synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
typical contexts
In a context of synthetic differential geometry or D-geometry, the de Rham space $dR(X)$ of a space $X$ is the quotient of $X$ that identifies infinitesimally close points.
It is the coreduced reflection of $X$.
Let CRing be the category of commutative rings. For $R \in CRing$, write $I \in R$ for the nilradical of $R$, the ideal consisting of the nilpotent elements. The canonical projection $R \to R/I$ to the quotient by the ideal corresponds in the opposite category $Ring^{op}$ to the inclusion
of the reduced part of $Spec R$.
For $X \in PSh(Ring^{op})$ a presheaf on $Ring^{op}$ (for instance a scheme), its de Rham space $X_{dR}$ is the presheaf defined by
If $X \in PSh(Ring^{op})$ is a smooth scheme then the canonical morphism
is an epimorphism (hence an epimorphism over each $Spec R$) and therefore in this case $X_{dR}$ is the quotient of the relation “being infinitesimally close” between points of $X$: we have that $X_{dR}$ is the coequalizer
of the two projections out of the formal neighbourhood of the diagonal.
For $E \to X$ a bundle over $X$, its direct image under base change along the projection map $X \longrightarrow \Pi_{inf} X$ yields its jet bundle. See there for more.
In terms of differential homotopy type theory this means that forming “jet types” of dependent types over $X$ is the dependent product operation along the unit of the infinitesimal shape modality
For $X : Ring \to Set$ a scheme, the big site $Ring^{op}/X_{dR}$ of $X_{dR}$, is the crystaline site of $X$.
Morphisms $X_{dR} \to Mod$ encode flat higher connections: local systems.
Accordingly, descent for de Rham spaces – sometimes called de Rham descent encodes flat 1-connections. This is described at Grothendieck connection,
The category of D-modules on a space is equivalent to that of quasicoherent sheaves on the corresponding de Rham space (Lurie, above theorem 0.4).
The term de Rham space or de Rham stack apparently goes back to
But actually there it just has the notation “$X_{dR}$” and then the functor it co-represents is called the “de Rham shape” of $X$.
A review of the constructions is on the first two pages of
The de Rham space construction on spaces (schemes) is described in section 3, p. 7
which goes on to assert the existence of its derived functor on the homotopy category $Ho Sh_\infty(C)$ of ∞-stacks in proposition 3.3. on the same page.
Similar discussion in a context of derived algebraic geometry is in
The characterization of formally smooth scheme as above is also on that page.
See also online comments by David Ben-Zvi here and here on the $n$Café. and here on MO.