Schreiber Seminar on Smooth Loci

A topic list and references for a seminar on synthetic differential geometry and smooth loci, held spring 2010.

See also the followup Seminar on derived differential geometry.

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Topic outline

Here is a bare list of possible topics. The items are repeated with background information and pointers to the literature below.

1. C∞-ring

   • Lawvere theory

2. smooth loci$\mathbb{L} := (C^\infty Ring^{fin})^{op}$

   • the embedding Diff $\hookrightarrow \mathbb{L}$.

3. infinitesimal objects

   • atomic object

   • amazing right adjoint

   • Kock-Lawvere axiom

4. infinitesimal singular simplicial complex

   • combinatorial differential forms

5. deformation theory of C∞-rings

   • Fermat theory

   • module

   • derivation

   • Kähler differential

   • quasicoherent sheaves of modules

6. synthetic differential geometry

   • topos, smooth topos

   • category of sheaves, Grothendieck topos

   • Models for Smooth Infinitesimal Analysis

7. derived synthetic differential geometry

   • simplicial C∞-ring

   • notions of space

     • structured (∞,1)-topos

     • geometry (for structured (∞,1)-toposes).

   • derived smooth manifold

Topics and literature

Basic concepts

Idea

The ring $C^\infty(X) = C^\infty(X,\mathbb{R})$ of smooth real-valued functions on a smooth manifold $X$ has considerably more structure than just being a ring: the ring multiplication itself on $C^\infty(X)$ may be thought of as induced from the multiplication $p : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ as

$$C^\infty(X,\mathbb{R}) \times C^\infty(X,\mathbb{R}) \stackrel{\simeq}{\to} C^\infty(X, \mathbb{R} \times \mathbb{R}) \stackrel{C^\infty(X,p)}{\to} C^\infty(C,\mathbb{R})$$

but similarly every smooth map $f : \mathbb{R}^n \to \mathbb{R}^m$ induces naturally a map

$$C^\infty(X,f) : C^\infty(X,\mathbb{R}^n) \to C^\infty(X,\mathbb{R}^m) \,.$$

Such a ring $K$, equipped with the structure that allows to operate with every smooth map $f$ on it

$$K(f) : K^n \to K^m$$

in a compatible way is called a $C^\infty$-ring: a ring equipped with a smooth structure.

More abstractly speaking, a $C^\infty$-ring is a product-preserving copresheaf on CartSp. This in turn means that it is a model for the Lawvere theory given by CartSp.

Not all $C^\infty$-rings are rings of smooth functions on a smooth manifold. We may however think of the opposite category $\mathbb{L} := C^\infty Ring^{op}$ as the category of generalized smooth spaces whose function rings are arbitrary $C^\infty$-ring: smooth loci. The ordinary category Diff of smooth manifolds is full and faithfully embedded into the category of smooth loci

$$Diff \hookrightarrow \mathbb{L}$$

but smooth loci crucially include also infinitesimal objects, such as the abstract tangent vector $D$, whose $C^\infty$-ring of smooth functions is $C^\infty(\mathbb{R})/(x^2)$: the ring of dual numbers.

Topics

1. C∞-ring

   • Lawvere theory

2. smooth loci$\mathbb{L} := (C^\infty Ring^{fin})^{op}$

   • the embedding Diff $\hookrightarrow \mathbb{L}$.

References

Special properties

Idea

Topics

1. infinitesimal objects

   • atomic object

   • amazing right adjoint

   • Kock-Lawvere axiom

2. infinitesimal singular simplicial complex

   • combinatorial differential forms

3. deformation theory of C∞-rings

   • Fermat theory

   • module

   • derivation

   • Kähler differential

   • quasicoherent sheaves of modules

References

Applications

Idea

Topics

1. synthetic differential geometry

   • topos, smooth topos

   • category of sheaves, Grothendieck topos

   • Models for Smooth Infinitesimal Analysis

2. derived synthetic differential geometry

   • simplicial C∞-ring

   • notions of space

     • structured (∞,1)-topos

     • geometry (for structured (∞,1)-toposes).

   • derived smooth manifold

References

References

A standard textbook reference is chapter 1 of

• Ieke Moerdijk and Gonzalo Reyes, Models for Smooth Infinitesimal Analysis Springer (1991)

The concept of $C^\infty$-rings in particular and that of synthetic differential geometry in general was introduced in

• Bill Lawvere, Categorical dynamics

  in Anders Kock (eds.) Topos theoretic methods in geometry, volume 30 of Various Publ. Ser., pages 1-28, Aarhus Univ. (1997)

but examples of the concept are older. A discussion from the point of view of functional analysis is in

• G. Kainz, A. Kriegl, Peter Michor, $C^\infty$-algebras from the functional analytic view point Journal of pure and applied algebra 46 (1987) (pdf)

A characterization of those $C^\infty$-rings that are algebras of smooth functions on some smooth manifold is given in

• Peter Michor, Jiri Vanzura, Characterizing algebras of $C^\infty$-functions on manifolds (pdf)

Lawvere’s ideas were later developed by Eduardo Dubuc, Anders Kock, Ieke Moerdijk, Gonzalo Reyes, and Gavin Wraith.

Studies of the properties of $C^\infty$-rings include

• Ieke Moerdijk, Gonzalo Reyes, Rings of Smooth Functions and Their Localizations, I , Journal of Algebra 99 (1986)

Synthetic spaces locally isomorphic to smooth loci were discussed in

• Eduardo Dubuc, $C^\infty$-schemes Amer. J. Math. 103 (1981) (pdf JSTOR).

and more recently in

• Dominic Joyce, Algebraic geometry over $C^\infty$-rings (arXiv:1001.0023)

The higher geometry generalization to a theory of derived smooth manifolds – spaces with structure sheaf taking values in simplicial C∞-rings – was initiated in

• David Spivak, Quasi-Smooth derived manifolds (pdf of original version, arXiv:0810.5174)

based on the general machinery of structured (∞,1)-toposes in

• Jacob Lurie, Structured Spaces

where this is briefly mentioned in the very last paragraph.