# nLab total complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

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 $k$.

## Definition

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.)

###### Definition

For $C_{\bullet, \bullet} \in Ch_\bullet(Ch_\bullet(\mathcal{A}))$ a double complex, it 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 give by the linear combination

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

## Properties

### Total homology and spectral sequences

###### Remark

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

###### Remark

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

### Exactness

###### Proposition

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

###### Proof

Use the acyclic assembly lemma.

### 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.

## References

For instance secton 1.2 of

Revised on November 24, 2013 23:28:35 by Tim Porter (2.26.15.29)