Shape, Cosmoi, and the Profinality¶
Job: develop the shape \(\int\mathcal{X}\) of an \(\infty\)-topos, the \(\infty\)-cosmos (Riehl–Verity), the profinite / Galois, and the étale homotopy — with the shape of the étale \(\infty\)-topos as the worked example.
Core¶
The shape of an ∞-topos¶
| Shape / \(\int\mathcal{X}\) / cohesive / HoTT | Description |
|---|---|
| Shape \(\int\mathcal{X}\) | \(\mathrm{Pro}(\mathcal{S})\) the shape (the topos-theory / cohesive, the HoTT) |
| Homotopy type | \(\int\mathcal{X}\) the homotopy type (the topos-theory / cohesive) |
The cosmoi and the ∞-cosmos¶
| ∞-cosmos / Riehl–Verity / higher | Description |
|---|---|
| ∞-cosmos | the Riehl–Verity: the \(\mathbf{Cat}_\infty\) the enriched (the higher categories) |
The profinite and the Galois¶
| Profinite / \(\mathrm{Pro}\) / condensed / Galois | Description |
|---|---|
| Profinite | \(\mathrm{Pro}(\mathbf{FinSet})\) the profinite (the condensed mathematics) |
| Galois | the Galois the profinite (the arithmetic geometry / étale) |
The shape and the étale homotopy¶
| Étale homotopy / Artin–Mazur / shape | Description |
|---|---|
| Étale homotopy | \(\int X_{\mathrm{\acute{e}t}}\) the étale homotopy type (the Artin–Mazur, the arithmetic geometry / étale) |
| Shape / étale | \(\int\mathbf{Shv}(X_{\mathrm{\acute{e}t}})\) the étale homotopy (the arithmetic geometry) |
See it¶
Shape of the étale ∞-topos and the étale homotopy type. The shape: \(\int\mathbf{Shv}(X_{\mathrm{\acute{e}t}},\mathcal{S})\) the étale homotopy type (the Artin–Mazur, the arithmetic geometry / étale). The profinite: \(\mathrm{Pro}(\mathbf{FinSet})\) the profinite (the condensed mathematics). The key point: the shape the étale homotopy, the profinite the condensed mathematics.
Check¶
Explain the shape of an \(\infty\)-topos and the étale homotopy type.
The shape: the \(\int\mathcal{X}=\mathrm{Pro}(\mathcal{S})\) the pro-spaces (the topos-theory / cohesive, the \(\int\dashv\Gamma\), the HoTT). The étale: the \(\int\mathbf{Shv}(X_{\mathrm{\acute{e}t}},\mathcal{S})\) the étale homotopy type (the Artin–Mazur, the \(\mathrm{Pro}(\mathbf{FinHomotopy})\), the arithmetic geometry / étale). The profinite: the \(\mathrm{Pro}(\mathbf{FinSet})\) the profinite (the condensed mathematics, the Galois the profinite). So the shape the pro-space, the étale the Artin–Mazur, the profinite the condensed.
Explain the \(\infty\)-cosmos and the Riehl–Verity.
The \(\infty\)-cosmos: the Riehl–Verity the \(\mathbf{Cat}_\infty\) the enriched (the \(\mathcal{K}\) the \(\infty\)-cosmos, the isofibrations, the higher categories). The \(\infty\)-cosmos the framework: the \(\mathcal{K}\) the \(\infty\)-cosmos the \(\mathbf{Cat}_\infty\)-enriched, the isofibrations the fibrations, the higher categories. The \(\infty\)-topos the \(\infty\)-cosmos: the \(\infty\)-topos the \(\infty\)-cosmos (the higher categories). So the \(\infty\)-cosmos the Riehl–Verity, the \(\infty\)-topos the \(\infty\)-cosmos.
Pitfalls¶
- The shape the pro-space (\(\int\mathcal{X}=\mathrm{Pro}(\mathcal{S})\), the topos-theory / cohesive) — the shape the pro, the \(\int\dashv\Gamma\).
- The ∞-cosmos the Riehl–Verity (the higher categories) — the \(\infty\)-cosmos the \(\mathbf{Cat}_\infty\)-enriched, the isofibrations.
- The profinite the [condensed] (the \(\mathrm{Pro}(\mathbf{FinSet})\), the condensed mathematics) — the profinite the Pro, the condensed.
- The étale homotopy the Artin–Mazur (the arithmetic geometry / étale) — the étale the Artin–Mazur, the \(\mathrm{Pro}(\mathbf{FinHomotopy})\).
Refs¶
- Lurie, Higher Topos Theory (2009), §6; the shape / the pro-spaces
- Artin–Mazur, Étale Homotopy (1969); the étale homotopy type
- Riehl–Verity, Elements of ∞-Category Theory (2022); the ∞-cosmos
- Barwick–Glasman, "A note on the étale homotopy type" (2017); the shape / the condensed
Next¶
∞-topoi index · Foundations · Descent & Stacks · Cohesive & Differential · Internal Language & HoTT · Topos theory / cohesive · HoTT · Condensed mathematics · Arithmetic geometry / étale