smooth map (Rev #5) in nLab

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⊣♭⊣♯)(\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⊣♭ dR\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)

  (ℜ⊣ℑ⊣&)(\Re \dashv \Im \dashv \&)

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality ⊣\dashv bosonic modality ⊣\dashv rheonomy modality

  (⇉⊣⇝⊣Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

singular cohesion

id ⊣ id ∨ ∨ fermionic ⇉ ⊣ ⇝ bosonic ⊥ ⊥ bosonic ⇝ ⊣ Rh rheonomic ∨ ∨ reduced ℜ ⊣ ℑ infinitesimal ⊥ ⊥ infinitesimal ℑ ⊣ & étale ∨ ∨ cohesive esh ⊣ ♭ discrete ⊥ ⊥ discrete ♭ ⊣ ♯ continuous ∨ ∨ ∅ ⊣ * \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 }

Models

• Models for Smooth Infinitesimal Analysis

  • smooth algebra (C ∞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