nLab
coboundary

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

In a cochain complex (V ,d)(V^\bullet,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\mathbf{H}, for XX and AA two objects, a cocycle on XX with coefficients in AA is an object in H(X,A)\mathbf{H}(X,A) and a coboundary between cocycles is a morphism in there.

H n=Z n/B nH_n = Z_n/B_n(chain-)homology(cochain-)cohomologyH n=Z n/B nH^n = Z^n/B^n
C nC_nchaincochainC nC^n
Z nC nZ_n \subset C_ncyclecocycleZ nC nZ^n \subset C^n
B nC nB_n \subset C_nboundarycoboundaryB nC nB^n \subset C^n

Revised on October 27, 2013 22:35:03 by Urs Schreiber (82.113.99.217)