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)
A lined topos $(\mathcal{T}, R)$ is
a ringed topos $(\mathcal{T}, k)$
(usually with the internal ring object $(k,+,\cdot)$ assumed to be commutative)
and equipped with a choice $(R,+,\cdot)$ of internal commutative algebra object $(R,+,\cdot)$ over $k$ – the line object.
A smooth topos is a lined topos where the line is required to be a smooth differentiable space with infinitesimal subspaces in a certain way. This is the basic type of object studied in synthetic differential geometry.
A super smooth topos is a lined topos that is a smooth topos and in which the $k$-algebra structure on $R$ is refined to that of a $k$-superalgebra.
The line object $R$ in a lined topos $\mathcal{T}$ canonically has the structure of a cartesian interval object.
As described there, this canonically induces
a cosimplicial object $\Delta_R : \Delta \to \mathcal{T}$
a functor $\Pi : \mathcal{T} \to S \mathcal{T}$
that sends each object in the topos to a simplicial object
( which may be interpreted as presenting the path ∞-groupoid of $X$).
The following terminology is sometimes useful.
(contractible object)
Let $(\mathcal{T} = Sh(C), R)$ be a lined Grothendieck topos with respect to a site $C$.
Call an object $X \in \mathcal{T}$ contractible with respect to the interval object $R$, if the simplicial sheaf $\Pi(X) = X^{\Delta_R^\bullet} : C^{op} \to$ SSet sends each object to a contractible simplicial set.
Examples
sheaves on topological spaces Let $Top'$ be a small version of the category of sufficiently nice topological spaces, for instance connected CW complexes. The canonical line object in $Sh(Top)$ is ${*} \stackrel{0}{\to} [0,1] \stackrel{1}{\leftarrow} {*}$ the standard topological interval. For $X \in Top$, $\Pi(X) = X^{\Delta_R^\bullet}$ is the singular simplicial complex of $X$. This is contractible in the above sense precisely if $X$ is a contractible space in the standard sense.
sheaves on cartesian spaces Let CartSp be the full subcategory of Diff on smooth manifolds of the form $\mathbb{R}^n$, for $n \in \mathbb{N}$. The canonical line object in $\mathcal{T} = Sh(CartSp)$ is the real line regarded as an interval object
In the lined topos $(\mathcal{T} = Sh(CartSp), R = \mathbb{R})$ the representable objects $\mathbb{R}^n$ are contractible with respect to $R$.
This is not quite as entirely trivial as it may seem on first sight, but follows directly from the Tietze extension theorem for smooth manifolds:
we check that for all $V \in$ CartSp every boundary of a simplex $\partial \Delta[k] \to \Pi(\mathbb{R}^n)(V)$ extends through $\partial \Delta[k] \hookrightarrow \Delta[k]$:
by the construction of the cosimplicial object $\Delta_R : \Delta \to Sh(CartSp)$ we have that morphisms $\partial \Delta[k] \to \Pi(\mathbb{R}^n)(V)$ correspond to smooth maps from the boundary of a $V$-cylinder over the standard $k$-simplex in $\mathbb{R}^k \times V \to \mathbb{R}^n$. Since this is a closed subset of $\mathbb{R}^k \times V$, by the Tietze extension theorem these maps extend (apply the theorem to each of their components) to all of $\mathbb{R}^k \times V$, hence in particular to the standard $k$-simplex inside $\mathbb{R}^k$ defined by the interval object. This constitutes the required extension to a $V$-family of $k$-simplices in $\mathbb{R}^n$
sheaves on cartesian smooth loci A small variation of the above example leads to smooth toposes with contractible representables:
let $CartSp_{synth} \subset \mathbb{L}$ be the full subcategory of smooth loci on those smooth loci of the form $\mathbb{R}^n \times D_k(r)$, where $D_k(r)$ is the infinitesimal space of $k$th order infinitesimal neighbours of the origin in $\mathbb{R}^r$.
The line object is again ${*} \stackrel{0}{\to} \mathbb{R} \stackrel{1}{\leftarrow} {*}$ as in the above example. Crucially, the infinitesimal spaces $D_k(r)$ all have a unique point ${*} \to D_k(r)$. Accordingly, there is also a unique morphism $R^n \to D_k(r)$ for all $n$. It follows that simplices in $R^n \times D_k(r)$ are simplices in $R^n$ as above, and trivial as maps to the $D_k(r)$-factor. Hence the above argument carries over to this case and shows that all the $\mathbb{R}^n \times D_k(r)$ are contractible.