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

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 $\mathbf{B}G$ of groups $G$.

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

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