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**
synthetic differential geometry**
**Introductions**
from point-set topology to differentiable manifolds
geometry of physics:
coordinate systems,
smooth spaces,
manifolds,
smooth homotopy types,
supergeometry
**Differentials**
*
differentiation,
chain rule
*
differentiable function
*
infinitesimal space,
infinitesimally thickened point,
amazing right adjoint
**
V-manifolds**
*
differentiable manifold,
coordinate chart,
atlas
*
smooth manifold,
smooth structure,
exotic smooth structure
*
analytic manifold,
complex manifold
*
formal smooth manifold,
derived smooth manifold
**
smooth space**
*
diffeological space,
Frölicher space
*
manifold structure of mapping spaces
**Tangency**
*
tangent bundle,
frame bundle
*
vector field,
multivector field,
tangent Lie algebroid;
*
differential forms,
de Rham complex,
Dolbeault complex
*
pullback of differential forms,
invariant differential form,
Maurer-Cartan form,
horizontal differential form,
*
cogerm differential form
*
integration of differential forms
*
local diffeomorphism,
formally étale morphism
*
submersion,
formally smooth morphism,
*
immersion,
formally unramified morphism,
*
de Rham space,
crystal
*
infinitesimal disk bundle
**The magic algebraic facts**
*
embedding of smooth manifolds into formal duals of R-algebras
*
smooth Serre-Swan theorem
*
derivations of smooth functions are vector fields
**Theorems**
*
Hadamard lemma
*
Borel's theorem
*
Boman's theorem
*
Whitney extension theorem
*
Steenrod-Wockel approximation theorem
*
Whitney embedding theorem
*
Poincare lemma
*
Stokes theorem
*
de Rham theorem
*
Hochschild-Kostant-Rosenberg theorem
*
differential cohomology hexagon
**Axiomatics**
*
Kock-Lawvere axiom
*
smooth topos,
super smooth topos
*
microlinear space
*
integration axiom
**
cohesion**
* (
shape modality $\dashv$
flat modality $\dashv$
sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
*
discrete object,
codiscrete object,
concrete object
*
points-to-pieces transform
*
structures in cohesion
*
dR-shape modality $\dashv$
dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
**
infinitesimal cohesion**
*
classical modality
**
tangent cohesion**
*
differential cohomology diagram
**
differential cohesion**
* (
reduction modality $\dashv$
infinitesimal shape modality $\dashv$
infinitesimal flat modality)
$(\Re \dashv \Im \dashv \)$
*
reduced object,
coreduced object,
formally smooth object
*
formally étale map
* [structures in differential cohesion](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion)
**
graded differential cohesion**
*
fermionic modality $\dashv$
bosonic modality $\dashv$
rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
**
singular cohesion
$$
\array{
id \dashv id
\\
\vee \vee
\\
\stackrel{fermionic}{} \rightrightarrows \dashv \rightsquigarrow \stackrel{bosonic}{}
\\
\bot \bot
\\
\stackrel{bosonic}{} \rightsquigarrow \dashv \mathrm{R}\!\!\mathrm{h} \stackrel{rheonomic}{}
\\
\vee \vee
\\
\stackrel{reduced}{} \Re \dashv \Im \stackrel{infinitesimal}{}
\\
\bot \bot
\\
\stackrel{infinitesimal}{} \Im \dashv \ \stackrel{\text{étale}}{}
\\
\vee \vee
\\
\stackrel{cohesive}{} \esh \dashv \flat \stackrel{discrete}{}
\\
\bot \bot
\\
\stackrel{discrete}{} \flat \dashv \sharp \stackrel{continuous}{}
\\
\vee \vee
\\
\emptyset \dashv \ast
}
$$
{#Diagram}
**Models**
{#models_2}
*
Models for Smooth Infinitesimal Analysis
*
smooth algebra ($C^\infty$-ring)
*
smooth locus
*
Fermat theory
*
Cahiers topos
*
smooth ∞-groupoid
*
formal smooth ∞-groupoid
*
super formal smooth ∞-groupoid
**
Lie theory,
∞-Lie theory**
*
Lie algebra,
Lie n-algebra,
L-∞ algebra
*
Lie group,
Lie 2-group,
smooth ∞-group
**
differential equations,
variational calculus**
*
D-geometry,
D-module
*
jet bundle
*
variational bicomplex,
Euler-Lagrange complex
*
Euler-Lagrange equation,
de Donder-Weyl formalism,
*
phase space
**
Chern-Weil theory,
∞-Chern-Weil theory**
*
connection on a bundle,
connection on an ∞-bundle
*
differential cohomology
*
ordinary differential cohomology,
Deligne complex
*
differential K-theory
*
differential cobordism cohomology
*
parallel transport,
higher parallel transport,
fiber integration in differential cohomology
*
holonomy,
higher holonomy
*
gauge theory,
higher gauge theory
*
Wilson line,
Wilson surface
**
Cartan geometry (
super,
higher)**
*
Klein geometry, (
higher)
*
G-structure,
torsion of a G-structure
*
Euclidean geometry,
hyperbolic geometry,
elliptic geometry
* (
pseudo-)
Riemannian geometry
*
orthogonal structure
*
isometry,
Killing vector field,
Killing spinor
*
spacetime,
super-spacetime
*
complex geometry
*
symplectic geometry
*
conformal geometry
Contents
Idea
A smooth topos or smooth lined toposlined topos is the kind of topos studied in synthetic differential geometry. a category of generalized smooth spaces for which a notion of infinitesimal space exists.
It is defined to be a category of objects that behave like spaces, one of which – the line object – is equipped with the structure of a commutative algebra, such that for infinitesimal objects all morphisms are linear – i.e. such that the Kock-Lawvere axiom holds.
Definition
There is a standard definition and various straightforward variations.
standard defnition
Definition
A Forlined topos is alined topos there is the obvious notion of -algebra objects in .
-
a topos
-
equipped with a choice of internal ring object
-
and equipped with a choice of internal commutative algebra object over .
For and any two -algebra objects, there is the subobject of morphisms that are algebra homomorphisms.
Write
for the algebra spectrum of in .
An -Weil algebra is an -algebra of the form , where is an -finite-dimensional nilpotent ideal.
Definition
For (smooth topos) a lined topos there is the obvious notion of -algebra objects in .
For Alined topos and any is two asmooth topos -algebra objects, if there is thesubobject of morphisms that are algebra homomorphisms.
Write
for the algebra spectrum of in .
An -Weil algebra is an -algebra of the form , where is an -finite-dimensional nilpotent ideal.
Definition
(smooth topos)
A lined topos is a smooth topos if
-
the algebra spectra of all Weil algebras in are infinitesimal objects in that the functor has a right adjoint;
-
it satisfies the Kock-Lawvere axiom in that for all -Weil algebra objects the canonical morphism
is an isomorphism in .
variants
There are various immediate variants of this concepts.
super smooth topos
In synthetic differential supergeometry one considers a notion of smooth topos that axiomatizes not just ordinary differential geometry but supergeometry.
A super smooth topos is defined as a smooth topos with the notion of algebra replaced everywhere by superalgebra.
So a super smooth topos is a topos equipped with a superalgebra object with even part and odd part etc.
An algebra spectrum object is now an internal object of superalgebra homomorphisms and the condition is that for every super Weil algebra we have that is an infinitesimal object and that is an isomorphism.
This means that essentially all the standard general theory of smooth toposes goes through literally for super smooth toposes, too. The main difference is that a super smooth topos contains more types of infinitesimal objects.
There is for instance still the standard even infinitesimal interval
but there is now also the odd infinitesimal interval
Notice that in the graded commutative algebra every odd element automatically squares to 0.
Urs Schreiber: I’d think that the cominatorial/simplicial definition of differential forms in synthetic differential geometry applied verbatim in a super smooth topos automatically yields the right/expected notion of differential forms in supergeometry.
Examples
Dubuc topos
Stein topos