nLab local coefficient bundle

Idea

In the context of twisted cohomology a cocycle on a space $X$ with coefficients in a coefficient object $V$ is not quite a direct morphism $X \to V$ as in ordinary $G$-cohomology, but is instead a section of a $V$-fiber ∞-bundle $E \to X$ over $X$. This is called the local coefficient bundle for the given twisted cohomology. Its class $[E] \in H^1(X, \mathbf{Aut}(V))$ is the twist.

Related concepts

• twisted cohomology, local system of coefficients

• twisted infinity-bundle

References

Local coefficient bundles in the context of twisted ordinary cohomology:

• Norman Steenrod, Homology With Local Coefficients, Annals of Mathematics, Second Series, Vol. 44, No. 4 (Oct., 1943), pp. 610-627 (jstor:1969099)

• M. Bullejos, E. Faro, M. A. García-Muñoz, Homotopy colimits and cohomology with local coefficients, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 no. 1 (2003), p. 63-80 (numdam:CTGDC_2003__44_1_63_0)

• Paul Goerss, Rick Jardine, p. 331 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (doi:10.1007/978-3-0346-0189-4)