Descent, Stacks, and the ∞-Topos of Higher Stacks¶
Job: develop the descent / hypercovers / \(\infty\)-stacks, the stacks / gerbes, the derived / spectral stacks, and the descent / spectral — with the étale \(\infty\)-topos / derived stacks as the worked example.
Core¶
The descent and the ∞-stacks¶
| Descent / hypercovers / ∞-stacks | Description |
|---|---|
| Descent | \(F(c)\xrightarrow{\sim}\lim F(c')\) the sheaf condition for hypercovers (the foundations) |
| ∞-stacks | sheaves of spaces (the higher categories) |
The stacks and the gerbes¶
| Stack / 1-truncated / gerbe | Description |
|---|---|
| Stack | sheaf of groupoids, \(1\)-truncated (the topos-theory) |
| Gerbe | \(1\)-truncated gerbe, \(H^2\) (the algebraic geometry / cohomology) |
The derived / spectral stacks¶
| Derived / spectral / DAG / SAG | Description |
|---|---|
| Derived stacks | the derived AG, sheaves of spaces on SCR |
| Spectral stacks | the SAG / foundations, \(\mathbb{E}_\infty\)-rings |
The descent and the spectral / the higher¶
| Descent / spectral / higher | Description |
|---|---|
| Descent / spectral | the SAG / spectral GAGA, the descent |
| Higher / descent | the higher categories, the descent the \(\infty\) |
See it¶
Descent for the étale ∞-topos of a scheme and the derived stacks. The descent: \(\mathbf{Shv}(X_{\mathrm{\acute{e}t}},\mathcal{S})\) the sheaf condition the hypercovers (the foundations). The derived: the DAG the sheaves of spaces on SCR (the \(\infty\)-stacks). The spectral: the SAG the \(\mathbb{E}_\infty\). The key point: the descent the hypercovers, the derived / spectral the \(\infty\)-stacks.
Check¶
Explain the descent condition for \(\infty\)-stacks and the hypercovers.
The descent: \(F(c)\xrightarrow{\sim}\lim_{c'\to c\in S} F(c')\) the sheaf condition (the \(S\) the covering sieve, the foundations). The hypercovers: the \(U_\bullet\to c\) the hypercover (the simplicial / the cosimplicial, the \(\check{C}\)ech + higher, the higher categories). The \(\infty\)-stack: the \(F:C^{\mathrm{op}}\to\mathcal{S}\) the sheaf of spaces (the descent the hypercovers, the higher categories). So the descent the hypercovers, the \(\infty\)-stacks.
Explain the derived stacks and the spectral stacks.
The derived: the DAG the sheaves of spaces on SCR (the \(X:\mathrm{SCR}^{\mathrm{op}}\to\mathcal{S}\), the \(\infty\)-stack, the higher categories). The spectral: the SAG the sheaves on \(\mathrm{CAlg}^{\mathrm{op}}\) (the \(\mathbb{E}_\infty\)-rings, the spectral DM, the SAG / foundations). The descent: the SAG / spectral GAGA the \(\infty\)-stack the descent (the higher categories). So the derived the SCR, the spectral the \(\mathbb{E}_\infty\).
Pitfalls¶
- The descent the hypercovers (the \(U_\bullet\to c\), the foundations) — the hypercovers the \(\check{C}\)ech + higher, the \(\infty\)-stacks.
- The stack the 1-truncated (the sheaf of groupoids, the topos-theory) — the stack the 1-truncated, the gerbe the \(H^2\).
- The derived the [DAG] (the SCR, the derived AG) — the derived the \(\infty\)-stacks, the higher categories.
- The spectral the [SAG] (the \(\mathbb{E}_\infty\), the spectral-algebraic-geometry/foundations) — the spectral the \(\mathrm{CAlg}\), the SAG.
Refs¶
- Lurie, Higher Topos Theory (2009), §6; the descent / the \(\infty\)-stacks
- Toën–Vezzosi, HAG II (2005–); the derived stacks
- Lurie, SAG (2016–2018); the spectral stacks / the SAG / foundations
- Hollander, "The sheaf condition for \(\infty\)-stacks" (2010); the descent
Next¶
∞-topoi index · Foundations · Shape & Cosmoi · Cohesive & Differential · Internal Language & HoTT · Higher categories · DAG · SAG