Topos Theory

topos theory



Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory




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.


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 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


Revised on November 11, 2017 13:04:09 by Urs Schreiber (