# nLab total complex

Contents

### Context

#### Homological algebra

homological algebra

Introduction

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, its associated total complex $Tot(C)_\bullet \in Ch_\bullet(\mathcal{A})$ is the chain complex whose components are the direct sums, direct products or a mixture of those

The sum total complex

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

the product total complex

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

the product-sum total complex

$Tot^{\pi\oplus}(C)_n = \prod_{k+l = n, k \lt 0} C_{k,l}\oplus\bigoplus_{k+l = n, k\geq 0} C_{k,l}$

and the sum-product total complex

$Tot^{\oplus\pi}(C)_n = \bigoplus_{k+l = n, k \lt 0} C_{k,l}\oplus\prod_{k+l = n, k\geq 0} C_{k,l}$

and whose differentials are given by the linear combination

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

Using the four different types of total complexes gives the flexibility to make more nuanced exactness statements for the total complex under various assumptions on the double complex.

## 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 the total homology of a double complex, is the spectral sequence of a double complex. See there for more details.

### Exactness

###### Proposition

First let $C$ be a double complex in any abelian category

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

Now let $C$ be a double complex of abelian groups.

• If $C_{\bullet,\bullet}$ has exact rows then the product-sum total complex is exact.
• If $C_{\bullet,\bullet}$ has exact rows and kernels (or equivalently) images between row complexes are exact, then the sum and product total complexes are exact
• If $C_{\bullet,\bullet}$ has exact rows and for each $i$ taking $H_i$ of the columns gives an exact complex, then the sum-product total complex 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.

For instance secton 1.2 of