nLab
synthetic differential topology

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

tangent cohesion

differential cohesion

graded differential 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& 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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

Synthetic differential topology (SDT) is a synthetic axiomatization of differential topology in analogy to how synthetic differential geometry (SDG) is synthetic axiomatization of differential geometry.

Where in SDG the concept of infinitesimal neighbourhoods is encoded by the axioms (the Kock-Lawvere axiom) in SDT it is germs of spaces that are being encoded by the axioms.

Notice that the germ of a manifold around a point is in general “larger” than the formal neighbourhood of that point, reflecting, dually, the fact that there are smooth functions which are non-vanishing in every open neighbourhood of that point but all whose partial derivatives vanish at that point (see also at bump function).

SDT uses the representability of germs by an object defined intrinsically as the points not well-separated from 0R n0 \in R^n, ¬¬{0}\neg \neg \{0\}.

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

References

  • Marta Bunge and Eduardo Dubuc, Local Concepts in Synthetic Differential Geometry and Germ Representability, article

  • Marta Bunge, Felipe Gago, Synthetic aspects of C C^\infty-mapping II: Mather’s theorem for infinitesimally represented germs, Journal of Pure and Applied Algebra 55 (1988) 213-250 North-Holland (doi:10.1016/0022-4049(88)90117-X)

  • Marta Bunge, Felipe Gago, Ana Maria San Luis, Synthetic Differential Topology, Cambridge, to appear April 2018

Revised on November 12, 2017 10:49:35 by David Corfield (209.93.199.52)