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.
Here is a bare list of possible topics. The items are repeated with background information and pointers to the literature below.
smooth loci $\mathrm{\pi \x9d\x95\x83}:=({C}^{\mathrm{\beta \x88\x9e}}{\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}$
The ring ${C}^{\mathrm{\beta \x88\x9e}}(X)={C}^{\mathrm{\beta \x88\x9e}}(X,\mathrm{\beta \x84\x9d})$ of smooth real-valued functions on a smooth manifold $X$ has considerably more structure than just being a ring: the ring multiplication itself on ${C}^{\mathrm{\beta \x88\x9e}}(X)$ may be thought of as induced from the multiplication $p:\mathrm{\beta \x84\x9d}\Gamma \x97\mathrm{\beta \x84\x9d}\beta \x86\x92\mathrm{\beta \x84\x9d}$ as
but similarly every smooth map $f:{\mathrm{\beta \x84\x9d}}^{n}\beta \x86\x92{\mathrm{\beta \x84\x9d}}^{m}$ induces naturally a map
Such a ring $K$, equipped with the structure that allows to operate with every smooth map $f$ on it
in a compatible way is called a ${C}^{\mathrm{\beta \x88\x9e}}$-ring: a ring equipped with a smooth structure.
More abstractly speaking, a ${C}^{\mathrm{\beta \x88\x9e}}$-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 ${C}^{\mathrm{\beta \x88\x9e}}$-rings are rings of smooth functions on a smooth manifold. We may however think of the opposite category $\mathrm{\pi \x9d\x95\x83}:={C}^{\mathrm{\beta \x88\x9e}}{\mathrm{Ring}}^{\mathrm{op}}$ as the category of generalized smooth spaces whose function rings are arbitrary ${C}^{\mathrm{\beta \x88\x9e}}$-ring: smooth loci. The ordinary category Diff of smooth manifolds is full and faithfully embedded into the category of smooth loci
but smooth loci crucially include also infinitesimal objects, such as the abstract tangent vector $D$, whose ${C}^{\mathrm{\beta \x88\x9e}}$-ring of smooth functions is ${C}^{\mathrm{\beta \x88\x9e}}(\mathrm{\beta \x84\x9d})/({x}^{2})$: the ring of dual numbers.
smooth loci $\mathrm{\pi \x9d\x95\x83}:=({C}^{\mathrm{\beta \x88\x9e}}{\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}$
A standard textbook reference is chapter 1 of
The concept of ${C}^{\mathrm{\beta \x88\x9e}}$-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
A characterization of those ${C}^{\mathrm{\beta \x88\x9e}}$-rings that are algebras of smooth functions on some smooth manifold is given in
Lawvereβs ideas were later developed by Eduardo Dubuc, Anders Kock, Ieke Moerdijk, Gonzalo Reyes, and Gavin Wraith.
Studies of the properties of ${C}^{\mathrm{\beta \x88\x9e}}$-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 β spaces with structure sheaf taking values in simplicial Cβ-rings β was initiated in
based on the general machinery of structured (β,1)-toposes in
where this is briefly mentioned in the very last paragraph.