# The Stacks Project

## Tag 019H

### 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 $$Q(U)^n = \mathop{\rm Coker}( \xymatrix{ \bigoplus_{i = 0}^{n - 1} U_{n - 1} \ar[r]^-{\delta^n_i} & U_n }).$$ 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 $s(U) = A(U) \oplus Q(U)$ 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 $\text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$.

Proof. Omitted. But the results are the exact dual statements to Lemmas 14.23.1, 14.23.4, 14.23.6, 14.23.7, 14.23.8, and Theorem 14.24.3. $\square$

The code snippet corresponding to this tag is a part of the file simplicial.tex and is located in lines 4236–4350 (see updates for more information).

\section{Dold-Kan for cosimplicial objects}
\label{section-dold-kan-cosimplicial}

\noindent
Let $\mathcal{A}$ be an abelian category.
According to Homology, Lemma \ref{homology-lemma-abelian-opposite}
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 \ref{theorem-dold-kan})
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 $Q$.

\medskip\noindent
First we define the
{\it 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 {\it Moore complex} associated
to $U$.

\medskip\noindent
On the other hand, given a
cosimplicial object $U$ of $\mathcal{A}$ set
$Q(U)^0 = U_0$ and
$$Q(U)^n = \Coker( \xymatrix{ \bigoplus_{i = 0}^{n - 1} U_{n - 1} \ar[r]^-{\delta^n_i} & U_n }).$$
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 \ref{lemma-N-d-in-N}.)
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 {\it normalized cochain complex associated
to $U$}.
The dual to the Dold-Kan Theorem \ref{theorem-dold-kan} is the following.

\begin{lemma}
\label{lemma-dual-dold-kan}
Let $\mathcal{A}$ be an abelian category.
\begin{enumerate}
\item The functor
$s : \text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$
is exact.
\item The maps $s(U)^n \to Q(U)^n$ define a morphism
of cochain complexes.
\item There exists a functorial direct sum decomposition
$s(U) = A(U) \oplus Q(U)$ in $\text{CoCh}_{\geq 0}(\mathcal{A})$.
\item The functor $Q$ is exact.
\item The morphism of complexes $s(U) \to Q(U)$ is a quasi-isomorphism.
\item The functor $U \mapsto Q(U)^\bullet$ defines
an equivalence of categories
$\text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$.
\end{enumerate}
\end{lemma}

\begin{proof}
Omitted. But the results are the exact dual statements to
Lemmas \ref{lemma-s-exact}, \ref{lemma-map-associated-complexes},
\ref{lemma-decompose-associated-complexes},
\ref{lemma-N-exact}, \ref{lemma-quasi-isomorphism}, and
Theorem \ref{theorem-dold-kan}.
\end{proof}

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