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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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