nLab synthetic differential topology

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 $0 \in R^n$, $\neg \neg \{0\}$ (“logical topology”).

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

Related concepts

• logical topology

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)

The logical topology goes back to

• Jacques Penon, Infinitésimaux et intuitionnisme, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 22 (1981) no. 1, p. 67-72 (numdam:CTGDC_1981__22_1_67_0)

Review in the context of cohesive toposes, modal type theory and cohesive homotopy type theory includes

• David Myers, Logical Topology and Axiomatic Cohesion, talk at Geometry in Modal Homotopy Type Theory 2019 (pdf slides, video recording)