Sectionnons

Sectionnons are a theoretical construct in differential geometry and higher category theory used to describe generalized sections of a fiber bundle that may not glue to a single global section in the conventional sense. They encode local section data together with a prescribed twisting that prevents global coherence, capturing obstructions to forming a true section.

Formal definition and structure: Let p: E → B be a fiber bundle with typical fiber F. A sectionnon of p consists of an open cover {U_i} of B, local sections s_i: U_i → E, and a collection of transition data t_ij in Aut(F) on overlaps U_i ∩ U_j, together with an obstruction class o in a suitable cohomology group that measures the failure of the s_i to glue into a global section. The s_i relate by s_i|_(U_i∩U_j) ≈ t_ij ∘ s_j. If the obstruction o vanishes, the data glue to a genuine section; otherwise, they define a sectionnon with twisting class o.

Relation to known concepts: Sectionnons generalize ordinary sections and are closely related to twisted bundles, gerbes, and obstruction theory. They are stable under pullback along maps between bases and can be composed with compatible fibration structures, yielding a web of related generalized sections.

Examples and applications: In complex and differential geometry, sectionnons appear when considering nontrivial twistings of bundles, where local trivializations exist but a global section cannot be chosen without accounting for the twisting data. In theoretical physics, similar structures arise in gauge theories with background flux or in the study of localized fields on twisted spaces, where obstructions play a central role.

See also: Section, Local section, Twisted bundle, Gerbe, Obstruction theory.