total complex

(also nonabelian homological algebra)

For $C_{\bullet, \bullet}$ a double complex (in some abelian category $\mathcal{A}$), its *total complex* $Tot(C)_\bullet$ is an ordinary complex which in degree $k$ is the direct sum of all components of total degree $k$.

Let $\mathcal{A}$ be an abelian category with arbitrary direct sums.

Write $Ch_\bullet(\mathcal{A})$ for the category of chain complexes in $\mathcal{A}$ and $C_{\bullet, \bullet} \in Ch_\bullet(Ch_\bullet(\mathcal{A}))$ 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 $C_{\bullet, \bullet} \in Ch_\bullet(Ch_\bullet(\mathcal{A}))$ a double complex, its associated **total complex** $Tot(C)_\bullet \in Ch_\bullet(\mathcal{A})$ is the chain complex whose components are the direct sums

$Tot(C)_n = \bigoplus_{k+l = n} C_{k,l}$

and whose differentials are given by the linear combination

$\partial^{Tot}
\coloneqq
\partial^C_{vert} + (-1)^{vertical\;degree} \partial^C_{hor}
\,.$

The chain homology of the total complex $Tot(C)_\bullet$ is sometimes called the **total homology** of the double complex $C_{\bullet, \bullet}$.

A tool for computing the homology of a total complex, hence for computing the total homology of a double complex, is the *spectral sequence of a double complex*. See there for more details.

If $C_{\bullet,\bullet}$ is bounded and has exact rows or columns then also $Tot(C)_\bullet$ is exact.

Use the acyclic assembly lemma.

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 January 18, 2017 11:52:21
by Anonymous
(192.76.8.11)