nLab partial section

Partial sections

Partial sections

Definition

Given a morphism f:ABf: A \to B in any category, a partial section of ff is a partial map s:BAs: B \nrightarrow A such that fsf \circ s equals the identity on the domain of ss. Explictly, this is a subobject i:DBi: D \to B of BB together with a morphism s:DAs: D \to A such that fs=if \circ s = i.

Examples

  • A local section of a continuous map f:ABf: A \to B is a partial section of ff whose domain is an open subset of BB.

  • In obstruction theory, one may be interested in how far one can construct a section of a bundle (say a principal bundle with group GG), p:EXp: E \to X, over a CW-complex XX. Given a section of the restriction of EE over the (k1)(k-1)-skeleton X k1X_{k-1} (i.e., a partial section of XX), obstructions to extending to a partial section over X kX_k are measured by a class in H k(X k,X k1;π k(G))H^k(X_k, X_{k-1}; \pi_k(G)).

Last revised on September 4, 2013 at 02:21:36. See the history of this page for a list of all contributions to it.