Lemma 14.29.3. Let \mathcal{A} be an abelian category. Let U, V be simplicial objects of \mathcal{A}. Let a, b : U \to V be a pair of morphisms. Assume the corresponding maps of chain complexes N(a), N(b) : N(U) \to N(V) are homotopic by a homotopy \{ N_ n : N(U)_ n \to N(V)_{n + 1}\} . Then there exists a homotopy from a to b as in Definition 14.26.1. Moreover, one can choose the homotopy h : U \times \Delta [1] \to V such that N_ n = N(h)_ n where N(h) is the homotopy coming from h as in Section 14.27.
Proof. Let (\diamond N(U), \diamond a, \diamond b, \diamond h) be as in Lemma 14.29.1 and its proof. By that lemma there exists a morphism \diamond N(U) \to N(V) representing the triple (N(a), N(b), \{ N_ n\} ). We will show there exists a morphism \psi : N(U \times \Delta [1]) \to \diamond {N(U)} such that \diamond a = \psi \circ N(e_0), and \diamond b = \psi \circ N(e_1). Moreover, we will show that the homotopy between N(e_0), N(e_1) : N(U) \to N(U \times \Delta [1]) coming from (14.27.0.1) and Lemma 14.27.1 with h = \text{id}_{U \times \Delta [1]} is mapped via \psi to the canonical homotopy \diamond h between the two maps \diamond a, \diamond b : N(U) \to \diamond {N(U)}. Certainly this will imply the lemma.
Note that N : \text{Simp}(\mathcal{A}) \to \text{Ch}_{\geq 0}(\mathcal{A}) as a functor is a direct summand of the functor s : \text{Simp}(\mathcal{A}) \to \text{Ch}_{\geq 0}(\mathcal{A}). Also, the functor \diamond is compatible with direct sums. Thus it suffices instead to construct a morphism \Psi : s(U \times \Delta [1]) \to \diamond {s(U)} with the corresponding properties. This is what we do below.
By Definition 14.26.1 the morphisms e_0 : U \to U \times \Delta [1] and e_1 : U \to U \times \Delta [1] are homotopic with homotopy \text{id}_{U \times \Delta [1]}. By Lemma 14.27.1 we get an explicit homotopy \{ h_ n : s(U)_ n \to s(U \times \Delta [1])_{n + 1}\} between the morphisms of chain complexes s(e_0) : s(U) \to s(U \times \Delta [1]) and s(e_1) : s(U) \to s(U \times \Delta [1]). By Lemma 14.29.2 above we get a corresponding morphism
\Phi : \diamond {s(U)} \to s(U \times \Delta [1])
According to the construction, \Phi _ n restricted to the summand s(U)[-1]_ n = s(U)_{n - 1} of \diamond {s(U)}_ n is equal to h_{n - 1}. And
h_{n - 1} = \sum \nolimits _{i = 0}^{n - 1} (-1)^{i + 1} s^ n_ i \cdot \alpha ^ n_{i + 1} : U_{n - 1} \to \bigoplus \nolimits _ j U_ n \cdot \alpha ^ n_ j.
with obvious notation.
On the other hand, the morphisms e_ i : U \to U \times \Delta [1] induce a morphism (e_0, e_1) : U \oplus U \to U \times \Delta [1]. Denote W the cokernel. Note that, if we write (U \times \Delta [1])_ n = \bigoplus _{\alpha : [n] \to [1]} U_ n \cdot \alpha , then we may identify W_ n = \bigoplus _{i = 1}^ n U_ n \cdot \alpha ^ n_ i with \alpha ^ n_ i as in Section 14.26. We have a commutative diagram
\xymatrix{ 0 \ar[r] & U \oplus U \ar[rd]_{(1, 1)} \ar[r] & U \times \Delta [1] \ar[d]^\pi \ar[r] & W \ar[r] & 0 \\ & & U & & }
This implies we have a similar commutative diagram after applying the functor s. Next, we choose the splittings \sigma _ n : s(W)_ n \to s(U \times \Delta [1])_ n by mapping the summand U_ n \cdot \alpha ^ n_ i \subset W_ n via (-1, 1) to the summands U_ n \cdot \alpha ^ n_0 \oplus U_ n \cdot \alpha ^ n_ i \subset (U \times \Delta [1])_ n. Note that s(\pi )_ n \circ \sigma _ n = 0. It follows that (1, 1) \circ \delta (\sigma )_ n = 0. Hence \delta (\sigma ) factors as in Lemma 14.29.2. By that lemma we obtain a canonical morphism \Psi : s(U \times \Delta [1]) \to \diamond {s(U)}.
To compute \Psi we first compute the morphism \delta (\sigma ) : s(W) \to s(U)[-1] \oplus s(U)[-1]. According to Homology, Lemma 12.14.4 and its proof, to do this we have compute
d_{s(U \times \delta [1]), n} \circ \sigma _ n - \sigma _{n - 1} \circ d_{s(W), n}
and write it as a morphism into U_{n - 1} \cdot \alpha ^{n - 1}_0 \oplus U_{n - 1} \cdot \alpha ^{n - 1}_ n. We only do this in case \mathcal{A} is the category of abelian groups. We use the short hand notation x_{\alpha } for x \in U_ n to denote the element x in the summand U_ n \cdot \alpha of (U \times \Delta [1])_ n. Recall that
d_{s(U \times \delta [1]), n} = \sum \nolimits _{i = 0}^ n (-1)^ i d^ n_ i
where d^ n_ i maps the summand U_ n \cdot \alpha to the summand U_{n - 1} \cdot (\alpha \circ \delta ^ n_ i) via the morphism d^ n_ i of the simplicial object U. In terms of the notation above this means
d_{s(U \times \delta [1]), n}(x_\alpha ) = \sum \nolimits _{i = 0}^ n (-1)^ i (d^ n_ i(x))_{\alpha \circ \delta ^ n_ i}
Starting with x_\alpha \in W_ n, in other words \alpha = \alpha ^ n_ j for some j \in \{ 1, \ldots , n\} , we see that \sigma _ n(x_\alpha ) = x_\alpha - x_{\alpha ^ n_0} and hence
(d_{s(U \times \delta [1]), n} \circ \sigma _ n)(x_\alpha ) = \sum \nolimits _{i = 0}^ n (-1)^ i (d^ n_ i(x))_{\alpha \circ \delta ^ n_ i} - \sum \nolimits _{i = 0}^ n (-1)^ i (d^ n_ i(x))_{\alpha ^ n_0 \circ \delta ^ n_ i}
To compute d_{s(W), n}(x_\alpha ), we have to omit all terms where \alpha \circ \delta ^ n_ i = \alpha ^{n - 1}_0, \alpha ^{n - 1}_ n. Hence we get
\begin{matrix} (\sigma _{n - 1} \circ d_{s(W), n})(x_\alpha ) =
\\ \sum \nolimits _{i = 0, \ldots , n\text{ and } \alpha \circ \delta ^ n_ i \not= \alpha ^{n - 1}_0\text{ or }\alpha ^{n - 1}_ n} \Big((-1)^ i (d^ n_ i(x))_{\alpha \circ \delta ^ n_ i} - (-1)^ i (d^ n_ i(x))_{\alpha ^{n - 1}_0} \Big)
\end{matrix}
Clearly the difference of the two terms is the sum
\sum \nolimits _{i = 0, \ldots , n\text{ and } \alpha \circ \delta ^ n_ i = \alpha ^{n - 1}_0\text{ or }\alpha ^{n - 1}_ n} \Big((-1)^ i (d^ n_ i(x))_{\alpha \circ \delta ^ n_ i} - (-1)^ i (d^ n_ i(x))_{\alpha ^{n - 1}_0} \Big)
Of course, if \alpha \circ \delta ^ n_ i = \alpha ^{n - 1}_0 then the term drops out. Recall that \alpha = \alpha ^ n_ j for some j \in \{ 1, \ldots , n\} . The only way \alpha ^ n_ j \circ \delta ^ n_ i = \alpha ^{n - 1}_ n is if j = n and i = n. Thus we actually get 0 unless j = n and in that case we get (-1)^ n(d^ n_ n(x))_{\alpha ^{n - 1}_ n} - (-1)^ n(d^ n_ n(x))_{\alpha ^{n - 1}_0}. In other words, we conclude the morphism
\delta (\sigma )_ n : W_ n \to (s(U)[-1] \oplus s(U)[-1])_ n = U_{n - 1} \oplus U_{n - 1}
is zero on all summands except U_ n \cdot \alpha ^ n_ n and on that summand it is equal to ((-1)^ nd^ n_ n, -(-1)^ nd^ n_ n). (Namely, the first summand of the two corresponds to the factor with \alpha ^{n - 1}_ n because that is the map [n - 1] \to [1] which maps everybody to 0, and hence corresponds to e_0.)
We obtain a canonical diagram
\xymatrix{ 0 \ar[r] & s(U) \oplus s(U) \ar[r] \ar[d] & \diamond {s(U)} \ar[r] \ar[d]^{\Phi }& s(U)[-1] \ar[r] \ar[d] & 0 \\ 0 \ar[r] & s(U) \oplus s(U) \ar[r] \ar[d] & s(U \times \Delta [1]) \ar[r] \ar[d]^\Psi & s(W) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & s(U) \oplus s(U) \ar[r] & \diamond {s(U)} \ar[r] & s(U)[-1] \ar[r] & 0 }
We claim that \Phi \circ \Psi is the identity. To see this it is enough to prove that the composition of \Phi and \delta (\sigma ) as a map s(U)[-1] \to s(W) \to s(U)[-1] \oplus s(U)[-1] is the identity in the first factor and minus identity in the second. By the computations above it is ((-1)^ nd^ n_0, -(-1)^ nd^ n_0) \circ (-1)^ n s^ n_ n = (1, -1) as desired. \square
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