nLab
cartesian space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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

          Definition

          Definition

          A Cartesian space is a finite Cartesian product of the real line \mathbb{R} with itself. Hence, a Cartesian space has the form n\mathbb{R}^n where nn is some natural number (possibly zero).

          This definition is silent on which category the real line \mathbb{R} is being considered as an object of. For instance, if \mathbb{R} is regarded as a topological space (hence an object in the category Top), then the topology on n\mathbb{R}^n is Euclidean topology the real line \mathbb{R} with itself where nn is some natural number. Another possibility is to regard \mathbb{R} as a smooth manifold (hence an object in the category Diff). The Cartesian space n\mathbb{R}^n with its standard topology (and sometimes smooth structure) is also called real nn-dimensional space (distinguish from “real nn-dimensional vector space” which is only isomorphic to it as a vector space).

          One may also speak of the complexified cartesian space n\mathbb{C}^n, or indeed of the cartesian space K nK^n for any field KK, or indeed for any set KK, or indeed for any object KK of any cartesian monoidal category.

          Example

          In particular:

          • 0\mathbb{R}^0 is the point,
          • 1\mathbb{R}^1 is the real line,
          • 2\mathbb{R}^2 is the real plane, which may be identified (in two canonical ways) with the complex plane \mathbb{C}.
          Remark

          Cartesian spaces carry plenty of further canonical structure:

          Sometimes one is interested in allowing nn to take other values, in which case one wants a product in some category that might not be the Cartesian product on underlying sets.

          For example, if one is studying Cartesian spaces as inner product spaces, then one might want an 0\aleph_0-dimensional Cartesian space to be the 0\aleph_0-dimensional Hilbert space l 2l^2, which is a proper subset of the cartesian product 0\mathbb{R}^{\aleph_0}.

          Properties

          Topological structure

          The open n-ball is homeomorphic Cartesian space n\mathbb{R}^n

          𝔹 n n. \mathbb{B}^n \simeq \mathbb{R}^n \,.

          Smooth structures

          For all nn \in \mathbb{N}, the open n-ball with its standard smooth structure is diffeomorphic to the Cartesian space n\mathbb{R}^n with its standard smooth structure

          𝔹 n n. \mathbb{B}^n \simeq \mathbb{R}^n \,.

          In fact, in d4d \neq 4 there is no choice:

          Theorem

          For nn \in \mathbb{N} a natural number with n4n \neq 4, there is a unique (up to isomorphism) smooth structure on the Cartesian space n\mathbb{R}^n.

          This was shown in (Stallings).

          Theorem

          In d=4d = 4 the analog of this statement is false. One says that on 4\mathbb{R}^4 there exist exotic smooth structures.

          Theorem

          In dimension dd \in \mathbb{N} for d4d \neq 4 we have:

          every open subset of d\mathbb{R}^d which is homeomorphic to 𝔹 d\mathbb{B}^d is also diffeomorphic to it.

          See the first page of (Ozols) for a list of references.

          Remark

          In dimension 4 the analog statement fails due to the existence of exotic smooth structures on 4\mathbb{R}^4.

          The category of Cartesian spaces

          See CartSp.

          In complex geometry for purposes of Cech cohomology the role of Cartesian spaces is played by Stein manifolds.

          References

          Named after René Descartes.

          • John Stallings, The piecewise linear structure of Euclidean space , Proc. Cambridge Philos. Soc. 58 (1962), 481-488. (pdf)

          • V. Ozols, Largest normal neighbourhoods , Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (jstor)

          There are various slight variations of the category CartSpCartSp that one can consider without changing its basic properties as a category of test spaces for generalized smooth spaces. A different choice that enjoys some popularity in the literature is the category of open (contractible) subsets of Euclidean spaces. For more references on this see diffeological space.

          The site ThCartSpThCartSp of infinitesimally thickened Cartesian spaces is known as the site for the Cahiers topos. It is considered

          in detal in section 5 of

          and briefly mentioned in example 2) on p. 191 of

          following the original article

          With an eye towards Frölicher spaces the site is also considered in section 5 of

          • Hirokazu Nishimura, Beyond the Regnant Philosophy of Manifolds (arXiv:0912.0827)

          Last revised on January 23, 2017 at 13:42:19. See the history of this page for a list of all contributions to it.