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
( $\dashv$ $\dashv$ )
$(ʃ \dashv \flat \dashv \sharp )$
, ,
$\dashv$
$ʃ_{dR} \dashv \flat_{dR}$
( $\dashv$ $\dashv$ )
$(\Re \dashv \Im \dashv \&)$
$\dashv$ $\dashv$
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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)
In synthetic differential geometry, the tangent bundle of an object $X$ is the internal hom $X^{\mathbb{D}^1}$ out of the 1d first order infinitesimal disk $\mathbb{D}^1$, equipped with the projection to $X$ induced from the unique point $\ast \to \mathbb{D}^1$:
(Here we are using that the internal hom-functor $(-)^{(-)}$ is a contravariant functor in its “exponent” variable.)
This makes manifest and precise the intuitive idea that a tangent vector on $X$ is an “infinitesimal curve” in $X$, see also the Examples below.
In this formulation the operation of differentiation is simply the internal hom-functor:
Given a function:
its differential is its image under the internal hom $(-)^{(\mathbb{D}^1)}$:
In a standard model for synthetic differential geometry/differential cohesion such as the Cahiers topos $\mathbf{H}$, for $X \in SmthMfd \overset{y}{\hookrightarrow} \mathbf{H}$ an ordinary smooth manifold, its synthetic tangent bundle coincides with the traditional tangent bundle $T X \overset{p}{\to} T X$:
Moreover, if $Y \in SmthMfd \hookrightarrow \mathbf{H}$ is another smooth manifold, then a morphism
is equivalently a smooth function in the traditional sense (i.e. the external Yoneda embedding-functor $SmthMfd \overset{y}{\hookrightarrow} \mathbf{H}$ is fully faithful ) and its image under the internal hom is its traditional differentiation $d f$:
This way the evident functoriality of the internal hom $(-)^{(\mathbb{D}^1)}$ is identified with the chain rule of traditional differentiation.
For more on this see
at geometry of physics – supergeometry the section Super mapping spaces
For $X$ a microlinear space the synthetic tangent bundle shares many of the expected properties of a tangent bundle.
odd tangent bundle (analog in supergeometry)
For lecture notes see
Last revised on September 14, 2018 at 07:29:39. See the history of this page for a list of all contributions to it.