In synthetic differential geometry one formulates differential geometry axiomatically in toposes – called smooth toposes – of generalized smooth spaces.
The main point of the axioms is to ensure that a well defined notion of infinitesimal spaces exist in the topos, whose existence concretely and usefully formalizes the wide-spread but often vague intuition about the role of infinitesimals in differential geometry.
In particular, in such toposes $E$ there exists an infinitesimal space $D$ that behaves like the infinitesimal interval in such a way that for any space $X \in E$ the tangent bundle of $X$, is, again as an object of the topos, just the internal hom $T X \;\text{:=}\; X^D$ (using the notation of exponential objects in the cartesian closed category $E$). So a tangent vector in this context literally is an infinitesimal path in $X$.
This way, in smooth toposes it is possible to give precise well-defined meaning to many of the familiar computations – wide-spread in particular in the physics literature – that compute with supposedly “infinitesimal” quantities.
As quoted by Anders Kock in his first book (p. 9), Sophus Lie – one of the founding fathers of differential geometry and, of course Lie theory – once said that he found his main theorems in Lie theory using “synthetic reasoning”, but had to write them up in non-synthetic style (see analytic versus synthetic) just due to lack of a formalized language:
“The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following. I found these theories originally by synthetic considerations. But I soon realized that, as expedient ( zweckmässig ) the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which deal with objects that till now have almost exclusively been considered analytically. After long vacillations, I have decided to use a half synthetic, half analytic form. I hope my work will serve to bring justification to the synthetic method besides the analytical one.” (Sophus Lie, Allgemeine Theorie der partiellen Differentialgleichungen erster Ordnung, Math. Ann. 9 (1876).)
Synthetic differential geometry provides this formalized language.
The axioms of synthetic differential geometry demand that the topos $E$ of smooth spaces is
in which in particular
infinitesimal spaces exist and
satisfy the Kock-Lawvere axiom.
Depending on applications one imposes further axioms, such as the
With that little bit of axiomatics alone, a large amount of differential geometry may be formulated. This has been carried through quite comprehensively by Anders Kock, see the reference below.
In his work he particularly makes use of the fact that as sophisticated as a smooth topos may be when explicitly constructed (see the section on models), being a topos means that one can reason inside it almost literally as in Set. Using this Kock’s work gives descriptions of synthetic differential geometry which are entirely intuitive and have no esoteric topos-theoretic flavor. All he needs is the assumption that the Kock-Lawvere axiom is satisfied for “numbers”. Here “numbers” is really to be interpreted in the topos, but if one just accepts that they satisfy the KL axiom, one may work with infinitesimals in this context in essentially precisely the naive way, with the topos theory in the background just ensuring that everything makes good sense.
Being axiomatic, reasoning in synthetic differential geometry applies in every model for the axioms, i.e. in every concrete choice of smooth topos $T$.
Models of smooth toposes tend to be inspired, but more general than, constructions familiar from algebraic geometry. In particular the old insight promoted by Grothendieck in his work, that nilpotent ideals in rings are formal duals of spaces with infinitesimal extension is typically used to model infinitesimal spaces in synthetic differential geometry.
The main difference between models for smooth toposes and algebraic geometry from this perspective is that models for smooth topos tend to employ test spaces that are richer than plain formal duals to commutative rings or algebras, as in algebraic geometry: typical models for synthetic differential geometry use test spaces given by formal duals of generalized smooth algebras that remember “smooth structure” in the usual sense of differential geometry (and different from, though not entirely unrelated, to the notion of smooth scheme in algebraic geometry). This is in particular true for the well adapted models.
However, with a a sufficiently general perspective on higher geometry one finds that algebraic geometry and synthetic differential geometry are both special cases of a more general notion of theories of generalized spaces. For more on this see generalized scheme.
A topos $E$ modelling the axioms of synthetic differential geometry is called (well) adapted if the ordinary differential geometry of manifolds embeds into it, in particular if there is a full and faithful functor Diff $\to E$ from the category of ordinary smooth manifolds into $E$.
A standard model for well adapted synthetic toposes is obtained in terms of sheaves on duals of “germ determined” $C^\infty$-rings. This is described in great detail in the textbook Models for Smooth Infinitesimal Analysis.
The conception and discussion of these well adapted toposes goes back to E. Dubuc, who studied them in a long series of articles. He asks people to refer to this topos as the Dubuc topos.
This theory of well-adapted models was later summarized and extended in the standard textbook
The notion of synthetic differential geometry extends to the context of supergeometry. See
The tangent bundle of an object $X$ in a smooth topos is just the exponential object $T X := X^D$. The unique inclusion ${*} \to D$ induces a canonical projection $T X \to X$. A section $X \to T X$ of that projection is a tangent vector on $X$. Its adjunct is a morphism $D \to X^X$ that sends the unique point of $D$ to the identity $Id_X \in X^X$.
A differential equation is an extension problem in a smooth topos along a morphism that includes an infinitesimal object into another object.
For instance the ordinary first order homogeneous differential equation that asks the derivative of a functor $f : X \to A$ along a vector field $v : D \to X^X$ to be given by a specified map $\alpha X \to T A$ is given by a diagram of the form
where we have freely identified morphisms with their adjuncts. See differential equation for details.
A differential 1-form is a morphism $\omega : T X \to R$ that is “fiberwise linear”. One elegant way to say this is obtained by considering all higher differential forms at once:
for a sufficiently well behaved object $X$ in a smooth topos, there is the simplicial object which is the infinitesimal singular simplicial complex $X^{(\Delta^\bullet_{inf})}$ of $X$. Taking functions on this produces the cosimplicial algebra $Hom(X^{\Delta^\bullet_{inf}}, R)$. Its normalized Moore cochain complex is isomorphic to the de Rham dg-algebra of differential forms on $X$:
This is discussed at
The idea of axiomatizing differential geometry using topos theory originates in
The first model for the axioms presented there served to demonstrate that the theory is non-empty, but was hard to work with. Much of the later work was concerned with refining the model-building. For instance
These models are constructed in terms of sheaf toposes on the category of smooth loci, formal duals to C∞-rings. See there for a detailed list of references.
Transcripts or notes of further talks by Bill Lawvere on the subject are
Bill Lawvere, Toposes of laws of motion , transcript of a talk in Montreal, Sept. 1997 (pdf)
(on the description of differential equations in terms of synthetic differential geometry)
Bill Lawvere, Outline of synthetic differential geometry , lectures in Buffalo (1998) (pdf)
See also
The textbooks
Anders Kock, Synthetic Differential Geometry, (pdf)
Anders Kock, Synthetic Geometry of Manifolds (pdf)
develop in great detail the theory of differential geometry using the axioms of synthetic differential geometry. The main goal in these books is to demonstrate how these axioms lead to a very elegant, very intuitive and very comprehensive conception of differential geometry. Accordingly, concrete models (whose explicit description is typically much more evolved than the nice axiomatics that holds once they have been constructed) play a minor role in these books.
Somewhat complementary to that, the book
focuses on concrete constructions of well-adapted models using the technology of generalized smooth algebras ($C^\infty$-rings).
Another textbook is
Two useful introductory expositions of basic ideas of synthetic differential geometry are
Mike Shulman, Chicago Pizza-Seminar: Synthetic Differential Geometry (pdf)
John Bell, An invitation to smooth infinitesimal analysis (pdf)