# nLab synthetic differential topology

### Context

#### Differential geometry

Introductions

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Differentials

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Tangency

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The magic algebraic facts

Theorems

Axiomatics

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$(ʃ \dashv \flat \dashv \sharp )$

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$ʃ_{dR} \dashv \flat_{dR}$

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

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

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

Models

• ($C^\infty$-ring)

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# 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 $0 \in R^n$, $\neg \neg \{0\}$.

Examples of sequences of local structures

pointfirst order $\subset$ = arbitrary order infinitesimal$\subset$local = wise$\subset$finite
$\leftarrow$ $\to$
(path) of
/
$\mathbb{F}_p$ $\mathbb{Z}_p$ $\mathbb{Z}_{(p)}$ $\mathbb{Z}$
local strict 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 $C^\infty$-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, 2018, (CUP) (excerpt)

Last revised on August 28, 2018 at 05:47:30. See the history of this page for a list of all contributions to it.