Why should we study the total complex?

Question

Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be either constructed with objects $(\operatorname{Tot}(C_{\bullet,\bullet}))_n=\bigoplus_{p+q=n}C_{p,q}$ or $(\operatorname{Tot}(C_{\bullet,\bullet}))_n=\prod_{p+q=n}C_{p,q}$, though for simplicity let us assume the first case. In both cases the differential is given by the sum of the two (the vertical and horizontal) differentials of the original double complex.

Having seen this construction "in nature", I have some idea of its purpose:

• If we want to compute the (singular) homology of a product space, the right result is obtained using the Künneth theorem. In the process, it turns out that the correct tensor product of two complexes should be the total complex obtained from tensoring the complexes naively.
• The spectral sequences associated to a (bounded) double complex both converge to the homology of the total complex. Moreover, when wanting to construct any homology of a double complex, a natural start would be taking either vertical or horizontal homology and then iterating (as we are given additional structure), which are precisely the first pages of the respective spectral sequences.
• Defining the hyperhomology of a complex $C_\bullet$ boils down to choosing an Cartan-Eilenberg resolution of $C_\bullet$ and then considering at the associated total complex.

However, starting from just a double complex, which is a natural enough concept, given their occurrence when studying, say, the Ext and Tor functors, why would one consider the homology of the total complex as its "natural" associated homology in the first place? It was mentioned to me, that one of the simple reasons is just the requirement of having a complex at all to define homological objects, though this is a bit unsatisfying (even it would turn out to be the whole answer).

I asked this question a few days ago on Math.SE (here) but have received no comment/answer yet.

Is there any instrinsic motivation for studying the total complex and its associated homology? Or is the very concept just a convenient short-hand derived from naturally occurring examples?

Thanks in advance!

Answer 1

Chain complexes arise from simplicial abelian groups via the Dold–Kan correspondence: the normalized chain complex functor establishes an equivalence of categories from simplicial abelian groups to (nonnegatively graded) chain complexes.

Likewise, bicomplexes arise from simplicial objects in chain complexes (or bisimplicial abelian groups): the normalized chain bicomplex functor establishes an equivalence of categories from simplicial objects in chain complexes (or bisimplicial abelian groups) to bicomplexes.

Under this correspondence, the totalization simply computes (up to a quasi-isomorphism) the realization (alias homotopy colimit) of the corresponding simplicial object. This is precisely the content of the Dold–Puppe version of the Eilenberg–Zilber theorem.

The result continues to hold for unbounded chain complexes: the totalization of a chain complex of unbounded chain complexes computes the homotopy colimit of the corresponding simplicial object.

Simplicial objects are very common in homological algebra (e.g., the bar construction), so bicomplexes and totalizations fall out naturally.