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 $(X,x_0)$, 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.

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.

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**

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 |

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).