Cohesive ∞-Topoi and the Differential Cohomology¶
Job: develop the cohesive \(\infty\)-topos (Schreiber), the differential cohomology hexagon, the smooth \(\infty\)-topos / infinitesimal, and the modalities — with the cohesive \(\infty\)-topos of smooth spaces as the worked example.
Core¶
The cohesive ∞-topos¶
| Cohesive ∞ / Schreiber / differential | Description |
|---|---|
| Cohesive \(\infty\)-topos | \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\) the \(\infty\) (the Schreiber, the topos-theory / cohesive) |
| Differential cohomology | the Schreiber: \(H_{\mathrm{diff}}\) the differential cohomology (the topos-theory / cohesive) |
The differential cohomology and the hexagon¶
| \(H_{\mathrm{diff}}\) / hexagon / higher | Description |
|---|---|
| Differential cohomology | the Schreiber: \(H_{\mathrm{diff}}\) the hexagon (the higher categories) |
| Hexagon | \(H_{\mathrm{diff}}\to H_{\mathrm{dR}}\to H^n\to\dots\) the hexagon (the higher categories) |
The smooth / the cohesive and the infinitesimal¶
| Smooth / \(\mathbf{Sh}(\mathrm{CartSp})\) / infinitesimal | Description |
|---|---|
| Smooth \(\infty\)-topos | \(\mathbf{Sh}(\mathrm{CartSp},\mathcal{S})\) the smooth (the topos-theory / synthetic) |
| Infinitesimal | the \(D\) the infinitesimal (the topos-theory / synthetic) |
The modalities and the flat / sharp¶
| Modalities / \(\flat\) / \(\sharp\) / HoTT | Description |
|---|---|
| \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\) | the modalities (the topos-theory / cohesive) |
| \(\flat\dashv\sharp\) | the flat / sharp (the HoTT) |
See it¶
Cohesive ∞-topos of smooth spaces and the Schreiber differential cohomology hexagon. The cohesive \(\infty\): the \(\mathbf{Sh}(\mathrm{CartSp},\mathcal{S})\) the smooth \(\infty\)-topos (the \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\), the topos-theory / cohesive). The Schreiber: the \(H_{\mathrm{diff}}\) the differential cohomology hexagon (the \(H_{\mathrm{diff}}\to H_{\mathrm{dR}}\to H^n\to\dots\), the higher categories). The key point: the cohesive \(\infty\) the smooth, the Schreiber the hexagon.
Check¶
Explain the cohesive \(\infty\)-topos and the \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\).
The cohesive \(\infty\): the \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\) the \(\infty\) (the Schreiber, the topos-theory / cohesive). The \(\mathrm{Disc}\): the discrete (the \(\mathrm{Disc}\dashv\Gamma\), the topos-theory / cohesive). The \(\Gamma\dashv\mathrm{Codisc}\): the \(\Gamma\) the global sections, the \(\mathrm{Codisc}\) the codiscrete (the topos-theory / cohesive). The \(\int\): the shape \(\int\dashv\Gamma\) (the HoTT). The \(\flat\dashv\sharp\): the flat / sharp (the HoTT). So the cohesive \(\infty\) the \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\), the \(\int\) the shape.
Explain the Schreiber differential cohomology hexagon.
The Schreiber: the cohesive \(\infty\)-topos the differential cohomology (the \(H_{\mathrm{diff}}\) the differential, the higher categories). The hexagon: the \(H_{\mathrm{diff}}\to H_{\mathrm{dR}}\to H^n\to\dots\to H^{n+1}\to H_{\mathrm{diff}}\) the hexagon (the \(H_{\mathrm{dR}}\) the de Rham, the \(H^n\) the cohomology, the higher categories). The smooth: the \(\mathbf{Sh}(\mathrm{CartSp},\mathcal{S})\) the smooth \(\infty\)-topos (the topos-theory / synthetic, the \(D\) the infinitesimal). So the Schreiber the \(H_{\mathrm{diff}}\), the hexagon the \(H_{\mathrm{dR}}\to H^n\).
Pitfalls¶
- The cohesive \(\infty\) the Schreiber (\(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\), the topos-theory / cohesive) — the cohesive the \(\infty\), the Schreiber.
- The hexagon the \(H_{\mathrm{diff}}\) (the \(H_{\mathrm{dR}}\to H^n\), the higher categories) — the hexagon the differential, the de Rham.
- The smooth \(\mathbf{Sh}(\mathrm{CartSp})\) (the topos-theory / synthetic) — the smooth the CartSp, the \(D\) the infinitesimal.
- The \(\flat\dashv\sharp\) (the HoTT) — the flat the \(\flat\), the sharp the \(\sharp\).
Refs¶
- Schreiber, Differential Cohomology in a Cohesive ∞-Topos (2013–); the cohesive \(\infty\) / the hexagon
- Lawvere, "Categories of spaces may not be concrete" (1986); the cohesive (the topos-theory / cohesive)
- Shulman, "All (∞,1)-toposes have strict univalent universes" (2019); the \(\flat\) / \(\sharp\) (the HoTT)
- Kock, Synthetic Differential Geometry (2006); the \(D\) (the topos-theory / synthetic)
Next¶
∞-topoi index · Foundations · Descent & Stacks · Shape & Cosmoi · Internal Language & HoTT · Topos theory / cohesive · Topos theory / synthetic · Higher categories