smooth map



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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& \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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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.


Basic facts about smooth functions are


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


An early account, in the context of Cohomotopy, cobordism theory and the Pontryagin-Thom construction:

  • Lev Pontrjagin, Chapter I of: Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001, pdf)

Last revised on February 3, 2021 at 10:56:41. See the history of this page for a list of all contributions to it.