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)
Let $X$ and $X$ be smooth manifolds of finite dimension. Let $f : X \to Y$ be a differentiable function.
A differentiable function $f : X \to Y$ is called an immersion precisely if the canonical morphism
is a monomorphism.
This morphism is the one induced from the universal property of the pullback by the commuting diagram
given by the differential of $f$ going between the tangent bundles.
Equivalently this means the following:
The function $f : X \to Y$ is an immersion precisely if for every point $x \in X$ the differential
between the tangent space of $X$ at $x$ and the tangent space of $Y$ at $f(y)$ is an injection.
(embedding of smooth manifolds)
An immersion whose underlying continuous function is an embedding of topological spaces is called an an embedding of smooth manifolds.
(immersions that are not embeddings)
Consider an immersion $f \;\colon\; (a,b) \to \mathbb{R}^2$ of an open interval into the Euclidean plane (or the 2-sphere) as shown on the right. This is not a embedding of smooth manifolds: around the points where the image crosses itself, the function is not even injective, but even at the points where it just touches itself, the pre-images under $f$ of open subsets of $\mathbb{R}^2$ do not exhaust the open subsets of $(a,b)$, hence do not yield the subspace topology.
As a concrete examples, consider the function $(sin(2-), sin(-)) \;\colon\; (-\pi, \pi) \longrightarrow \mathbb{R}^2$. While this is an immersion and injective, it fails to be an embedding due to the points at $t = \pm \pi$ “touching” the point at $t = 0$.
graphics grabbed from Lee
An immersion $f : X \to Y$ is precisely a local embeddings: for every point $x \in X$ there is an open neighbourhood $x \in U \subset X$ such that $f|_U : U \to Y$ is an embedding of smooth manifolds.
A smooth function $f : X \to Y$ between smooth manifolds is canonically regarded as a morphism in the cohesive (∞,1)-topos SynthDiff∞Grpd. With respect to the canonical infinitesimal neighbourhood inclusion $i :$ Smooth∞Grpd $\hookrightarrow$ SynthDiff∞Grpd there is a notion of formally unramified morphism in $SynthDiff\infty Grpd$.
$f$ is an immersion precisely if it is formally unramified with respect to this infinitesimal cohesion.
See the discussion at SynthDiff∞Grpd for details.
The algebraic geometry analogue of a submersion is a smooth morphism.
The analogue between arbitrary topological spaces (not manifolds) is simply an open map. There is also topological submersion, of which there are two versions.