nLab Banach fixed-point theorem

tangent bundle, frame bundle
- vector field, multivector field, tangent Lie algebroid;
- differential forms, de Rham complex, Dolbeault complex
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

cohesion

(shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$
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)

$(\Re \dashv \Im \dashv \&)$

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{&#233;tale}}{} \\ && \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 ( $C^\infty$ -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
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
complex geometry
symplectic geometry
conformal geometry

Statement
Related concepts
References

Statement

The Banach fixed-point theorem or contraction mapping theorem states:

Let $(X, \rho)$ be a sequentially Cauchy complete metric space with a point $x_0:X$ and a rational number $C : \mathbb{Q}$ such that for all $x:X$ and $y:X$ , $\rho(x, y) \leq C$ . Let $T : X \to X$ be an endomap with a rational Lipschitz constant $0 \lt c \lt 1$ . Then $X$ has a unique fixed point, a point $x$ with $\rho(T (x), x) = 0$ , such that for any $y : X$ with $\rho(T (y), y) = 0$ , $x = y$ .

Related concepts

inverse function theorem
ordinary differential equation

References

Auke B. Booij, Analysis in univalent type theory (pdf)