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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In differential topology, given a pair $X$, $Y$ of smooth manifolds, and a pair of parallel smooth functions $X \underoverset{f_1}{f_0}{\rightrightarrows} Y$ between them, a smooth homotopy between these is (typically defined to be) a left homotopy by a smooth function, hence a smooth function $\eta \;\colon\; X \times \mathbb{R} \longrightarrow Y$ on the product manifold of $X$ with the real line, which restricts to $f_i$ at $X \times \{i\}$, for $i \in \{0,1\} \subset \mathbb{R}$.
More generally, for $X$, $Y$ differentiable manifolds and $f_0$ and $f_1$ differentiable functions, one may ask for differentiable homotopies, all to any given order(s) of differentiability.
At least if $X$ and $Y$ are closed manifolds, then a smooth homotopy exists between any pair of parallel smooth functions $(f_0, f_1)$ between them as soon as there exists an ordinary (i.e. continuous) left homotopy between their underlying continuous functions.
More generally, if $(f_0, f_1)$ are $n+2$-fold differentiable, then an ordinary continuous homotopy between them implies an $n$-fold differentiable homotopy.
(Pontrjagin 55, Thm. 8, p. 41)
Created on February 3, 2021 at 10:43:11. See the history of this page for a list of all contributions to it.