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 $Y$ be two smooth manifolds of finite dimension and let $f : X \to Y$ be a differentiable function between them
In components, the definition of submersion reads as follows.
The function $f : X \to Y$ is called a submersion precisely if its differential $d f\colon T X \to T Y$ is for every point $x \in X$ a surjection $d f_x\colon T_x X \to T_{f(x)} Y$.
More abstractly formulated, this means equivalently the following.
The function $f : X \to Y$ is a submersion precisely if the canonical morphism
from the tangent bundle of $X$ to the pullback of the tangent bundle of $Y$ along $f$ is a surjection.
This morphism is the one induced by the universal property of the pullback from the commuting diagram
In terms of coordinates, the map $f$ is a submersion at a point $p\colon X$ if and only if there exists a coordinate chart on $X$ near $p$ and a coordinate chart on $Y$ near $f(p)$ relative to which $f$ is the projection $f(x_1,\ldots,x_n) = (x_1,\ldots,x_m)$. This definition applies to infinite-dimensional manifolds, to non-differentiable maps, even between non-differentiable manifolds.
While the category Diff of (finite dimensional) smooth manifolds does not have all pullbacks, the pullback along a submersion always exists. This is because a submersion is transversal to every other smooth map into its codomain. Moreover, submersions are stable under pullback.
The surjective submersions (that is the submersions that are also epimorphisms in Diff) are regular epimorphisms.
Surjective submersions form a singleton Grothendieck pretopology on Diff, and so may be used in internal category theory when using $Diff$ as the ambient category. They appear notably in the definition of Lie groupoids.
Ehresmann's theorem states that a proper submersion is a locally trivial fibration.
For $f : X \to Y$ a submersion, then around every point of $X$ there is an open neighbourhood on which $f$ restricts to a projection.
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 smooth morphism in $SynthDiff\infty Grpd$.
$f$ is a submersion precisely if it is formally smooth 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.
For instance chapter XIV
Ehresmann’s theorem is due to