nLab germ of a space

Idea

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.

Definition

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

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

Examples

Let $f$ be a germ of a holomorphic function on a complex manifold $X$, defined on an open neighbourhood $U$ of some point $x_0$. Then the vanishing set of $f$, $V(f) \coloneqq \{ x \in U \,|\, f(x) = 0 \}$, 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.

Related concepts

• germ

• microbundle

• synthetic differential 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

References

A short exposition is contained in the textbook

• Daniel Huybrechts, Complex geometry - an introduction. Springer (2004). Universitext. 309 pages. (pdf)

For germs in deformation theory, see for instance

• Marco Manetti, Deformation theory via differential graded Lie algebras (arXiv:0507284).