higher geometry / derived geometry
geometric little (∞,1)-toposes
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Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
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(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)
For $X$ a scheme, analogous to how an $X$-scheme is a scheme $E \to X$ over $X$, a $\mathcal{D}_X$-scheme is a scheme over the de Rham space $\mathbf{\Pi}_{inf}(X)$ of $X$.
See also diffiety.
For $X$ a scheme, a $\mathcal{D}_X$-scheme is a scheme $E \to \mathbf{\Pi}_{inf}(X)$ over the de Rham space $\mathbf{\Pi}_{inf}(X)$ of $X$.
This definition makes sense in much greater generality: in any context of differential cohesion.
In the sheaf topos over affine schemes, an $X$-affine $\mathcal{D}_X$-scheme is a commutative monoid object in the monoidal category of quasicoherent sheaves $QC(\mathbf{\Pi}_{inf}(X))$, which is equivalently the category of D-modules over $X$:
This is (BeilinsonDrinfeld, section 2.3).
This is indeed equivalent to the above abstract definition
This appears as (Lurie, theorem, 0.6 and below remark 0.7)
The free $\mathcal{D}_X$-scheme on a given $X$-scheme $E \to X$ is the jet bundle of $E$.
This is (BeilinsonDrinfeld, 2.3.2).
This fact makes $\mathcal{D}$-geometry a natural home for variational calculus.
The definition in terms of monoids in D-modules is in section 2.3 in
Alexander Beilinson and Vladimir Drinfeld, Chiral Algebras
chapter 2, Geometry of D-schemes (pdf)
The observation that this is equivalent to the abstract definition given above appears in pages 5 and 6 of