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)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(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 topos can be viewed as a generalization of a topological space. A smooth structure on a topos is a corresponding generalization of the notion of smooth manifold, in that to put a smooth structure on $Sh(X)$, for a topological space $X$, is (at least closely related to) putting the structure of a smooth manifold on $X$. However, since it is phrased internally with reference to the (Cauchy and Dedekind) real numbers objects, it is applicable to any topos.
Given a topos $\mathbf{H}$, write
$\mathbb{N} \in \mathbf{H}$ for its natural numbers object;
$\mathbb{R}_{C} \in \mathbf{H}$ for its Cauchy real numbers object;
$\mathbb{R}_{D} \in \mathbf{H}$ for its Dedekind real numbers object;
$\mathbb{R} \in$ Set for the external real numbers.
There are canononical subobject inclusions
(Fourman 75, def. 3.6) A morphism
in the topos $\mathbf{H}$ is called a standard function, precisely if
it is a continuous function in the internal sense that
it respects the sub-object of Cauchy reals, in that
We will furthermore consider smooth standard functions, meaning standard functions that satisfy the internalized ordinary definition of smooth function (i.e. $C^\infty$).
We define an equivalence relation on $\mathbb{R}_D^n$ by taking two elements to be equivalent if there are smooth standard functions taking them into each other:
(Fourman 75, def. 4.1) A smooth structure of dimension $n$ on a topos $\mathbf{H}$ is an equivalence class for the above equivalence relation on $\mathbb{R}_D^n$. In other words, the object of smooth structures of dimension $n$ is the quotient of $\mathbb{R}_D^n$ by this equivalence relation.
(Fourman 75, def. 4.4) Given a smooth structure $S$ of dimension $n$ on the topos $\mathbf{H}$ according to def. , then the smooth real number object is the subobject
defined internally as the set of all images of points in $S\subseteq \mathbb{R}_D^n$ under smooth standard functions $\mathbb{R}_D^n \to \mathbb{R}_D$. In symbols, it is given by
Note that any function constant at a Cauchy real is standard. Therefore, every Cauchy real is a smooth real.
(Fourman 75, example 4.3 1) Let $X$ be a smooth manifold of dimension $n$, with sheaf topos $\mathbf{H} \coloneqq Sh(X)$. As shown at real numbers object, $\mathbb{R}_D$ is then the sheaf of continuous real-valued functions on $X$.
Let $S\subseteq \mathbb{R}_D^n$ be the sheaf of local coordinate systems, i.e. $S(U)$ is the set of real-valued functions $U\to \mathbb{R}^n$ that are smooth and are locally diffeomorphisms onto their images. Then $S$ is a smooth structure on $Sh(X)$ of dimension $n$, according to def. .
The corresponding object $\mathbb{R}_S$ of smooth reals is the sheaf of smooth real-valued functions $X\to \mathbb{R}$. Note that since $X$ is locally connected, the Cauchy real numbers object $\mathbb{R}_C$ in $Sh(X)$ is the sheaf of locally constant real-valued functions, so $\mathbb{R}_S$ sits strictly in between $\mathbb{R}_C$ and $\mathbb{R}_D$.
Let $h:1\to \mathbb{R}_D^n$ be any global section of $\mathbb{R}_D^n$. Then the equivalence class of $h$, under the above equivalence relation, is a smooth structure. In particular, if $h$ is a tuple of Cauchy real numbers (such as $0$), then so is every point in $S$, and thus $\mathbb{R}_S = \mathbb{R}_C$. This gives the “discrete” smooth structure on $\mathbf{H}$.
For $\mathbf{H}=Sh(X)$, such a global section is a continuous map $h:X\to \mathbb{R}^n$, and this gives the smooth structure “cogenerated by” $h$, in that it makes $h$ smooth and only those other functions that must be smooth if $h$ is. The case when $h$ is a Cauchy real corresponds to $h:X\to \mathbb{R}^n$ being locally constant, in which case all smooth functions are locally constant; this is the “minimal” smooth structure on a topological space $X$.
Last revised on June 28, 2024 at 07:35:58. See the history of this page for a list of all contributions to it.