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.


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

  2. smooth loci 𝕃:=(C ∞Ring fin) op

  3. infinitesimal objects

  4. infinitesimal singular simplicial complex

  5. deformation theory of C∞-rings

  6. synthetic differential geometry

  7. derived synthetic differential geometry

Topics and literature

Basic concepts


The ring C ∞(X)=C ∞(X,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 ∞(X) may be thought of as induced from the multiplication p:ℝ×ℝ→ℝ as

C ∞(X,ℝ)Γ—C ∞(X,ℝ)→≃C ∞(X,ℝ×ℝ)β†’C ∞(X,p)C ∞(C,ℝ)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:ℝ n→ℝ m induces naturally a map

C ∞(X,f):C ∞(X,ℝ n)β†’C ∞(X,ℝ m).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→K mK(f) : K^n \to K^m

in a compatible way is called a C ∞-ring: a ring equipped with a smooth structure.

More abstractly speaking, a C ∞-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 ∞-rings are rings of smooth functions on a smooth manifold. We may however think of the opposite category 𝕃:=C ∞Ring op as the category of generalized smooth spaces whose function rings are arbitrary C ∞-ring: smooth loci. The ordinary category Diff of smooth manifolds is full and faithfully embedded into the category of smooth loci

Diffβ†ͺ𝕃Diff \hookrightarrow \mathbb{L}

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


  1. C∞-ring

  2. smooth loci 𝕃:=(C ∞Ring fin) op


Special properties



  1. infinitesimal objects

  2. infinitesimal singular simplicial complex

  3. deformation theory of C∞-rings





  1. synthetic differential geometry

  2. derived synthetic differential geometry



A standard textbook reference is chapter 1 of

The concept of C ∞-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 ∞-algebras from the functional analytic view point Journal of pure and applied algebra 46 (1987) (pdf)

A characterization of those C ∞-rings that are algebras of smooth functions on some smooth manifold is given in

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

Studies of the properties of C ∞-rings include

Synthetic spaces locally isomorphic to smooth loci were discussed in

and more recently in

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

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

where this is briefly mentioned in the very last paragraph.

Revised on June 14, 2011 17:14:06 by Urs Schreiber (