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

Models

• 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

∞-Lie theory (higher geometry)

Background

Smooth structure

• generalized smooth space

  • smooth manifold

  • diffeological space

  • Frölicher space

• smooth topos

  • Cahiers topos

• smooth ∞-groupoid, concrete smooth ∞-groupoid

• synthetic differential ∞-groupoid

Higher groupoids

• ∞-groupoid

  • groupoid

  • 2-groupoid

  • strict ∞-groupoid

    • crossed complex

• ∞-group

  • simplicial group

Lie theory

• Lie theory

  • Lie integration, Lie differentiation

  • Lie's three theorems

  • Lie theory for stacky Lie groupoids

∞-Lie groupoids

• ∞-Lie groupoid

  • strict ∞-Lie groupoid

  • Lie groupoid

    • differentiable stack

    • orbifold

• ∞-Lie group

  • Lie group

    • simple Lie group, semisimple Lie group

  • Lie 2-group

∞-Lie algebroids

• ∞-Lie algebroid

  • Lie algebroid

• Lie ∞-algebroid representation

• L-∞-algebra

  • model structure for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras

  • Lie algebra

    • semisimple Lie algebra, compact Lie algebra

  • Lie 2-algebra

    • strict Lie 2-algebra

    • differential crossed module

• Lie 3-algebra

  • differential 2-crossed module

• dg-Lie algebra, simplicial Lie algebra

• super L-∞ algebra

  • super Lie algebra

Formal Lie groupoids

• formal group, formal groupoid

Cohomology

• Lie algebra cohomology

  • Chevalley-Eilenberg algebra

  • Weil algebra

  • invariant polynomial

    • Killing form

• nonabelian Lie algebra cohomology

Homotopy

• homotopy groups of a Lie groupoid

Related topics

• ∞-Chern-Weil theory

Examples

$\infty$-Lie groupoids

• Atiyah Lie groupoid

• fundamental ∞-groupoid

  • path groupoid

  • path n-groupoid

• smooth principal ∞-bundle

$\infty$-Lie groups

• orthogonal group

  • special orthogonal group

  • spin group

  • string 2-group

  • fivebrane 6-group

• unitary group

  • special unitary group

• circle Lie n-group

  • circle group

$\infty$-Lie algebroids

• tangent Lie algebroid

• action Lie algebroid

• Atiyah Lie algebroid

• symplectic Lie n-algebroid

  • symplectic manifold

  • Poisson Lie algebroid

  • Courant Lie algebroid

    • generalized complex geometry

$\infty$-Lie algebras

• general linear Lie algebra

• orthogonal Lie algebra, special orthogonal Lie algebra

• endomorphism L-∞ algebra

• automorphism ∞-Lie algebra

• string Lie 2-algebra

• fivebrane Lie 6-algebra

• supergravity Lie 3-algebra

• supergravity Lie 6-algebra

• line Lie n-algebra