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 , where space may depend on the context (topological space, complex manifold, …).
Morphisms are equivalence classes of basepoint-preserving morphisms defined on arbitrary open neighbourhoods of . Two such morphisms are considered equal if they agree on an open neighbourhood of contained in the intersection of their domains.
Let be a germ of a holomorphic function on a complex manifold , defined on an open neighbourhood of some point . Then the vanishing set of , , 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
A short exposition is contained in the textbook
For germs in deformation theory, see for instance
Last revised on February 23, 2023 at 12:02:09. See the history of this page for a list of all contributions to it.