nLab
coboundary

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

In a cochain complex (V •,d) a coboundary is an element in the image of the differential.

More generally, in the context of the intrinsic cohomology of an (∞,1)-topos H, for X and A two objects, a cocycle on X with coefficients in A is an object in H(X,A) and a coboundary between cocycles is a morphism in there.

H n=Z n/B n(chain-)homology(cochain-)cohomologyH n=Z n/B n
C nchaincochainC n
Z n⊂C ncyclecocycleZ n⊂C n
B n⊂C nboundarycoboundaryB n⊂C n

Revised on August 27, 2012 00:54:04 by Urs Schreiber (89.204.139.66)
Edit | Back in time (1 revision) | See changes | History | Views: Print | TeX | Source | Linked from: BRST complex, nonabelian cohomology, torsor, chain complex, homology, abelian sheaf cohomology, cohomology, twisting function, bundle gerbe, local system, motivation for sheaves, cohomology and higher stacks, Eckmann-Hilton duality, principal bundle, connection on a bundle, vector bundle, string structure, Eilenberg-Mac Lane spectrum, tmf, gerbe, Hochschild-Serre spectral sequence, fiber sequence, principal 2-bundle, principal infinity-bundle, group cohomology, twisted cohomology, generalized (Eilenberg-Steenrod) cohomology, generalized cohomology, Sweedler coring, algebraic K-theory, bialgebra cocycle, chain homology and cohomology, gauge theory, K-theory, Karoubi K-theory, cup product, field strength, twisted bundle, bundle gerbe module, twisted K-theory, topological K-theory, fiber bundle, singular cohomology, cohomology - contents, equivariant cohomology, A Survey of Elliptic Cohomology, elliptic cohomology, line bundle, cohomology ring, de Rham complex, reduced cohomology, multiplicative cohomology theory, Fivebrane structure, spin structure, orientation, de Rham theorem, Gromov-Witten invariants, Brown representability theorem, cochain on a simplicial set, nonabelian Hodge theory, string 2-group, string group, Lie algebra cohomology, fivebrane 6-group, Lie algebra extension, Hodge theory, motive, topological chiral homology, etale cohomology, motivic cohomology, nonabelian Lie algebra cohomology, group extension, nonabelian group cohomology, monadic descent, cocycle, Chern character, lifting gerbe, higher monadic descent, bibundle, infinity-Lie algebra cohomology, circle bundle, relative cohomology, characteristic class, Hochschild cohomology, cyclic cohomology, constant sheaf, locally constant infinity-stack, crystalline cohomology, Maslov index, differential function complex, simplicial principal bundle, G-structure, fiber integration, Thom isomorphism, orientation in generalized cohomology, ordinary cohomology, Poincare lemma, Floer homology, Witten genus, harmonic form, cohomological descent, groupal model for universal principal infinity-bundles, universal coefficient theorem, groupoid principal bundle, associated infinity-bundle, hypercohomology, universal principal infinity-bundle, Pontryagin class, secondary characteristic class, Lie group cohomology, bundle 2-gerbe, gauge transformation, principal 3-bundle, Chern class, vielbein, boundary, Stokes theorem, differential characteristic class, quantum sheaf cohomology, Dixmier-Douady class, n-vector bundle, KK-theory, coboundary, cohesive (infinity,1)-topos -- structures, cohomological dimension, charge of a conserved current, cohomology operation, infinity-gerbe, discrete torsion, cohomology localization, E-theory, Deligne conjecture, Gerstenhaber-Schack cohomology, Gelfand-Fuks cohomology, cycle, Parametrized Homotopy Theory, fundamental class, (infinity,1)-vector bundle, twisted spin structure, Stiefel-Whitney class, contents of contents, 2-gerbe, twisted spin^c structure, universal characteristic class, instanton, Euler class, Yang-Mills instanton Floer homology, spin^c structure, integral Stiefel-Whitney class, Bockstein homomorphism, cohomology in a local net of observables, 2-framing, cohomology of operads, twisted Wu structure, double cover, Hodge conjecture, first Chern class, chains and cochains - table, universal principal bundle, Mayer-Vietoris sequence, intersection pairing, Wu class, Grothendieck-Riemann-Roch theorem, reduction and lift of structure groups, Steenrod square, integral Wu structure, string^c 2-group, K-theory of a bipermutative category, obstruction, projective resolution, twisted smooth cohomology in string theory, Kadeishvili's theorem, canonical bundle, Theta characteristic, compactly supported cohomology, KR-theory, ring extension, local coefficient bundle, fiber infinity-bundle, K-theory of a permutative category, cochain, infinity-group extension, geometry of physics, Hilbert's Theorem 90, groupoid-principal infinity-bundle, Hodge structure, field (physics), long exact sequence in homology, Hodge theorem, Galois cohomology, suspension isomorphism, Massey product, k-invariant, 2-vector bundle, line 2-bundle, instanton sector, cofiber sequence, extended Lagrangian, level (Chern-Simons theory), gauge field - table, universal Chern-Simons n-bundle, centrally extended groupoid, cohomological integration, Poisson cohomology