cubical / presheaf model, and the set-theoretic / constructive — with the simplicial / ∞-topos as the worked example.
Core¶
The simplicial model and the Kan¶
| Simplicial / Voevodsky / Kan / univalent | Description |
|---|---|
| Simplicial model | Voevodsky, \(\mathbf{sSet}\) the Kan, the univalent (the foundations) |
| Kan / univalent | the Kan fibrations the univalent (the foundations) |
The ∞-topos semantics¶
| ∞-topos / Shulman / internal language | Description |
|---|---|
| ∞-topos semantics | Shulman: the \(\infty\)-topos the univalent (the infinity-topoi) |
| Internal language | the infinity-topoi / internal language the HoTT (the foundations) |
The cubical model and the presheaf¶
| Cubical / presheaf / Cohen–Coquand–Huber–Mörtberg | Description |
|---|---|
| Cubical model | \(\mathbf{Set}^{\square^{\mathrm{op}}}\) the presheaf (the foundations) |
| Presheaf / cubical | the cubical presheaf the univalent (the foundations) |
The set-theoretic and the constructive¶
| Set-theoretic / ZFC / constructive | Description |
|---|---|
| Set-theoretic | the ZFC the model (the foundations) |
| Constructive | the constructive model (the foundations) |
See it¶
Simplicial model \(\mathbf{sSet}\) (Voevodsky) and the \(\infty\)-topos semantics (Shulman). The simplicial: the Voevodsky the \(\mathbf{sSet}\) the Kan the univalent (the foundations). The \(\infty\)-topos: the Shulman the infinity-topoi the univalent (the infinity-topoi / internal language). The key point: the simplicial the Voevodsky, the \(\infty\)-topos the Shulman.
Check¶
Explain the simplicial model and the univalent universes in \(\mathbf{sSet}\).
The simplicial: the \(\mathbf{sSet}\) the Kan fibrations (the foundations, the Voevodsky). The univalent: the \(\mathcal{U}\) the universe (the Kan fibration the classifying, the \(\mathcal{U}\to\mathcal{U}\) the universal, the foundations). The univalence: the \(\mathsf{idtoequiv}:(A=B)\to(A\simeq B)\) the equivalence (the Voevodsky, the foundations). So the simplicial the Kan, the univalent the \(\mathcal{U}\).
Explain the \(\infty\)-topos semantics (Shulman) and the cubical model.
The \(\infty\)-topos: the Shulman the infinity-topoi the univalent (the every \(\infty\)-topos has a univalent universe, the infinity-topoi / internal language). The cubical: the \(\mathbf{Set}^{\square^{\mathrm{op}}}\) the presheaf (the Cohen–Coquand–Huber–Mörtberg, the foundations). The constructive: the cubical the constructive (the canonicity, the foundations). So the \(\infty\)-topos the Shulman, the cubical the presheaf.
Pitfalls¶
- The simplicial the Kan (the Voevodsky, the foundations) — the simplicial the Kan, the univalent the \(\mathcal{U}\).
- The ∞-topos the Shulman (the infinity-topoi) — the \(\infty\)-topos the univalent, the Shulman.
- The cubical the presheaf (the \(\square^{\mathrm{op}}\), the foundations) — the cubical the constructive, the canonicity.
- The set-theoretic the ZFC (the classical, the foundations) — the ZFC the model, the constructive the different.
Refs¶
- Voevodsky, "The simplicial model of univalent foundations" (2009); the \(\mathbf{sSet}\)
- Shulman, "All (∞,1)-toposes have strict univalent universes" (2019); the ∞-topos
- Cohen–Coquand–Huber–Mörtberg, "Cubical type theory" (2016); the cubical / the presheaf
- Bezem–Coquand–Huber, "A model of type theory in cubical sets" (2014); the cubical model
Next¶
HoTT index · Foundations · Cubical & Computational · Synthetic Homotopy · Univalent Categories · Infinity-topoi