Remark 12.18.6. Let $\mathcal{A}$ be an additive category with countable direct sums. Let $\text{DoubleComp}(\mathcal{A})$ denote the category of double complexes. We can consider an object $A^{\bullet , \bullet }$ of $\text{DoubleComp}(\mathcal{A})$ as a complex of complexes as follows

For the variant where we switch the role of the indices, see Remark 12.18.7. In this remark we show that taking the associated total complex is compatible with all the structures on complexes we have studied in the chapter so far.

First, observe that the shift functor on double complexes viewed as complexes of complexes in the manner given above is the functor $[0, 1]$ defined in Remark 12.18.5. By Remark 12.18.5 the functor

is compatible with shift functors, in the sense that we have a functorial isomorphism $\gamma : \text{Tot}(A^{\bullet , \bullet })[1] \to \text{Tot}(A^{\bullet , \bullet }[0, 1])$.

Second, if

are homotopic when $f$ and $g$ are viewed as morphisms of complexes of complexes in the manner given above, then

are homotopic maps of complexes. Indeed, let $h = (h^ q)$ be a homotopy between $f$ and $g$. If we denote $h^{p, q} : A^{p, q} \to B^{p, q - 1}$ the component in degree $p$ of $h^ q$, then this means that

The fact that $h^ q : A^{\bullet , q} \to B^{\bullet , q - 1}$ is a map of complexes means that

Let us define $h' = ((h')^ n)$ the homotopy given by the maps $(h')^ n : \text{Tot}^ n(A^{\bullet , \bullet }) \to \text{Tot}^{n - 1}(B^{\bullet , \bullet })$ using $(-1)^ ph^{p, q}$ on the summand $A^{p, q}$ for $p + q = n$. Then we see that

restricted to the summand $A^{p, q}$ is equal to

which evaluates to $f^{p, q} - g^{p, q}$ by the equations given above. This proves the second compatibility.

Third, suppose that in the paragraph above we have $f = g$. Then the assignment $h \leadsto h'$ above is compatible with the identification of Lemma 12.14.9. More precisely, if we view $h$ as a morphism of complexes of complexes $A^{\bullet , \bullet } \to B^{\bullet , \bullet }[0, -1]$ via this lemma then

is equal to $h'$ viewed as a morphism of complexes via the lemma. Here $\gamma $ is the identification of Remark 12.18.5. The verification of this third point is immediate.

Fourth, let

be a complex of double complexes and suppose we are given splittings $s^ q : C^{\bullet , q} \to B^{\bullet , q}$ and $\pi ^ q : B^{\bullet , q} \to A^{\bullet , q}$ of this as in Lemma 12.14.10 when we view double complexes as complexes of complexes in the manner given above. This on the one hand produces a map

by the procedure in Lemma 12.14.10. On the other hand taking $\text{Tot}$ we obtain a complex

which is termwise split (see below) and hence comes with a morphism

well defined up to homotopy by Lemmas 12.14.10 and 12.14.12. Claim: these maps agree in the sense that

is equal to $\delta '$ where $\gamma $ is as in Remark 12.18.5. To see this denote $s^{p, q} : C^{p, q} \to B^{\bullet , q}$ and $\pi ^{p, q} : B^{p, q} \to A^{p, q}$ the components of $s^ q$ and $\pi ^ q$. As splittings $(s')^ n : \text{Tot}^ n(C^{\bullet , \bullet }) \to \text{Tot}^ n(B^{\bullet , \bullet })$ and $(\pi ')^ n : \text{Tot}^ n(B^{\bullet , \bullet }) \to \text{Tot}^ n(A^{\bullet , \bullet })$ we use the maps whose components are $s^{p, q}$ and $\pi ^{p, q}$ for $p + q = n$. We recall that

The restriction of this to the summand $C^{p, q}$ is equal to

The equality holds because $s^ q$ is a morphism of complexes (with $d_1$ as differential) and because $\pi ^{p + 1, q} \circ s^{p + 1, q} = 0$ as $s$ and $\pi $ correspond to a direct sum decomposition of $B$ in every bidegree. On the other hand, for $\delta $ we have

whose restriction to the summand $C^{p, q}$ is equal to $\pi ^{p, q + 1} \circ d_2^{p, q} \circ s^{p, q}$. The difference in signs is exactly canceled out by the sign of $(-1)^ p$ in the isomorphism $\gamma $ and the fourth claim is proven.

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