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total complex

Idea

For $C_{•,•}$ a double complex (in some abelian category $𝒜$), its total complex $\mathrm{Tot}(C{)}_{•}$ 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 ${\mathrm{Ch}}_{•}(𝒜)$ for the category of chain complexes in $𝒜$ and $C_{•,•}\in {\mathrm{Ch}}_{•}({\mathrm{Ch}}_{•}(𝒜))$ 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.)

Defintition

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

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

and whose differentials are give by the linear combination

$${\partial }^{\mathrm{Tot}}≔{\partial }_{\mathrm{vert}}^C+(-1{)}^{\mathrm{vertical}\mathrm{degree}}{\partial }_{\mathrm{hor}}^C.$$\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 $\mathrm{Tot}(C{)}_{•}$ 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 $\mathrm{Tot}(C{)}_{•}$ 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

• Charles Weibel, An Introduction to Homological Algebra