Cohesive Toposes, Higher Toposes, and the ∞-Topos¶
Job: develop the cohesive topos (Lawvere), the ∞-topos (Lurie), the cohesive ∞-topos / differential cohomology (Schreiber), and the shape / flat / sharp — with the cohesive ∞-topos of smooth spaces as the worked example.
Core¶
The cohesive topos¶
| Cohesive / \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\) / Lawvere | Description |
|---|---|
| Cohesive topos | \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\) (the cohesion, the Lawvere, the category theory) |
The higher topos and the ∞-topos¶
| ∞-topos / Lurie / higher | Description |
|---|---|
| ∞-topos | the infinity-topoi, the Lurie, the higher categories |
The cohesive ∞-topos and the differential cohomology¶
| Cohesive ∞ / Schreiber / differential | Description |
|---|---|
| Cohesive ∞-topos | the Schreiber: the cohesive ∞-topos the differential cohomology (the higher categories) |
The shape and the flat / the sharp¶
| Shape \(\int\) / flat \(\flat\) / sharp \(\sharp\) | Description |
|---|---|
| Shape \(\int\) | \(\int\dashv\Gamma\) (the homotopy type theory) |
| Flat \(\flat\) / sharp \(\sharp\) | \(\flat\dashv\sharp\) (the homotopy type theory) |
See it¶
Cohesive ∞-topos of smooth spaces and the Schreiber differential cohomology. The cohesive: the \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\) (the Lawvere, the category theory). The ∞: the infinity-topoi the Lurie (the higher categories). The Schreiber: the cohesive ∞-topos the differential cohomology (the higher categories). The key point: the cohesive ∞-topos the smooth, the Schreiber the differential cohomology.
Check¶
Explain the cohesive topos and the \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\).
The cohesive: the \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\) (the \(\mathrm{Disc}\) the discrete, \(\Gamma\) the global sections, \(\mathrm{Codisc}\) the codiscrete, the Lawvere, the category theory). The \(\mathrm{Disc}\dashv\Gamma\): the \(\mathrm{Disc}\) the embedding the sets the spaces, \(\Gamma\) the points (the category theory). The \(\Gamma\dashv\mathrm{Codisc}\): the \(\Gamma\) the points, \(\mathrm{Codisc}\) the codiscrete (the category theory). The shape \(\int\): the \(\int\dashv\Gamma\) (the homotopy type theory, the \(\int\) the homotopy type). So the cohesive the \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\), the \(\int\) the shape.
Explain the ∞-topos and the cohesive ∞-topos / Schreiber differential cohomology.
The ∞-topos: the infinity-topoi the Lurie (the \((\infty,1)\)-topos, the higher categories, the \(\infty\)-Grothendieck). The cohesive ∞: the Schreiber (the cohesive \(\infty\)-topos the \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\) the \(\infty\), the higher categories). The differential cohomology: the Schreiber the cohesive \(\infty\)-topos the differential cohomology (the \(H_{\mathrm{dR}}\to H_{\mathrm{diff}}\) the higher categories). So the ∞-topos the Lurie, the cohesive \(\infty\) the Schreiber.
Pitfalls¶
- The cohesive \(\mathrm{Disc}\dashv\Gamma\dashv\mathrm{Codisc}\) (the Lawvere, the category theory) — the cohesive the adjoints, the shape \(\int\).
- The ∞-topos the Lurie (the infinity-topoi, the higher categories) — the ∞-topos the \((\infty,1)\), the Lurie.
- The cohesive ∞ the Schreiber (the differential cohomology, the higher categories) — the cohesive ∞ the differential, the Schreiber.
- The shape / flat / sharp (\(\int\dashv\Gamma\), \(\flat\dashv\sharp\), the HoTT) — the \(\int\) the homotopy type, the \(\flat\) the flat, the \(\sharp\) the sharp.
Refs¶
- Lawvere, "Categories of spaces may not be concrete" (1986); the cohesive
- Lurie, Higher Topos Theory (2009); the ∞-topos (the infinity-topoi)
- Schreiber, Differential Cohomology in a Cohesive ∞-Topos (2013–); the cohesive ∞ / the differential
- Shulman, "All (∞,1)-toposes have strict univalent universes" (2019); the HoTT
Next¶
TT index · Foundations · Classifying & Geometric · Boolean & Forcing · Synthetic & Infinitesimal · Infinity-topoi · Higher categories · HoTT