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14.25 Dold-Kan for cosimplicial objects

Let $\mathcal{A}$ be an abelian category. According to Homology, Lemma 12.5.2 also $\mathcal{A}^{opp}$ is abelian. It follows formally from the definitions that

\[ \text{CoSimp}(\mathcal{A}) = \text{Simp}(\mathcal{A}^{opp})^{opp}. \]

Thus Dold-Kan (Theorem 14.24.3) implies that $\text{CoSimp}(\mathcal{A})$ is equivalent to the category $\text{Ch}_{\geq 0}(\mathcal{A}^{opp})^{opp}$. And it follows formally from the definitions that

\[ \text{CoCh}_{\geq 0}(\mathcal{A}) = \text{Ch}_{\geq 0}(\mathcal{A}^{opp})^{opp}. \]

Putting these arrows together we obtain an equivalence

\[ Q : \text{CoSimp}(\mathcal{A}) \longrightarrow \text{CoCh}_{\geq 0}(\mathcal{A}). \]

In this section we describe $Q$.

First we define the cochain complex $s(U)$ associated to a cosimplicial object $U$. It is the cochain complex with terms zero in negative degrees, and $s(U)^ n = U_ n$ for $n \geq 0$. As differentials we use the maps $d^ n : s(U)^ n \to s(U)^{n + 1}$ defined by $d^ n = \sum _{i = 0}^{n + 1} (-1)^ i \delta ^{n + 1}_ i$. In other words the complex $s(U)$ looks like

\[ \xymatrix{ 0 \ar[r] & U_0 \ar[rr]^{\delta ^1_0 - \delta ^1_1} & & U_1 \ar[rr]^{\delta ^2_0 - \delta ^2_1 + \delta ^2_2} & & U_2 \ar[r] & \ldots } \]

This is sometimes also called the Moore complex associated to $U$.

On the other hand, given a cosimplicial object $U$ of $\mathcal{A}$ set $Q(U)^0 = U_0$ and

\[ Q(U)^ n = \mathop{\mathrm{Coker}}( \xymatrix{ \bigoplus _{i = 0}^{n - 1} U_{n - 1} \ar[r]^-{\delta ^ n_ i} & U_ n }). \]

The differential $d^ n : Q(U)^ n \to Q(U)^{n + 1}$ is induced by $(-1)^{n + 1}\delta ^{n + 1}_{n + 1}$, i.e., by fitting the morphism $(-1)^{n + 1}\delta ^{n + 1}_{n + 1}$ into a commutative diagram

\[ \xymatrix{ U_ n \ar[rr]_{(-1)^{n + 1}\delta ^{n + 1}_{n + 1}} \ar[d] & & U_{n + 1} \ar[d] \\ Q(U)^ n \ar[rr]^{d_ n} & & Q(U)^{n + 1}. } \]

We leave it to the reader to show that this diagram makes sense, i.e., that the image of $\delta ^ n_ i$ maps into the kernel of the right vertical arrow for $i = 0, \ldots , n - 1$. (This is dual to Lemma 14.18.8.) Thus our cochain complex $Q(U)$ looks like this

\[ 0 \to Q(U)^0 \to Q(U)^1 \to Q(U)^2 \to \ldots \]

This is called the normalized cochain complex associated to $U$. The dual to the Dold-Kan Theorem 14.24.3 is the following.

Lemma 14.25.1. Let $\mathcal{A}$ be an abelian category.

  1. The functor $s : \text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$ is exact.

  2. The maps $s(U)^ n \to Q(U)^ n$ define a morphism of cochain complexes.

  3. There exists a functorial direct sum decomposition $s(U) = D(U) \oplus Q(U)$ in $\text{CoCh}_{\geq 0}(\mathcal{A})$.

  4. The functor $Q$ is exact.

  5. The morphism of complexes $s(U) \to Q(U)$ is a quasi-isomorphism.

  6. The functor $U \mapsto Q(U)^\bullet $ defines an equivalence of categories $\text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$.

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