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nonabelian groupoid cohomology

**cohomology** * cocycle, coboundary, coefficient * homology * chain, cycle, boundary * characteristic class * universal characteristic class * secondary characteristic class * differential characteristic class * fiber sequence/long exact sequence in cohomology * fiber ∞-bundle, principal ∞-bundle, associated ∞-bundle, twisted ∞-bundle * ∞-group extension * obstruction ### Special and general types ### * cochain cohomology * ordinary cohomology, singular cohomology * group cohomology, nonabelian group cohomology, Lie group cohomology * Galois cohomology * groupoid cohomology, nonabelian groupoid cohomology * generalized (Eilenberg-Steenrod) cohomology * cobordism cohomology theory * integral cohomology * K-theory * elliptic cohomology, tmf * taf * abelian sheaf cohomology * Deligne cohomology * de Rham cohomology * Dolbeault cohomology * etale cohomology * group of units, Picard group, Brauer group * crystalline cohomology * syntomic cohomology * motivic cohomology * cohomology of operads * Hochschild cohomology, cyclic cohomology * string topology * nonabelian cohomology * principal ∞-bundle * universal principal ∞-bundle, groupal model for universal principal ∞-bundles * principal bundle, Atiyah Lie groupoid * principal 2-bundle/gerbe * covering ∞-bundle/local system * (∞,1)-vector bundle / (∞,n)-vector bundle * quantum anomaly * orientation, Spin structure, Spin^c structure, String structure, Fivebrane structure * cohomology with constant coefficients / with a local system of coefficients * ∞-Lie algebra cohomology * Lie algebra cohomology, nonabelian Lie algebra cohomology, Lie algebra extensions, Gelfand-Fuks cohomology, * bialgebra cohomology ### Special notions * Čech cohomology * hypercohomology ### Variants ### * equivariant cohomology * equivariant homotopy theory * Bredon cohomology * twisted cohomology * twisted bundle * twisted K-theory, twisted spin structure, twisted spin^c structure * twisted differential c-structures * twisted differential string structure, twisted differential fivebrane structure * differential cohomology * differential generalized (Eilenberg-Steenrod) cohomology * differential cobordism cohomology * Deligne cohomology * differential K-theory * differential elliptic cohomology * differential cohomology in a cohesive topos * Chern-Weil theory * ∞-Chern-Weil theory * relative cohomology ### Extra structure * Hodge structure * orientation, in generalized cohomology ### Operations ### * cohomology operations * cup product * connecting homomorphism, Bockstein homomorphism * fiber integration, transgression * cohomology localization ### Theorems * universal coefficient theorem * Künneth theorem * de Rham theorem, Poincare lemma, Stokes theorem * Hodge theory, Hodge theorem nonabelian Hodge theory, noncommutative Hodge theory * Brown representability theorem * hypercovering theorem * Eckmann-Hilton-Fuks duality

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Contents

Idea

Nonabelian groupoid cohomology is the nonabelian cohomology of groupoids.

This generalizes nonabelian group cohomology, which is the special case of nonabelian group cohomology for groupoids that are deloopings BG\mathbf{B}G of groups GG.

Examples

Atiyah Lie groupoid

The Atiyah Lie groupoid At(P)Π(X)At(P) \to \Pi(X) of a principal bundle PXP \to X is the extension of Π(X)\Pi(X) classified by the nonabelian groupoid cocycle Π(X)BAUT(G)\Pi(X) \to \mathbf{B} AUT(G) coming from a connection on PP.

Revised on January 11, 2010 23:45:35 by David Roberts (203.24.207.198)