nLab
germ

Contents

Definition

A germ is an element of (a total space of) an etale space or equivalently an element in some stalk of a sheaf (all stalks together form the total space of the etale space of the sheaf). Exactly what this means depends on which sheaf one is considering.

This general description of what a germ of some sheaf at some point is can be extracted from stalk, although that article is pretty abstract right now.

More generally, the notion of stalk makes sense in any topos that need not be a Grothendieck topos of sheaves by way of the notion of point of a topos. Generally a germ is an element in the stalk of an object of a topos over some point of the topos.

Examples

Originally, the term came from geometry, where sheaves of (continuous, smooth, holomorphic etc.) functions or more generally, sections of a (say fibre) bundle ξ:EB\xi:E\to B were considered; germs in geometrical intuition are always germs of something (of a section of the sheaf in some neighborhood of a point, but also elements of a colimit at a point of local sections of an original presheaf which is not always a sheaf; though the germs a posteriori make a sheaf, they can be considered or defined in relation to an original presheaf which is not necessarily a sheaf). Germs of sections are defined as the elements of the colimit sets of the appropriate sets of sections Γ Uξ\Gamma_U \xi where the colimit is over all open sets containing xBx\in B with inverse inclusion (inverse because a presheaf of sections of a bundle is a contravariant functor).

In other words, germs of sections at a point xBx\in B are equivalence classes [U,s][U,s] of pairs of the form (U,s)(U,s), where UxU\ni x is an open set in the base BB and s:UEs:U\to E is a section defined over UU; two sections (U,s)(U,s), (U,s)(U',s') in colim UxΓ Uξcolim_{U\ni x} \Gamma_U \xi are equivalent if there is a smaller WUUW\subset U\cap U' and s W=s Ws|_W=s'|_W. This construction of germs of sections of a bundle over BB (object of the slice category Top/BTop/B) leads to an adjoint pair of functors between Top/BTop/B and the category of presheaves of sets over BB which restricts to the equivalence of the category of etale spaces Et BEt_B and the category of sheaves over BB.

For example, take the sheaf of continuous (say, real-valued) functions on some space XX. Then every partial function ff defined on a neighbourhood of any given point aa in XX defines a germ at aa. Furthermore, the germ of ff equals the germ of gg if and only if f=gf = g on some neighbourhood UU of aa; note that UU must be contained in the intersection of the domains of ff and gg, but it may be smaller yet.

For a modern example of this kind, with only 11 stalk, consider the nonarchimedean field of germs of holomorphic functions at the origin of the field \mathbb{C} of complex numbers, which plays an important role in mirror symmetry as the base field for the geometry of families of Calabi-Yau manifolds in the large volume limit (cf. Kontsevich, Soibelman doi:10.1007/0-8176-4467-9, arxiv:math/0406564v1).

Examples of sequences of infinitesimal and local structures

first order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
square-0 ring extensionnilpotent ring extensionring extension
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

  • R. Godement, Topologie algebrique et theorie de faisceaux, Paris 1958

  • M. Kashiwara, P. Shapira: Categories and sheaves, Springer 2006

  • S. MacLane, I. Moerdijk: Sheaves in geometry and logic, Springer 1992

Revised on February 6, 2013 18:32:46 by Urs Schreiber (82.113.106.234)