For a double complex (in some abelian category ), its total complex is an ordinary complex which in degree is the direct sum of all components of total degree .
Let be an abelian category with arbitrary direct sums.
Write for the category of chain complexes in and for the category of double complexes. (Hence we use the convention that in a double complex the vertical and horizontal differential commute with each other.)
For a double complex, it total complex is the chain complex whose components are the direct sums
and whose differentials are give by the linear combination
Total homology and spectral sequences
If is bounded and has exact rows or coloumns then also is exact.
Relation to total simplicial sets and homotopy colimits
The total chain complex is, under the Dold-Kan correspondence, equivalent to the diagonal of a bisimplicial set – see Eilenberg-Zilber theorem. As discussed at bisimplicial set, this is weakly homotopy equivalent to the total simplicial set of a bisimplicial set.
For instance secton 1.2 of
Revised on November 24, 2013 23:28:35
by Tim Porter