Synthetic Differential Geometry, the Infinitesimal, and the Kock¶
Job: develop the SDG (Kock), the infinitesimal \(D=\{d:d^2=0\}\), the Kock–Lawvere axiom, and the smooth / Cahiers topos — with the \(D\) / Kock–Lawvere as the worked example.
Core¶
The synthetic differential geometry (SDG)¶
| SDG / \(D\) / Kock | Description |
|---|---|
| SDG | the Kock: \(D=\{d:d^2=0\}\) the infinitesimal, the nilpotent (the category theory / categorical logic) |
The infinitesimal and the nilpotent¶
| \(D\) / nilpotent / Kock–Lawvere | Description |
|---|---|
| \(D\) | \(\{d:d^2=0\}\) the nilpotent, \(D^n\) the \(n\)-dim |
| Kock–Lawvere axiom | \(f:D\to R\) the \(f(d)=a+bd\) (every function on \(D\) affine) |
The smooth topos and the Cahiers¶
| Smooth / Cahiers / \(C^\infty\) | Description |
|---|---|
| Smooth topos | the Cahiers topos, \(C^\infty\)-rings (the category theory) |
| \(C^\infty\)-rings | the smooth algebras (the category theory) |
The synthetic and the classical¶
| Synthetic / classical / SDG / differential | Description |
|---|---|
| Synthetic / classical | the SDG the synthetic, the classical the geometry / differential |
| Synthetic / differential | the \(D\) the synthetic, the geometry / differential |
See it¶
\(D=\{d:d^2=0\}\) the Kock–Lawvere axiom and the SDG. The \(D\): the \(\{d:d^2=0\}\) the infinitesimal (the nilpotent, the \(D^n\) the \(n\)-dim). The Kock–Lawvere: the \(f:D\to R\) the \(f(d)=a+bd\) (the every function on \(D\) the affine, the Kock–Lawvere axiom). The SDG: the \(D\) the synthetic, the geometry / differential the classical. The key point: the \(D\) the Kock–Lawvere, the SDG the synthetic — the infinitesimal the topos.
Check¶
Explain the Kock–Lawvere axiom and the \(D\), and the SDG.
The \(D\): the \(\{d\in R:d^2=0\}\) the infinitesimal (the nilpotent, the \(D^n=\{d\in R^n:d_i d_j=0\}\) the \(n\)-dim). The Kock–Lawvere: the \(f:D\to R\) the \(f(d)=a+bd\) (the every function on \(D\) the affine, the axiom the \(R^D\cong R\times R\) the linear). The SDG: the \(D\) the synthetic (the category theory / categorical logic, the smooth topos the \(D\)), the geometry / differential the classical (the \(D\) the non-standard, the smooth the topos). So the \(D\) the Kock–Lawvere, the SDG the synthetic.
Explain the Cahiers topos and the \(C^\infty\)-rings, and the smooth.
The Cahiers: the smooth topos (the \(C^\infty\)-rings the smooth algebras, the category theory). The \(C^\infty\)-rings: the \(A\) the \(C^\infty\)-algebra (the smooth operations, the \(F:C^\infty(\mathbb{R}^n)\to A\), the category theory). The smooth: the Cahiers the \(C^\infty\) the \(D\) (the \(D\) the infinitesimal the \(C^\infty\), the SDG). The synthetic: the \(D\) the synthetic, the geometry / differential the classical. So the Cahiers the \(C^\infty\), the \(D\) the smooth.
Pitfalls¶
- The \(D=\{d:d^2=0\}\) the nilpotent (the infinitesimal, the Kock–Lawvere) — the \(D\) the nilpotent, the \(f(d)=a+bd\) the affine.
- The Kock–Lawvere (\(f:D\to R\) the \(a+bd\), the \(R^D\cong R\times R\)) — the Kock–Lawvere the axiom, the SDG.
- The Cahiers the \(C^\infty\) (the smooth algebras, the category theory) — the Cahiers the smooth, the \(C^\infty\)-rings.
- The synthetic vs the classical (the SDG the \(D\), the geometry / differential) — the synthetic the \(D\), the classical the standard.
Refs¶
- Kock, Synthetic Differential Geometry (2006); the SDG / the \(D\)
- Kock–Lawvere, "Taylor series and infinitesimals" (1981); the Kock–Lawvere axiom
- Moerdijk–Reyes, Models for Smooth Infinitesimal Analysis (1991); the Cahiers / the \(C^\infty\)
- Lavendhomme, Basic Concepts of Synthetic Differential Geometry (1996); the SDG
Next¶
TT index · Foundations · Classifying & Geometric · Boolean & Forcing · Cohesive & Higher · Category theory · Geometry / differential