nLab
smooth map

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • 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

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          Contents

          Idea

          A function which is differentiable function to arbitrary order is called a smooth function.

          Between Cartesian spaces

          A function on (some open subset of) a cartesian space n\mathbb{R}^n with values in the real line \mathbb{R} is smooth, or infinitely differentiable, if all its derivatives exist at all points. More generally, if A nA \subseteq \mathbb{R}^n is any subset, a function f:Af: A \to \mathbb{R} is defined to be smooth if it has a smooth extension to an open subset containing AA.

          By coinduction: A function f:f : \mathbb{R} \to \mathbb{R} is smooth if (1) its derivative exists and (2) the derivative is itself a smooth function.

          For A nA \subseteq \mathbb{R}^n, a smooth map ϕ:A m\phi: A \to \mathbb{R}^m is a function such that πϕ\pi \circ \phi is a smooth function for every linear functional π: m\pi: \mathbb{R}^m \to \mathbb{R}. (In the case of finite-dimensional codomains as here, it suffices to take the π\pi to range over the mm coordinate projections.)

          The concept can be generalised from cartesian spaces to Banach spaces and some other infinite-dimensional spaces. There is a locale-based analogue suitable for constructive mathematics which is not described as a function of points but as a special case of a continuous map (in the localic sense).

          Between smooth manifolds

          A topological manifold whose transition functions are smooth maps is a smooth manifold. A smooth function between smooth manifolds is a function that (co-)restricts to a smooth function between subsets of Cartesian spaces, as above, with respect to any choice of atlases, hence which is a kk-fold differentiable function (see there for more details), for all kk The category Diff is the category whose objects are smooth manifolds and whose morphisms are smooth maps betweeen them.

          Between generalized smooth spaces

          There are various categories of generalised smooth spaces whose morphisms are generalized smooth functions.

          For details see for example at smooth set.

          Properties

          Basic facts about smooth functions are

          Examples

          Every analytic functions (for instance a holomorphic function) is also a smooth function.

          A crucial property of smooth functions, however, is that they contain also bump functions.

          Examples of sequences of local structures

          geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
          \leftarrow differentiationintegration \to
          smooth functionsderivativeTaylor seriesgermsmooth function
          curve (path)tangent vectorjetgerm of curvecurve
          smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
          function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
          arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
          Lie theoryLie algebraformal grouplocal Lie groupLie group
          symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

          Last revised on September 15, 2017 at 03:50:17. See the history of this page for a list of all contributions to it.