nLab partial section

Definition

Given a morphism $f: A \to B$ in any category, a partial section of $f$ is a partial map $s: B \nrightarrow A$ such that $f \circ s$ equals the identity on the domain of $s$. Explictly, this is a subobject $i: D \to B$ of $B$ together with a morphism $s: D \to A$ such that $f \circ s = i$.

Examples

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

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