# nLab total complex

### Context

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Idea

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

## Definition

Let $𝒜$ be an abelian category with arbitrary direct sums.

Write ${\mathrm{Ch}}_{•}\left(𝒜\right)$ for the category of chain complexes in $𝒜$ and ${C}_{•,•}\in {\mathrm{Ch}}_{•}\left({\mathrm{Ch}}_{•}\left(𝒜\right)\right)$ 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.)

###### Defintition

For ${C}_{•,•}\in {\mathrm{Ch}}_{•}\left({\mathrm{Ch}}_{•}\left(𝒜\right)\right)$ a double complex, it total complex $\mathrm{Tot}\left(C{\right)}_{•}\in {\mathrm{Ch}}_{•}\left(𝒜\right)$ is the chain complex whose components are the direct sums

$\mathrm{Tot}\left(C{\right)}_{n}=\underset{k+l=n}{⨁}{C}_{k,l}$Tot(C)_n = \bigoplus_{k+l = n} C_{k,l}

and whose differentials are give by the linear combination

${\partial }^{\mathrm{Tot}}≔{\partial }_{\mathrm{vert}}^{C}+\left(-1{\right)}^{\mathrm{vertical}\phantom{\rule{thickmathspace}{0ex}}\mathrm{degree}}{\partial }_{\mathrm{hor}}^{C}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\mathrm{Tot}\left(C{\right)}_{•}$ is sometimes called the total homology of the double complex ${C}_{•,•}$.

###### 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 ${C}_{•,•}$ is bounded and has exact rows or coloumns then also $\mathrm{Tot}\left(C{\right)}_{•}$ is exact.

###### Proof

Use the acyclic assembly lemma?.

### Relation to total simplicial sets and homotopy colimits

The total chain complex is under Dold-Kan correspondence to the diagonal of a bisimplicial set – see Eilenberg-Zilber theorem. As discussed at bisimplicial set, this is weakly homotopy equivalent to the called the total simplicial set of a bisimplicial set.

## References

For instance secton 1.2 of

Revised on September 3, 2012 22:26:15 by Urs Schreiber (131.174.188.82)