germ of a space



A germ of a space is an equivalence class of pointed spaces, where two such spaces regarded as equivalent if they are isomorphic on small open neighbourhoods of the base points.


The category of germs of spaces has as objects pointed spaces (X,x 0)(X,x_0), where space may depend on the context (topological space, complex manifold, …).

Morphisms (X,x 0)(Y,y 0)(X,x_0) \to (Y,y_0) are equivalence classes of basepoint-preserving morphisms UYU \to Y defined on arbitrary open neighbourhoods UU of x 0x_0. Two such morphisms are considered equal if they agree on an open neighbourhood of x 0x_0 contained in the intersection of their domains.


Let ff be a germ of a holomorphic function on a complex manifold XX, defined on an open neighbourhood UU of some point x 0x_0. Then the vanishing set of ff, V(f){xU|f(x)=0}V(f) \coloneqq \{ x \in U \,|\, f(x) = 0 \}, is not well-defined as a space. However, it gives rise to a well-defined germ of a space.

The base space of a universal deformation? of a complex manifold is usually considered as a germ of a space.

As a localization

The category of germs of space is the localization of the category of pointed spaces at the class of those morphisms which restrict to isomorphisms on open neighbourhoods of the basepoints.

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


A short exposition is contained in the textbook

For germs in deformation theory, see for instance